{ "schema_version": 1, "title": "The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane Re(s)>7/8", "source": { "name": "paper.tex", "sha256": "42a5ee0febca59fd1def55cfd6c6808c322ef4237d7726303ea52f711deac6a3", "characters": 766316, "lines": 16678 }, "argument_gist": "The paper tries to contradict the existence of a rightmost possible zero in an entire Hecke L-function family. It constructs a completed cubic-theta sum with two exact representations: reflection gives a small direct bound, while Poisson summation exposes a principal Mellin signal containing the reciprocal of the target L-function. Bounds for the other rows would continue that reciprocal across the possible zeros. A balanced first stage claims a zero-free half-plane to the right of 11/12; prime compensation and two additional moment inductions claim 7/8. The full Hecke family is needed because Poisson introduces twists, even when the final target is zeta. These are the manuscript’s proposed steps, not a completed verification of its difficult inputs.", "review_provenance": {}, "phases": [ { "title": "§§1–2 — continuation", "gist": "Replace nonvanishing with a uniform reciprocal-continuation problem: a direct bound and a common power saving approximation force every reciprocal through the rightmost family zeros.", "claims": [ "thm:main", "lem:continuation-criterion" ] }, { "title": "§§3–6 — base reflection", "gist": "Build one completed cubic-theta probe and bound it through its reflected form; balanced scales give the low Z^(1/4+epsilon) estimate.", "claims": [ "prop:completed-reflection", "lem:unmarked-completed-moment", "lem:balanced-additive-norm", "prop:balanced-low" ] }, { "title": "§§7–11 — first high", "gist": "Poisson-expand the same probe, isolate its principal reciprocal signal, count troublesome nonprincipal rows through saturated inverse witnesses and the sextic sieve, then close 11/12.", "claims": [ "lem:local-euler", "prop:detector-witness", "lem:sextic-large-sieve", "prop:sextic-row-count", "prop:balanced-high", "thm:eleven-twelfths" ] }, { "title": "§§12–16 — compensated low", "gist": "Insert independent selected prime slots and asymmetric scales; obtain a new reflected low bound and an exact holomorphic local correction with controlled errors.", "claims": [ "cor:marked-reflection", "lem:reflected-energy", "prop:probe-low", "prop:probe-errors" ] }, { "title": "§17 — marked inverse", "gist": "Prove the selected prime inverse moment by a canonical invariant, terminal reflected energy and two masked Poisson transforms; initialize it and amplify the no-slot estimate.", "claims": [ "lem:marked", "lem:canonical-moment", "lem:finite-seminorm-propagation", "lem:masked-poisson", "lem:inverse-amplification" ] }, { "title": "§18 — plain fourth", "gist": "Prove the plain-witness fourth moment by decreasing effective width; center equal-product-scale comparisons so fixed inducing-character main terms cancel.", "claims": [ "lem:plain", "lem:centered-coefficient-invariant", "lem:centered-lattice-cancellation" ] }, { "title": "§19 — refined counts", "gist": "Turn the two simultaneous witnesses and independent short prime slots into complementary inverse/plain row counts, with explicit slot capacity and supply margins.", "claims": [ "lem:bin-prime-bound", "prop:detector-counts" ] }, { "title": "§20 — final closure", "gist": "Balance the complementary counts, certify a small positive endpoint margin, reserve all losses before choosing target dependent heights, and apply continuation plus quadratic transfer.", "claims": [ "lem:high-bin", "lem:balanced-endpoint", "prop:parameter-order", "lem:continuation-criterion", "prop:hecke-dirichlet-transfer" ] } ], "nodes": [ { "id": "sec:introduction", "kind": "section", "label": "Section 1", "title": "Introduction", "section": "sec:introduction", "argument_gist": "State the uniform 7/8 claim, contrast it with zero-density results, outline the two stages, and distinguish unconditional inputs from explicitly excluded results.", "anchor": { "source": "paper.tex", "line_start": 48, "line_end": 363, "char_start": 1590, "char_end": 18865 }, "gist_evidence": "paper.tex lines 48–363 (section scope)", "gist_origin": "source_audit_annotation" }, { "id": "thm:main", "kind": "theorem", "label": "Theorem 1.1", "title": "", "section": "sec:introduction", "statement": "Every finite-order Hecke $L$-function over $F=\\mathbb Q(\\sqrt{-3})$ has no zero in $\\Re s>7/8$. The same holds for every Dirichlet $L$-function, including $\\zeta(s)$. A pole at $s=1$ for a principal character is allowed.", "argument_gist": "All finite-order Hecke L-functions over Q(sqrt(-3)) and all Dirichlet L-functions are claimed zero-free to the right of 7/8; principal poles are allowed.", "gist_origin": "source_audit_annotation", "anchor": { "source": "paper.tex", "line_start": 106, "line_end": 112, "char_start": 4567, "char_end": 4833 }, "gist_evidence": "paper.tex lines 106–111; paper.tex lines 106–119" }, { "id": "cor:nonresidues-square-roots", "kind": "corollary", "label": "Corollary 1.2", "title": "", "section": "sec:introduction", "statement": "There are absolute constants $A,C>0$ such that, for every odd prime $p$, the least positive quadratic nonresidue $n(p)$ satisfies \\[ n(p)\\le C(\\log p)^A. \\] In particular, $n(p)\\ll_\\delta p^\\delta$ for every $\\delta>0$, proving Vinogradov's least quadratic nonresidue conjecture \\cite[Conjecture 1.1]{TaoNonresidues}. Given an odd prime $p$ and $a\\in\\mathbb F_p$ in binary representation, there is a deterministic algorithm, with running time polynomial in $\\log p$, that returns a square root of $a$ or reports that none exists.", "argument_gist": "Deduce a polylogarithmic bound for the least quadratic nonresidue and a deterministic algorithm for square roots modulo a prime from the claimed zero-free half-plane.", "gist_origin": "source_audit_annotation", "proof_anchor": { "source": "paper.tex", "line_start": 339, "line_end": 362, "char_start": 17631, "char_end": 18863 }, "proof_opening": "By Theorem~\\ref{thm:main} and the functional equation, the nontrivial zeros of every primitive Dirichlet $L$-function lie in $1/8\\le\\Re s\\le7/8$. They therefore lie in the strictly larger strip $1/16<\\Re s<15/16$. This supplies the weak-GRH hypothesis of Bhargava, Ivanyos, Mittal, and Saxena \\cite[Conjecture 6.3 and Theorem 6.7]{BIMS} with $\\epsilon=7/16$. Their bound for the least quadratic nonresidue gives the asserted inequality, for example with $A=32$; no optimization is needed here. The assertion for each $\\delta>0$ follows because every fixed power of $\\log p$ is $O_\\delta(p^\\delta)$. For the algorithm, first handle $a=0$ and test a nonzero $a$ by Euler's criterion. If $a$ is a square", "anchor": { "source": "paper.tex", "line_start": 325, "line_end": 338, "char_start": 17029, "char_end": 17629 }, "gist_evidence": "paper.tex lines 325–338 (full statement)" }, { "id": "sec:analytic-reduction", "kind": "section", "label": "Section 2", "title": "From a common signal to a zero-free half-plane", "section": "sec:analytic-reduction", "argument_gist": "Show that common positive low and approximation savings would continue every target reciprocal past the rightmost family zeros.", "anchor": { "source": "paper.tex", "line_start": 363, "line_end": 525, "char_start": 18865, "char_end": 25154 }, "gist_evidence": "paper.tex lines 363–525 (section scope)", "gist_origin": "source_audit_annotation" }, { "id": "lem:continuation-criterion", "kind": "proposition", "label": "Proposition 2.1", "title": "Continuation from a common signal", "section": "sec:analytic-reduction", "statement": "Fix $\\sigma_0\\in(1/2,1)$ and suppose $\\Delta_0=\\beta_*-\\sigma_0>0$. Let $C(s)=s+c$, where $c\\in\\mathbb R$ is fixed. Suppose that numbers $\\omega,\\sigma$ satisfying \\[ 0<\\omega<\\Delta_0,\\qquad \\sigma>0 \\] can be chosen independently of the target character. For every primitive finite-order Hecke character $\\eta$, suppose there exist a finite set $\\mathcal S$, a holomorphic function $H_\\eta$ on $\\Re s>\\sigma_0$, and a function $J_\\eta(Z)$ defined for all sufficiently large real $Z$, such that \\sup_{\\Re s>\\sigma_0}|H_\\eta(s)-1|\\le\\frac12. For $Z>0$, define f_\\eta(Z)=\\frac1{2\\pi i}\\int_{\\Re s=2} Z^{C(s)}e^{(s-5/6)^2} \\frac{H_\\eta(s)}{L_F^{\\mathcal S}(s,\\eta)}\\,ds. Assume that, as $Z\\to\\infty$, |J_\\eta(Z)|&\\ll_\\eta Z^{C(\\sigma_0)+\\omega}, \\\\ |J_\\eta(Z)-f_\\eta(Z)|&\\ll_\\eta Z^{C(\\beta_*)-\\sigma}. The implied constants, lower thresholds, excluded set, and function $H_\\eta$ may depend on $\\eta$. Then these assumptions contradict $\\beta_*>\\sigma_0$.", "argument_gist": "A common positive saving between a bounded probe and a reciprocal-L Mellin signal continues every target reciprocal across the family zero supremum.", "gist_origin": "source_audit_annotation", "proof_anchor": { "source": "paper.tex", "line_start": 434, "line_end": 503, "char_start": 21787, "char_end": 24372 }, "proof_opening": "Put \\[ \\epsilon_*:=\\min\\{\\Delta_0-\\omega,\\sigma\\}>0. \\] Because $C$ has slope one, the triangle inequality gives |f_\\eta(Z)|\\ll_\\eta Z^{C(\\beta_*)-\\epsilon_*} \\qquad (Z\\ge Z_{0,\\eta}). Moreover $\\epsilon_*\\le\\Delta_0-\\omega<\\Delta_0$, so $\\beta_*-\\epsilon_*>\\sigma_0$. We also need control as $Z\\downarrow0$. By Equation~\\eqref{eq:signal-euler-contract}, $H_\\eta$ is bounded on $\\Re s>\\sigma_0$. The reciprocal Euler product is absolutely and uniformly bounded on $\\Re s\\ge2$. On every fixed strip $2\\le\\Re s\\le B$, the Gaussian in Equation~\\eqref{eq:common-residue} is $O_B(e^{-(\\Im s)^2})$. Cauchy's theorem on rectangles therefore moves the contour to any fixed $B>2$, with horizontal integrals te", "anchor": { "source": "paper.tex", "line_start": 400, "line_end": 433, "char_start": 20492, "char_end": 21785 }, "gist_evidence": "paper.tex lines 400–432; paper.tex lines 400–502" }, { "id": "sec:first-stage", "kind": "section", "label": "Section 3", "title": "A first zero-free half-plane", "section": "sec:first-stage", "argument_gist": "State the weaker 11/12 result that the balanced first stage will prove.", "anchor": { "source": "paper.tex", "line_start": 525, "line_end": 557, "char_start": 25154, "char_end": 26639 }, "gist_evidence": "paper.tex lines 525–557 (section scope)", "gist_origin": "source_audit_annotation" }, { "id": "thm:eleven-twelfths", "kind": "theorem", "label": "Theorem 3.1", "title": "The $11/12$ half-plane", "section": "sec:first-stage", "statement": "Every finite-order Hecke $L$-function over $F=\\mathbb Q(\\sqrt{-3})$ has no zero in $\\Re s>11/12$. The same holds for every Dirichlet $L$-function, including $\\zeta(s)$. A pole at $s=1$ for a principal character is allowed.", "argument_gist": "The balanced base probe first establishes the weaker 11/12 family bound, providing the bootstrap for Part II.", "gist_origin": "source_audit_annotation", "proof_anchor": { "source": "paper.tex", "line_start": 6783, "line_end": 6806, "char_start": 309770, "char_end": 311024 }, "proof_opening": "Suppose that $\\beta_*>11/12$. Lemma~\\ref{lem:local-euler} and Equation~\\eqref{eq:shared-principal-nonvanishing} give the required holomorphic, nonzero $H_\\eta$ on $\\Re s>11/12$ for every primitive target. The low and high bounds established in Proposition~\\ref{prop:balanced-high} and its proof verify Proposition~\\ref{lem:continuation-criterion} with \\[ \\sigma_0=\\frac{11}{12},\\qquad C=C_{\\mathrm I},\\qquad \\omega=\\frac{\\Delta_1}{2},\\qquad \\sigma=\\frac1{4800}. \\] These two positive losses are independent of the target, while the allowed constants, excluded sets, and thresholds may depend on it. The continuation criterion contradicts the supposition. Therefore $\\beta_*\\le11/12$. By its definitio", "anchor": { "source": "paper.tex", "line_start": 532, "line_end": 538, "char_start": 25434, "char_end": 25738 }, "gist_evidence": "paper.tex lines 532–537; paper.tex lines 6783–6804" }, { "id": "sec:arithmetic", "kind": "section", "label": "Section 4", "title": "Arithmetic and analytic preliminaries", "section": "sec:arithmetic", "argument_gist": "Fix arithmetic, character zeros, reciprocity phases, smooth-profile conventions, and analytic bounds used throughout both stages.", "anchor": { "source": "paper.tex", "line_start": 557, "line_end": 1648, "char_start": 26639, "char_end": 77367 }, "gist_evidence": "paper.tex lines 557–1648 (section scope)", "gist_origin": "source_audit_annotation" }, { "id": "lem:fixed-numerator-ray", "kind": "lemma", "label": "Lemma 4.1", "title": "Fixed numerators give ray characters", "section": "sec:arithmetic", "statement": "Fix $0\\ne a\\in\\mathcal O$. At a prime ideal $\\mathfrak p\\nmid6a$, let $(a/\\mathfrak p)_6$ be the unique sixth root of unity congruent to $a^{(q_{\\mathfrak p}-1)/6}\\pmod{\\mathfrak p}$; at a prime outside $S$ this is $\\chi_p(a)$. The extension $F(a^{1/6})/F$ is finite abelian and unramified outside the primes dividing $6a$, and \\[ \\frac{\\operatorname{Frob}_{\\mathfrak p}(a^{1/6})}{a^{1/6}} =(a/\\mathfrak p)_6, \\] where $\\operatorname{Frob}_{\\mathfrak p}$ is arithmetic Frobenius. Hence the multiplicative extension of this symbol to ideals coprime to $6a$ is a finite-order ray character whose conductor is supported on the primes dividing $6a$. No bound on its conductor exponents at those primes is asserted. This identification is only on ideals coprime to $6a$; it does not erase the prescribed zero of $\\chi_A(a)$ when a good ideal $A$ meets $a$. In particular, choose one generator $\\pi_{\\mathfrak p}$ for each $\\mathfrak p\\in S$. On primary ideals outside $S$, the characters \\[ A\\longmapsto\\chi_A\\left(u\\prod_{\\mathfrak p\\in S} \\pi_{\\mathfrak p}^{v_{\\mathfrak p}}\\right), \\qquad u\\in\\mathcal O^\\times,\\quad v_{\\mathfrak p}\\ge0, \\] belong to one finite family of ray characters with a common modulus supported on $S$; this family is determined by $u$ and the residues $v_{\\mathfrak p}\\pmod6$.", "argument_gist": "Identify a fixed-numerator sixth-power residue symbol with a finite-order ray character through its Kummer extension, retaining the original excluded primes.", "gist_origin": "source_audit_annotation", "proof_anchor": { "source": "paper.tex", "line_start": 676, "line_end": 700, "char_start": 32724, "char_end": 34137 }, "proof_opening": "Let $\\xi^6=a$. Since $F$ contains all sixth roots of unity, all roots of $X^6-a$ lie in $F(\\xi)$, and $\\sigma\\mapsto\\sigma(\\xi)/\\xi$ embeds its Galois group into $\\mu_6$. The extension is therefore abelian. A prime outside $6a$ is unramified; for example this follows from the discriminant of $X^6-a$, which is supported on $6a$. At such a prime, arithmetic Frobenius is characterized on the residue field by $x\\mapsto x^{q_{\\mathfrak p}}$. Consequently its quotient on $\\xi$ reduces to $a^{(q_{\\mathfrak p}-1)/6}$. Reduction is injective on $\\mu_6$ outside $6$, proving the displayed identity. Multiplicativity of the Artin map gives the assertion for ideals. Artin reciprocity makes this map factor", "anchor": { "source": "paper.tex", "line_start": 646, "line_end": 675, "char_start": 31287, "char_end": 32722 }, "gist_evidence": "paper.tex lines 646–675 (full statement)" }, { "id": "lem:prime-gauss-identities", "kind": "lemma", "label": "Lemma 4.2", "title": "Prime Gauss identities", "section": "sec:arithmetic", "statement": "Let $p$ be the primary generator of a prime ideal outside $S$, and put $H=\\chi_p$ and $P=q_p$. Then \\[ \\gamma_2(p)^3=-\\alpha(p),\\qquad \\gamma_1(p)\\gamma_2(p) =\\overline{H(4)}\\gamma_3(p)\\gamma_2(p)^3. \\] For $j\\not\\equiv0\\pmod6$, one has $|\\gamma_j(p)|=1$.", "argument_gist": "Establish the prime Gauss-sum identities and absolute-value-one normalizations used to convert cubic coefficients into sextic notation.", "gist_origin": "source_audit_annotation", "proof_anchor": { "source": "paper.tex", "line_start": 777, "line_end": 837, "char_start": 37941, "char_end": 40300 }, "proof_opening": "Let $k=\\mathcal O/(p)$ and let $\\psi(x)=e(x/p)$ be its nontrivial additive character. For a multiplicative character $A$ of $k^\\times$, extended by zero, put \\[ \\tau(A)=\\sum_{x\\in k}A(x)\\psi(x),\\qquad J(A,B)=\\sum_{x\\in k}A(x)B(1-x). \\] If $A$ is nonprincipal, the change of variables $x=ty$ gives \\[ |\\tau(A)|^2 =\\sum_{t\\in k^\\times}A(t)\\sum_{y\\in k^\\times}\\psi((t-1)y)=P. \\] The inner sum is $P-1$ for $t=1$ and $-1$ otherwise, and $\\sum_{t\\in k^\\times}A(t)=0$. Conjugation also gives $\\tau(A)\\tau(\\overline A)=A(-1)P$. When $AB$ is nonprincipal, grouping the product $\\tau(A)\\tau(B)$ by $x+y$ gives \\[ \\tau(A)\\tau(B)=J(A,B)\\tau(AB). \\] The group with $x+y=0$ vanishes because $AB$ is nonprincipal; ", "anchor": { "source": "paper.tex", "line_start": 766, "line_end": 776, "char_start": 37594, "char_end": 37939 }, "gist_evidence": "paper.tex lines 766–776 (full statement)" }, { "id": "lem:quadratic-four-term", "kind": "lemma", "label": "Lemma 4.3", "title": "Quadratic four-term formula", "section": "sec:arithmetic", "statement": "For a nonzero odd $c\\in\\mathcal O$, define \\[ \\Gamma(c)=|c|^{-1}\\sum_{x\\bmod c}e(x^2/c). \\] Then \\Gamma(c)=\\frac12\\sum_{y\\bmod2\\mathcal O}e(-cy^2/4). For $c=a+b\\omega$ the right side is $(1+i^{-b}+i^a+i^{b-a})/2$. In particular $\\Gamma$ depends only on $c\\bmod4\\mathcal O$ and $\\Gamma(cv^2)=\\Gamma(c)$ for every odd $v$. The units modulo $4$ have square subgroup $\\{1,\\omega,\\omega^2\\}$ and square-class representatives $1,-1,\\lambda,-\\lambda$. The function $\\Gamma$ never vanishes on these units, and $\\mathfrak r(a,b)=\\Gamma(ab)/(\\Gamma(a)\\Gamma(b))$ satisfies {c|rrrr} c&1&-1&\\lambda&-\\lambda\\\\ \\hline \\Gamma(c)&1&1&i&-i \\\\ \\mathfrak r((-1)^e\\lambda^f,(-1)^g\\lambda^h) =(-1)^{eh+fg+fh},\\qquad e,f,g,h\\in\\{0,1\\}. Thus $|\\Gamma(c)|=1$ and $\\mathfrak r$ is a symmetric $\\{\\pm1\\}$-valued bicharacter of the square-class group.", "argument_gist": "Evaluate the quadratic Gauss sum through four residue classes modulo two; its values supply the fixed reciprocity phase modulo four.", "gist_origin": "source_audit_annotation", "proof_anchor": { "source": "paper.tex", "line_start": 871, "line_end": 933, "char_start": 41542, "char_end": 44436 }, "proof_opening": "Use the Fourier transform with kernel $e(-zy)$ and measure $d\\mu$. For $\\varepsilon>0$ apply Poisson summation on $\\mathcal O$ to $f_\\varepsilon(z)=e(z^2/c)e^{-\\pi\\varepsilon|z|^2}$. Direct Gaussian integration gives, with $r_\\varepsilon=1+3q_c\\varepsilon^2/16$, \\[ \\widehat f_\\varepsilon(y) =\\frac{|c|}{2\\sqrt{r_\\varepsilon}} e^{-\\pi\\varepsilon q_c|y|^2/(4r_\\varepsilon)} e\\bigl(-cy^2/(4r_\\varepsilon)\\bigr). \\] To check the normalization, rotate $z$ by half the argument of $c$. The quadratic matrix of the resulting real two-variable Gaussian is \\[ \\varepsilon&-4i/(\\sqrt3|c|)\\\\ -4i/(\\sqrt3|c|)&\\varepsilon , \\qquad \\det=\\frac{16r_\\varepsilon}{3q_c}. \\] Its eigenvalues are conjugates with positiv", "anchor": { "source": "paper.tex", "line_start": 841, "line_end": 870, "char_start": 40426, "char_end": 41540 }, "gist_evidence": "paper.tex lines 841–870 (full statement)" }, { "id": "lem:fixed-gauss-phase", "kind": "lemma", "label": "Lemma 4.4", "title": "Sextic reciprocity and the fixed Gauss phase", "section": "sec:arithmetic", "statement": "For coprime primary $a,b$ outside $S$, \\[ \\chi_b(a)=\\mathcal R(a,b)\\chi_a(b),\\qquad \\mathcal R(a,b)=\\mathfrak r(a,b). \\] On all primary pairs outside $S$, including noncoprime pairs, define $\\mathcal R$ by the bicharacter $\\mathfrak r$ of Equation~\\eqref{eq:reciprocity-four-class}. Define, on every primary index outside $S$, \\[ G(c)=\\overline{\\chi_c(4)}\\Gamma(c). \\] It has modulus one and factors through a fixed ray group supported at $2,3$. For all such $v,w$, G(vw)=G(v)G(w)\\mathcal R(v,w),\\qquad G(1)=1. Moreover $G(v^2)=\\chi_v(4)$, and at good primes G(p^3)=\\gamma_3(p),\\qquad \\mathcal R(p,p)=\\gamma_3(p)^2=\\chi_p(-1). On every primary $n$ outside $S$, one has $\\chi_n(-1)=\\mathcal R(n,n)$. The diagonal $t\\mapsto\\mathcal R(t,t)$ is a character of the fixed ray group. In the square-class notation of Equation~\\eqref{eq:reciprocity-four-class}, it is $\\mathfrak r(-1,n)$, where $-1$ denotes its residue square class modulo $4$, not the ideal ray class of the unit ideal $(-1)$. For squarefree $c$ the complete Gauss identities are \\gamma_2(c)^3=\\mu(c)\\alpha(c),\\qquad \\gamma_1(c)\\gamma_2(c)=\\mu(c)\\alpha(c)G(c),\\qquad G(c)=\\overline{\\chi_c(4)}\\gamma_3(c),\\quad |G(c)|=1. All occurrences of $G$ on nonsquarefree indices mean the finite-ray function just defined, not a nonsquarefree Gauss sum.", "argument_gist": "Record sextic reciprocity and a fixed ray-class Gauss phase so later transforms can track phases separately from moving residue symbols.", "gist_origin": "source_audit_annotation", "proof_anchor": { "source": "paper.tex", "line_start": 974, "line_end": 1053, "char_start": 46029, "char_end": 49576 }, "proof_opening": "At a good prime, counting square roots in the residue field gives \\[ \\sum_{x\\bmod p}e(x^2/p) =\\sum_{y\\bmod p}(1+\\chi_p(y)^3)e(y/p) =\\sum_{y\\bmod p}\\chi_p(y)^3e(y/p). \\] The additive sum without the character is zero. Hence $\\Gamma(p)=\\gamma_3(p)$, and replacing $x^2/p$ by $ux^2/p$ for a unit $u$ multiplies the sum by $\\chi_p(u)^3$. For squarefree $c$, representing a residue as $\\sum_{p\\mid c}(c/p)x_p$ gives, in both sums, \\[ \\Gamma(c)=\\prod_{p\\mid c}\\chi_p(c/p)^3\\Gamma(p),\\qquad \\gamma_3(c)=\\prod_{p\\mid c}\\chi_p(c/p)^3\\gamma_3(p). \\] The cross terms in the square are integral in the additive character. Thus $\\Gamma(c)=\\gamma_3(c)$ for squarefree $c$. In particular, for distinct good primes $", "anchor": { "source": "paper.tex", "line_start": 934, "line_end": 973, "char_start": 44438, "char_end": 46027 }, "gist_evidence": "paper.tex lines 934–973 (full statement)" }, { "id": "lem:smooth-calculus", "kind": "lemma", "label": "Lemma 4.5", "title": "Smooth calculus", "section": "sec:arithmetic", "statement": "For integers $J\\ge0$, \\[ \\|w\\|_{J,\\mathrm{sep}}\\ll_{J,d,\\Omega}p_{J+d+2}(w). \\] Logarithmic Fourier inversion separates any fixed norm monomial. More precisely, if $x_j=c_j\\prod_{\\ell=1}^r y_\\ell^{a_{j\\ell}}$ with $c_j,y_\\ell>0$ and fixed real exponents $a_{j\\ell}$, then \\[ w(x_1,\\ldots,x_d)=\\frac1{(2\\pi)^d}\\int_{\\mathbb R^d} \\widehat w(\\boldsymbol t)\\prod_j c_j^{it_j} \\prod_\\ell y_\\ell^{i\\sum_j a_{j\\ell}t_j}\\,d\\boldsymbol t. \\] If the coefficient measure in this formula is common to a collection of rows, Minkowski's inequality passes any separated row $\\ell^2$ bound through the integral using $\\|w\\|_{J,\\mathrm{sep}}$ whenever the separated bound has height growth at most $(1+|\\boldsymbol t|)^J$. For a unit box $I\\subset\\mathbb R^d$ and a smooth scalar function $F$ on $I+[-1,1]^d$, \\sup_I|F|^2\\ll_d \\sum_{\\alpha\\in\\{0,1\\}^d} \\int_{I+[-1,1]^d}|\\partial^\\alpha F|^2. Consequently, under the corresponding derivative bounds, rowwise choices of scales in a polynomial range cost powers of $\\log Z$, and rowwise choices of norm-twist heights of absolute value at most $T_1$ cost a fixed power of $1+T_1$. For $T_1\\ge1$ and integers $J,N\\ge0$, \\int_{|\\boldsymbol t|>T_1}|\\widehat w(\\boldsymbol t)| (1+|\\boldsymbol t|)^J\\,d\\boldsymbol t \\le T_1^{-N}\\|w\\|_{J+N,\\mathrm{sep}}. With the Mellin convention $\\mathcal MW(s)=\\int_0^\\infty W(y)y^s\\,dy/y$, a pure twist obeys \\[ \\mathcal M[W(y)y^{i\\omega}](s)=\\mathcal MW(s+i\\omega). \\] Let $D_y=y\\partial_y$, let $I\\subset\\mathbb R$ be compact, and let $N\\ge0$ be an integer. Suppose the integrals below are finite and the boundary terms in $N$ logarithmic integrations by parts vanish. Then, uniformly for $\\sigma\\in I$ and $T\\in\\mathbb R$, |\\mathcal MW(\\sigma+iT)| \\ll_{I,N}(1+|T|)^{-N} \\sum_{j=0}^N\\int_0^\\infty y^\\sigma|D_y^jW(y)|\\,\\frac{dy}{y}. These hypotheses hold for annular $W$ on every such $I$. They also hold when $W$ is smooth at zero and Schwartz at infinity and $I$ is a compact subset of $(0,\\infty)$. In particular a horizontal contour join, whose imaginary coordinate is fixed, is estimated by this pointwise bound, not solely by the integrated tail in Equation~\\eqref{eq:late-fourier-tail}. The following joint version will also be useful. For measurable functions $a(v),b(w)$ for which the right side is finite, and integers $J,N\\ge0$, &\\int_{\\mathbb R^2}e^{-(T+v)^2}|a(v)b(w)| (1+|T|+|v|+|w|)^J\\,dv\\,dw\\\\ &\\quad\\ll_{J,N}(1+|T|)^{-N} \\sup_v(1+|v|)^{J+N}|a(v)| \\int_{\\mathbb R}(1+|w|)^J|b(w)|\\,dw. Thus the product of a Gaussian in the sum of two heights and rapidly decreasing Mellin transforms in the other two heights has rapid pointwise decay on a fixed-height slice. For integrated tails, the linear map $(t_1,t_2,t_3)\\mapsto(t_1+t_2,t_2,t_3)$ is invertible, so all fixed weighted $L^1$ moments of this product also control complements of boxes in the original heights. Appending finitely many separating frequencies and fixed linear translations of these three arguments gives the same conclusion by a block triangular change of variables. After translating a pure twist in the Mellin variable, a discarded separated integrand bounded by $Z^B(1+|\\boldsymbol t|)^J$ has absolute tail at most $Z^BT_1^{-N}\\|w\\|_{J+N,\\mathrm{sep}}$. Here $B$ and $J$ must be fixed before $N$ is chosen. Further derivatives of a uniformly smooth separating profile change this last seminorm, not the previously fixed height order. On a join of bounded real length at height comparable to $T_1$, Equation~\\eqref{eq:pointwise-mellin-tail} with order $N+\\lceil J\\rceil$ gives $Z^BT_1^{-N}$ times the corresponding finite weighted derivative norm of the external test, provided the other factors have the stated bound there. This is $O(Z^BT_1^{-N})$ when those external norms are uniform.", "argument_gist": "Separate joint smooth norm profiles by logarithmic Fourier inversion and control Sobolev bounds, Mellin tails, and norm-twist height costs.", "gist_origin": "source_audit_annotation", "proof_anchor": { "source": "paper.tex", "line_start": 1212, "line_end": 1287, "char_start": 56969, "char_end": 60795 }, "proof_opening": "For an integer $m$ with $2m>J+d$, integration by parts gives \\[ (1+|\\boldsymbol t|^2)^m\\widehat w(\\boldsymbol t) =\\int_{\\mathbb R^d}(1-\\Delta)^m w_{\\log}(\\boldsymbol u) e^{-i\\boldsymbol t\\cdot\\boldsymbol u}\\,d\\boldsymbol u. \\] Its absolute value is bounded by the volume of $\\Omega$ times $C_{m,d}p_{2m}(w)$. The weighted integral is finite because $2m>J+d$; one can choose $2m\\le J+d+2$. Fourier inversion gives the monomial formula. For row vectors $B(\\boldsymbol t)$ in $\\ell^2$, the precise inequality used there is \\[ \\left\\|\\frac1{(2\\pi)^d}\\int\\widehat w(\\boldsymbol t) B(\\boldsymbol t)\\,d\\boldsymbol t\\right\\|_{\\ell^2} \\le\\frac1{(2\\pi)^d}\\int|\\widehat w(\\boldsymbol t)| \\|B(\\boldsymbol t)\\|_{\\", "anchor": { "source": "paper.tex", "line_start": 1123, "line_end": 1211, "char_start": 52813, "char_end": 56967 }, "gist_evidence": "paper.tex lines 1123–1211 (full statement)" }, { "id": "lem:gaussian-annular", "kind": "lemma", "label": "Lemma 4.6", "title": "Gaussian annular decomposition", "section": "sec:arithmetic", "statement": "Put \\[ W_{\\mathrm G}(y)=\\frac1{2\\sqrt\\pi} \\exp\\bigl(-\\tfrac14(\\log y)^2\\bigr). \\] This function is not annular. Choose a fixed $\\chi\\in C_c^\\infty(\\mathbb R)$ such that $\\sum_{k\\in\\mathbb Z}\\chi(u-k)=1$, and define \\[ W_{{\\mathrm G},k}(y)=W_{\\mathrm G}(y)\\chi(\\log y-k),\\qquad w_k(x)=W_{\\mathrm G}(e^k x)\\chi(\\log x). \\] The profiles $w_k$ have one fixed compact logarithmic support and, for every fixed $A\\ge0$ and integer $j\\ge0$, \\sum_{k\\in\\mathbb Z}e^{A|k|}p_j(w_k)<\\infty. Consequently Lemma~\\ref{lem:smooth-calculus} may be applied on each annulus after $y=e^k x$, and the resulting weighted separation norms are summable. For fixed $\\kappa>0$, $A,B,C_0\\ge0$, and $M>0$ one also has \\[ Z^B\\sum_{|k|>\\kappa\\log Z-C_0}e^{A|k|}p_j(w_k)\\ll Z^{-M} \\qquad(Z\\to\\infty). \\] Directly, $\\mathcal MW_{\\mathrm G}(s)=e^{s^2}$, and the weighted logarithmic derivatives in Equation~\\eqref{eq:pointwise-mellin-tail} are finite uniformly on every compact real strip.", "argument_gist": "Split the noncompact logarithmic Gaussian into smooth annular profiles whose weighted seminorms remain summable.", "gist_origin": "source_audit_annotation", "proof_anchor": { "source": "paper.tex", "line_start": 1317, "line_end": 1341, "char_start": 61941, "char_end": 63260 }, "proof_opening": "Such a partition is obtained by normalizing the integer translates of a nonnegative compactly supported smooth function positive on $[-1/2,1/2]$. Let $\\operatorname{supp}\\chi\\subset[-C,C]$. Each logarithmic derivative of $W_{\\mathrm G}(e^k x)$ is a polynomial in $k+\\log x$ times the same Gaussian. The product rule therefore gives \\[ p_j(w_k)\\le C_j(1+|k|)^j \\exp\\bigl(-\\tfrac14((|k|-C)_+)^2\\bigr). \\] This is summable after multiplication by $e^{A|k|}$ for every fixed $A$. The first assertion follows, and the separation norms follow from Lemma~\\ref{lem:smooth-calculus}. In the profile's logarithmic Fourier transform, rescaling an annulus inserts only the unit phase $e^{-ikt}$, so it does not c", "anchor": { "source": "paper.tex", "line_start": 1288, "line_end": 1316, "char_start": 60797, "char_end": 61939 }, "gist_evidence": "paper.tex lines 1288–1316 (full statement)" }, { "id": "lem:kernel-seminorms", "kind": "lemma", "label": "Lemma 4.7", "title": "Finite seminorms for Fourier and Mellin kernels", "section": "sec:arithmetic", "statement": "For a Schwartz function $f$ on $\\mathbb R^d$, put \\[ s_j(f)=\\sum_{|\\alpha|,|\\beta|\\le j} \\|x^\\alpha\\partial^\\beta f(x)\\|_{L^1(\\mathbb R^d)}. \\] For every $A\\ge0$ and multi-index $\\beta$, the ordinary Fourier transform satisfies \\[ \\sup_\\xi(1+|\\xi|)^A|\\partial^\\beta\\widehat f(\\xi)| \\ll_{A,\\beta,d}s_{\\lceil A\\rceil+|\\beta|}(f). \\] The same conclusion, with a fixed change in the constant, holds for any fixed nondegenerate linear Fourier pairing. For a radial transform written as $\\widehat f(\\xi)=F(|\\xi|^2)$, any prescribed bound \\[ \\sup_{r\\ge0}(1+r)^A|(r\\partial_r)^jF(r)| \\] is therefore controlled by finitely many Schwartz seminorms of $f$. For the Mellin assertion, let $m(s)$ be holomorphic on a fixed closed vertical strip and suppose, uniformly there, $|m(\\sigma+it)|\\le C(1+|t|)^h$ for a fixed $h\\ge0$. On an available line $\\Re s=\\sigma$ in that strip define \\[ K_W(x)=\\frac1{2\\pi i}\\int_{(\\sigma)}\\mathcal MW(-s)m(s)x^{-s}\\,ds, \\] where $W$ has the finite weighted logarithmic derivatives needed to make the integral absolutely convergent, with the boundary terms in the logarithmic integrations by parts vanishing. For every integer $j\\ge0$, \\[ |(x\\partial_x)^jK_W(x)| \\ll x^{-\\sigma}\\int_{\\mathbb R} |\\mathcal MW(-\\sigma-it)|(1+|t|)^{h+j}\\,dt. \\] The integral on the right is bounded by finitely many integrals of $y^{-\\sigma}|(y\\partial_y)^kW(y)|\\,dy/y$, uniformly for $\\sigma$ in the fixed strip. For $W(y)y^{i\\omega}$ the same bound costs at most an additional factor $(1+|\\omega|)^{h+j}$. After multiplication by a fixed annular cutoff, these conclusions also bound every fixed logarithmic seminorm of $y\\mapsto K_W(Ry)$, with the factor $R^{-\\sigma}$.", "argument_gist": "Bound Fourier and Mellin kernels through finitely many weighted smooth seminorms, including annular pullbacks and separated height parameters.", "gist_origin": "source_audit_annotation", "proof_anchor": { "source": "paper.tex", "line_start": 1391, "line_end": 1414, "char_start": 65318, "char_end": 66659 }, "proof_opening": "Differentiating $\\widehat f$ inserts $x^\\beta$, and multiplication by a monomial in $\\xi$ differentiates $x^\\beta f$ before Fourier transformation. The $L^1$ bound for the Fourier transform, the product rule, and the bound of $(1+|\\xi|)^A$ by a finite sum of monomials of degrees at most $\\lceil A\\rceil$ give the first assertion. For a radial function, $r\\partial_r$ corresponds to $(\\xi\\cdot\\partial_\\xi)/2$. Expanding its $j$th power gives finitely many polynomial multiples of Fourier derivatives, so the first assertion gives the radial one. A fixed linear change of coordinates changes only its constants. For the Mellin assertion, differentiation under the absolutely convergent integral inser", "anchor": { "source": "paper.tex", "line_start": 1347, "line_end": 1390, "char_start": 63508, "char_end": 65316 }, "gist_evidence": "paper.tex lines 1347–1390 (full statement)" }, { "id": "lem:hecke-strip-growth", "kind": "lemma", "label": "Lemma 4.8", "title": "Hecke strip growth", "section": "sec:arithmetic", "statement": "Let $\\psi$ be a primitive nonprincipal finite-order Hecke character of $F$, with conductor norm $Q$. Then |L(\\sigma+it,\\psi)|\\ll Q^{3/5}(3+|t|)^2, \\qquad-1/10\\le\\sigma\\le11/10. For the principal character, the same bound with $Q=1$ holds for $(s-1)\\zeta_F(s)/(s+1)$, with the removable value used at $s=1$.", "argument_gist": "Give polynomial conductor and height bounds on a fixed Hecke L-function strip, removing the principal pole before applying the same bound.", "gist_origin": "source_audit_annotation", "proof_anchor": { "source": "paper.tex", "line_start": 1436, "line_end": 1530, "char_start": 67638, "char_end": 72072 }, "proof_opening": "A finite-order character has trivial infinite type, since $\\mathbb C^\\times$ is connected. The primitive Hecke functional equation in this case is \\[ \\Lambda(s,\\psi)=(3Q)^{s/2}(2\\pi)^{-s}\\Gamma(s)L(s,\\psi),\\qquad \\Lambda(s,\\psi)=\\varepsilon(\\psi)\\Lambda(1-s,\\overline\\psi), \\quad |\\varepsilon(\\psi)|=1. \\] This is the functional equation for primitive characters of trivial infinite type stated in \\cite[Equation (1.1)]{GZ}; a nonprincipal $L(s,\\psi)$ is entire. Absolute Euler convergence bounds $L(11/10+it,\\psi)$ uniformly in $Q,t$. Applying the functional equation and Stirling's formula on $\\Re s=-1/10$ gives \\[ |L(-1/10+it,\\psi)|\\ll Q^{3/5}(3+|t|)^{6/5}. \\] The estimate for bounded $t$ follow", "anchor": { "source": "paper.tex", "line_start": 1425, "line_end": 1435, "char_start": 67196, "char_end": 67636 }, "gist_evidence": "paper.tex lines 1425–1435 (full statement)" }, { "id": "lem:logarithmic-control", "kind": "lemma", "label": "Lemma 4.9", "title": "Logarithmic control", "section": "sec:arithmetic", "statement": "Let $\\psi$ be a primitive nonprincipal finite-order Hecke character of conductor norm $Q$, and put $\\mathcal L(s)=L(s,\\psi)$. For the principal character put $Q=1$ and $\\mathcal L(s)=(s-1)\\zeta_F(s)/(s+1)$, with its removable value at one. Fix $a\\in[1/2,1]$ and $00$, these bounds hold on $\\Re s\\ge b+v$, with constants depending on $v,\\epsilon$ (and with the height of $s$ in place of $t$). In the principal case the reciprocal bound passes to $1/\\zeta_F$; the upper and logarithmic-derivative bounds are for the regularized function $\\mathcal L$.", "argument_gist": "A zero-free buffered disk yields subpower bounds for L and its reciprocal, plus logarithmic derivative control.", "gist_origin": "source_audit_annotation", "proof_anchor": { "source": "paper.tex", "line_start": 1554, "line_end": 1601, "char_start": 73165, "char_end": 75493 }, "proof_opening": "Write $R_j=2-a-je$ and $\\mathcal C=2Q(3+|t|)^2$. All the disks used here lie in $\\Re s>1/2$, where the principal regularizer is holomorphic. On the zero-free disk choose a holomorphic logarithm $g=\\log\\mathcal L$ whose value near the center is the Euler logarithm, together with the logarithm of the regularizing factor in the principal case. The value $g(2+it)$ is uniformly bounded. Lemma~\\ref{lem:hecke-strip-growth} and Euler convergence to the right of $11/10$ give \\[ \\Re g(s)=\\log|\\mathcal L(s)|\\ll\\log\\mathcal C \\qquad(|s-(2+it)|\\le R_3). \\] The possible heights differ from $t$ by at most $3/2$, which is included in $\\mathcal C$. Borel--Carath\\'eodory on radii $R_3,R_4$, whose difference i", "anchor": { "source": "paper.tex", "line_start": 1531, "line_end": 1553, "char_start": 72074, "char_end": 73163 }, "gist_evidence": "paper.tex lines 1531–1552; paper.tex lines 1531–1600" }, { "id": "lem:deleted-euler-factors", "kind": "lemma", "label": "Lemma 4.10", "title": "Deleted Euler factors", "section": "sec:arithmetic", "statement": "Let $R$ be a squarefree ideal and let $|a_p|\\le1$ for $p\\mid R$. Put $D_R(s)=\\prod_{p\\mid R}(1-a_pq_p^{-s})$. For fixed $\\sigma_0>0$ and every $\\epsilon>0$, \\[ |D_R(s)|+|D_R(s)^{-1}|\\ll_{\\sigma_0,\\epsilon}q_R^\\epsilon \\qquad(\\Re s\\ge\\sigma_0). \\] On a bounded real strip one has, uniformly in the height, \\[ |D_R(s)|\\ll_\\epsilon q_R^{(-\\Re s)_++\\epsilon}, \\qquad \\left|\\frac{D_R'(s)}{D_R(s)}\\right| \\ll_{\\sigma_0}\\log(2q_R)\\quad(\\Re s\\ge\\sigma_0). \\] In particular a support $q_R\\le Z^B$ with bounded $B$ costs any prescribed positive power of $Z$ on a fixed positive real half-plane. For an imprimitive function $L(s,\\psi^*)D_R(s)$, both the primitive conductor and the deletion radical $R$ must be included when applying logarithmic control.", "argument_gist": "Bound finite deleted Euler products and their reciprocals in a positive half-plane; include their radical explicitly when they vary with the scale.", "gist_origin": "source_audit_annotation", "proof_anchor": { "source": "paper.tex", "line_start": 1623, "line_end": 1647, "char_start": 76337, "char_end": 77365 }, "proof_opening": "Every factor is nonzero on $\\Re s>0$. For $\\Re s\\ge\\sigma_0$, both its absolute value and the absolute value of its inverse are at most $(1-q_p^{-\\sigma_0})^{-1}$. For sufficiently large $q_p$, depending on $\\sigma_0,\\epsilon$, $-\\log(1-q_p^{-\\sigma_0})\\le\\epsilon\\log q_p$. The finitely many smaller primes contribute a fixed constant. This proves the first estimate. For any real $\\sigma$, \\[ \\prod_{p\\mid R}(1+q_p^{-\\sigma}) \\le q_R^{(-\\sigma)_+}2^{\\#\\{p:p\\mid R\\}} \\ll_\\epsilon q_R^{(-\\sigma)_++\\epsilon}. \\] The last inequality follows by separating the finitely many primes with $q_p<2^{1/\\epsilon}$. Finally, logarithmic differentiation gives \\[ \\left|\\frac{D_R'}{D_R}(s)\\right| \\le\\sum_{p\\mid", "anchor": { "source": "paper.tex", "line_start": 1602, "line_end": 1622, "char_start": 75495, "char_end": 76335 }, "gist_evidence": "paper.tex lines 1602–1622 (full statement)" }, { "id": "sec:completed-reflection", "kind": "section", "label": "Section 5", "title": "Completed cubic reflection and unmarked row energy", "section": "sec:completed-reflection", "argument_gist": "Derive the completed cubic reflection and estimate its unmarked row energy without removing masks or overlaps in the completed index.", "anchor": { "source": "paper.tex", "line_start": 1648, "line_end": 3313, "char_start": 77367, "char_end": 155614 }, "gist_evidence": "paper.tex lines 1648–3313 (section scope)", "gist_origin": "source_audit_annotation" }, { "id": "prop:completed-reflection", "kind": "proposition", "label": "Proposition 5.1", "title": "Completed cubic reflection", "section": "sec:completed-reflection", "statement": "Fix a finite family $\\mathcal X$ of finite-ray characters whose conductors are supported on $S$. Choose one fixed integral modulus $\\mathfrak E$ with prime support exactly $S$, divisible by every conductor in this family. On primary elements extend each member of $\\mathcal X$ by zero away from the $\\mathfrak E$-units. Choose a single $L\\in\\mathcal O\\setminus\\{0\\}$ with prime support $S$, with $(18)\\mid(L)$ and $\\mathfrak E\\mid(L)$, large enough that, for every $\\Psi_0\\in\\mathcal X$, the function on $\\mathcal O$ \\[ \\phi(x)= \\chi_x(\\lambda)^2\\Psi_0(x),&x\\equiv1\\pmod3,\\ (x,\\mathfrak E)=1,\\\\ 0,&\\text{otherwise}, \\qquad \\widehat\\phi(h)=q_L^{-1}\\sum_{x\\bmod L}\\phi(x)e(-hx/L) \\] is $L$-periodic. The zero branch is evaluated without evaluating either character. Such a choice of $L$ exists by the cubic supplementary law and the ray periodicity, as verified below. Both $\\mathfrak E$ and $L$ are fixed for the entire family before any moving prime is chosen. Fix a member $\\Psi_0\\in\\mathcal X$ and put $\\mathfrak L=(L)$. Let $\\mathcal P$ be any finite set of distinct primary primes not dividing $L$, choose $j_p\\in\\{0,1,\\ldots,5\\}$ for each $p\\in\\mathcal P$, and set, on primary elements, \\[ \\Psi(n)=\\Psi_0(n)\\prod_{p\\in\\mathcal P}\\chi_p(n)^{j_p}. \\] No coprimality between the two variables in the following product is imposed: \\[ D(s,\\Psi)&=\\sum_{\\substack{n\\equiv1\\ (3)\\\\(n,\\mathfrak E)=1}}^{\\rm sf} \\gamma_2(n)\\bar\\alpha(n)\\Psi(n)q_n^{-s},\\\\ L_0(s,\\Psi)&=\\sum_{\\substack{b\\equiv1\\ (3)\\\\(b,\\mathfrak E)=1}} \\bar\\alpha(b)^3\\Psi(b)^3q_b^{-3s+1/2}, \\qquad T(s,\\Psi)=L_0(s,\\Psi)D(s,\\Psi). \\] Both series and their product converge absolutely for $\\Re s>1$. Let $X>0$. Suppose that $V\\in C^\\infty(0,\\infty)$ and that, for every $C>0$ and every integer $j\\ge0$, $(x\\partial_x)^jV(x)$ is $O_{C,j}(x^C)$ as $x\\to0$ and $O_{C,j}(x^{-C})$ as $x\\to\\infty$. Define \\[ \\widehat V(t)=\\int_0^\\infty V(x)x^t\\,\\frac{dx}{x}. \\] The smoothed expression to be transformed is the completed sum \\[ &\\frac1{2\\pi i}\\int_{(a)}\\widehat V(s-1/2)X^{s-1/2}T(s,\\Psi)\\,ds\\\\ &\\quad=\\sum_{\\substack{n\\ {\\rm sf},\\ b\\\\n,b\\equiv1\\ (3)\\\\(nb,\\mathfrak E)=1}} \\frac{\\gamma_2(n)\\overline{\\alpha(nb^3)}\\Psi(nb^3)} {\\sqrt{q_n}\\,q_b} V\\!\\left(\\frac{q_nq_b^3}{X}\\right),\\qquad a>1. \\] This equality follows by absolute convergence and Mellin inversion. The character acts on the whole index $nb^3$, with its zero values retained; $n$ and $b$ may share prime factors. The reflected test is defined by \\[ K=\\frac{(2\\pi)^4}{27},\\qquad R(t)=\\prod_{\\pm}\\frac{\\Gamma(1+t\\pm1/6)}{\\Gamma(1-t\\pm1/6)}, \\] \\[ V^\\sharp(x)=\\frac1{2\\pi i}\\int_{(\\sigma)} \\widehat V(-t)R(t)(Kx)^{-t}\\,dt\\qquad(\\sigma\\ge0). \\] The integral defining $V^\\sharp$ is absolutely convergent and is independent of $\\sigma\\ge0$. The product $T$ has an entire continuation. On every fixed vertical strip it has polynomial growth, with constants allowed to depend on $L,\\Psi_0,\\mathcal P,(j_p)$ and the strip. There is one fixed triple of cusp coefficient functions from $\\lambda^{-4}\\mathcal O\\setminus\\{0\\}$ to $\\mathbb C$, constructed from the theta expansions in the proof. The same triple is used for all choices of the arithmetic data, scale, and test profile. For each fixed arithmetic choice above, there is a finite family of terms, indexed as in (1), independent of $X$, $V$, and the contour parameter $a>1$. In each term, $d$ is selected from this triple and $\\vartheta$ is an additive character of $\\mathcal O$ modulo a fixed modulus depending only on $L$; their sectorwise dependence is exactly that stated in (3), and the remaining data have properties (1)--(3). For these terms and every $a>1$, the identity is &\\frac1{2\\pi i}\\int_{(a)}\\widehat V(s-1/2)X^{s-1/2}T(s,\\Psi)\\,ds\\\\ &\\quad=\\sum_{\\text{terms}}\\zeta \\sum_{0\\ne\\mu\\in\\lambda^{-4}\\mathcal O} \\frac{d(\\mu)\\alpha(\\mu)\\vartheta(\\lambda^4\\mu)}{\\sqrt{q_\\mu}} \\prod_{p\\ {\\rm active}}B_p(\\lambda^4\\mu) V^\\sharp\\!\\left(\\frac{q_\\mu X}{q_c^2}\\right). Every inner sum is absolutely convergent. The terms have the following precise properties. \\item A term is specified by $h_0\\bmod L$ and a subset of active primes. Every $p$ with $j_p\\ne0$ is active, while a prime with $j_p=0$ may be active or inactive. Write $r=\\prod_{p\\ {\\rm active}}p$. Then $c=c_Fr$, where $c_F$ is the reduced denominator of $\\lambda^2h_0/L$, with a normalizing unit. Thus $c_F$ ranges over a fixed finite set and its prime divisors divide $L$. There are $O_{\\mathfrak L}(2^{|\\mathcal P|})$ terms, or $O_{\\mathfrak L}(4^{|\\mathcal P|})$ after splitting each Ramanujan factor in the next display into its two summands. \\item The local column factors are \\[ B_p(x)= \\chi_p(x)^{-j_p-2},&j_p\\ne0,4,\\\\ q_p^{-1/2}(-1+q_p1_{p\\mid x}),&j_p=4,\\\\ q_p^{-1/2}\\chi_p(x)^{-2},&j_p=0\\text{ and }p\\text{ is active}. \\] The function $d$ is one of three fixed cusp coefficient functions. Writing $u$ for an Eisenstein unit, their support and size satisfy \\operatorname{supp}d\\subset \\{u\\lambda^k n b^3:k\\ge-4,\\ n,b\\equiv1\\pmod3,\\ n\\ {\\rm sf}\\}, \\qquad |d(\\mu)|\\le27\\,3^{k/6}|b|. The bound refers to the displayed representation of a supported index. It permits common primes of $n$ and $b$, and it imposes no exclusion at the primes of $L$ other than the displayed restriction at $\\lambda$. The character $\\vartheta$ is an additive character of a quotient of $\\mathcal O$ by a fixed modulus depending only on $L$. \\item The scalar $\\zeta$ is independent of $\\mu$ and satisfies $|\\zeta|\\le1/81$. Its moving-prime dependence can be specified exactly. For $m\\not\\equiv0\\pmod6$, put \\[ \\tau_{p,m}^{\\pm}=q_p^{-1/2}\\sum_{y\\bmod p}\\chi_p(y)^m e(\\pm y/p), \\qquad |\\tau_{p,m}^{\\pm}|=1. \\] For each active $p$, define units modulo $p$ by \\[ \\sigma_p=\\lambda^2c/p,\\qquad \\epsilon_p=-\\big((\\lambda^3c/p)\\sigma_p\\big)^{-1}, \\] and put \\[ \\omega_{p,j}= \\tau_{p,j}^{-}\\tau_{p,j+2}^{+}\\chi_p(\\epsilon_p)^{-j-2},&j\\ne0,4,\\\\ \\tau_{p,4}^{-},&j=4,\\\\ -\\tau_{p,2}^{+}\\chi_p(\\epsilon_p)^{-2},&j=0. \\] The indices of $\\tau$ are read modulo six. With $M=\\lambda^{12}L^4$, there is a number $\\kappa_F$ of absolute value one, depending only on $h_0$ and the class of $r$ modulo $M^2$, such that \\zeta=-\\frac i{81}\\bar\\alpha(c)^2\\widehat\\phi(h_0)\\bar\\kappa_F \\prod_{p\\ {\\rm inactive}}(1-q_p^{-1}) \\prod_{p\\ {\\rm active}}\\chi_p(\\sigma_p)^{-2}\\omega_{p,j_p}. In each of the finitely many classes of $h_0$ and $r\\bmod M^2$, both $d$ and $\\vartheta$ are fixed. Consequently these are common column factors within that sector. Apart from the displayed scale $q_c^2$, all other moving-prime dependence outside the $B_p$ is in the scalar $\\zeta$, which may depend on the whole active set. The branch conventions are simultaneous for all local prime sets with the same $L$ and $\\Psi_0$. In particular, if a good prime $p$ is added to the local set with exponent zero and is declared inactive, the resulting branch has the same $c_F,c,d,\\vartheta,\\kappa_F$, the same data at every other active prime, and the same kernel as the branch in which $p$ is absent. Its scalar is multiplied by $1-q_p^{-1}$. This compatibility includes every $h_0$, not only the unit classes modulo $L$. Bounds for the transformed test. For $A\\ge0$ and an integer $B\\ge0$, put \\[ \\mathfrak M_{A,B}(V)= \\sup_{-A\\le\\eta\\le1/4}\\int_{\\mathbb R} (1+|u|)^B|\\widehat V(\\eta+iu)|\\,du. \\] For every integer $j\\ge0$, |(x\\partial_x)^jV^\\sharp(x)| \\ll_{A,j}\\min\\{x^{1/4},(1+x)^{-A}\\} \\mathfrak M_{A,\\lceil4A\\rceil+j+2}(V). If $\\chi\\in C_c^\\infty(0,\\infty)$ is fixed, $Y>0$, and $J\\ge0$ is an integer, then \\int_{\\mathbb R}(1+|u|)^J \\left|\\int_0^\\infty\\chi(y)V^\\sharp(Yy)y^{-iu}\\frac{dy}{y}\\right|du \\ll_{A,J,\\chi}\\min\\{Y^{1/4},(1+Y)^{-A}\\} \\mathfrak M_{A,\\lceil4A\\rceil+J+4}(V). These are finite smooth seminorms. More precisely, \\mathfrak M_{A,B}(V)\\ll_{A,B} \\int_0^\\infty(1+x^{-A}+x^{1/4}) \\bigl(|V(x)|+|(x\\partial_x)^{B+2}V(x)|\\bigr)\\frac{dx}{x}. Replacing $V(x)$ by $x^{iu_0}V(x)$ multiplies $\\mathfrak M_{A,B}$ by at most $(1+|u_0|)^B$.", "argument_gist": "Transform the whole completed index nb^3 while retaining all fixed and moving nonunit zeros; expose three fixed cusp coefficient families and explicit active-prime factors.", "gist_origin": "source_audit_annotation", "proof_anchor": { "source": "paper.tex", "line_start": 1883, "line_end": 2463, "char_start": 87445, "char_end": 114553 }, "proof_opening": "We first express the character masks through finitely many translated theta values. We then transform their rational cusps and compute the resulting automorphy multipliers and local Fourier factors; the final Mellin comparison gives the reflection identity and the bounds for its transformed test. The fixed theta input. Let $\\Gamma=\\mathrm{SL}_2(\\mathcal O)$, $\\Gamma_1(3)=\\{g\\in\\Gamma:g\\equiv I\\pmod3\\}$, and $\\Gamma_2=\\langle\\mathrm{SL}_2(\\mathbb Z),\\Gamma_1(3)\\rangle$. The unconditional formulas of \\cite[Equations (5.4), (5.7), (5.9), and (5.12)--(5.17)]{DR} give a smooth cubic theta function $\\theta$ on $\\mathbb H^3=\\mathbb C\\times\\mathbb R_{>0}$ with \\[ \\theta(gw)=\\kappa(g)\\theta(w)\\quad(g", "anchor": { "source": "paper.tex", "line_start": 1676, "line_end": 1882, "char_start": 78720, "char_end": 87443 }, "gist_evidence": "paper.tex lines 1676–1881; paper.tex lines 1676–1881" }, { "id": "lem:reflection-row-sectors", "kind": "lemma", "label": "Lemma 5.2", "title": "Fixed ray sectors for nonzero rows", "section": "sec:completed-reflection", "statement": "Let $\\Psi_{\\rm base}$ range over a fixed finite family of finite-ray characters with conductors supported on $S$, extended by zero off the primary elements prime to $S$. Use the fixed generators $\\pi_{\\mathfrak l}$ for $\\mathfrak l\\in S$ from Lemma~\\ref{lem:fixed-numerator-ray}. For $0\\ne m\\in\\mathcal O$, write uniquely \\[ m=u m_Sm_{\\rm good},\\qquad m_S=\\prod_{\\mathfrak l\\in S}\\pi_{\\mathfrak l}^{v_{\\mathfrak l}(m)},\\qquad m_{\\rm good}=\\prod_{p\\notin S}p^{v_p(m)}, \\] where the good prime generators are primary and $u$ is a unit. On every primary element $A$ prime to $S$ one has \\chi_A(m)=\\chi_A(u m_S)\\mathcal R(A,m_{\\rm good}) \\prod_{p\\mid m_{\\rm good}}\\chi_p(A)^{v_p(m)}. All powers on the right retain their zero values, including when $v_p(m)\\equiv0\\pmod6$. Fix $u$, the valuations $v_{\\mathfrak l}(m)$ modulo six, and the class of $m_{\\rm good}$ in the fixed ray group through which $\\mathcal R$ factors. In such a sector define \\Psi_0^{[m]}(A)= \\Psi_{\\rm base}(A)\\chi_A(u m_S)\\mathcal R(A,m_{\\rm good}), &A\\equiv1\\pmod3,\\ (A,S)=1,\\\\ 0,&\\text{otherwise}. The nonzero branch is a member of a fixed finite family of finite-ray characters with conductors supported on $S$. Therefore a single $\\mathfrak E,L$ in Proposition~\\ref{prop:completed-reflection} works for all these sectors. The local prime set contains every $p\\mid m_{\\rm good}$, with exponent $v_p(m)$ modulo six, even when this exponent is zero. No moving good prime is absorbed into $L$.", "argument_gist": "Reduce arbitrary nonzero element rows to finitely many fixed ray sectors, separating primes in the fixed excluded set from the moving good part.", "gist_origin": "source_audit_annotation", "proof_anchor": { "source": "paper.tex", "line_start": 2508, "line_end": 2533, "char_start": 116555, "char_end": 118051 }, "proof_opening": "If $(A,m_{\\rm good})=1$, multiplicativity and the symmetric reciprocity factor in Lemma~\\ref{lem:fixed-gauss-phase} give Equation~\\eqref{eq:reflection-row-reduction}. If a good prime is shared, both sides are zero: the left side is the zero extension of $\\chi_A(m)$, and its local factor on the right is zero even for a six-divisible exponent. This proves the identity on the entire stated domain. On this domain $\\chi_A(u m_S)$ depends only on $u$ and the $S$-valuations modulo six, because every numerator factor is a unit modulo $A$. For each of the finitely many representatives $d=u\\prod_{\\mathfrak l\\in S}\\pi_{\\mathfrak l}^{e_{\\mathfrak l}}$, $0\\le e_{\\mathfrak l}<6$, Lemma~\\ref{lem:fixed-nume", "anchor": { "source": "paper.tex", "line_start": 2470, "line_end": 2507, "char_start": 114805, "char_end": 116553 }, "gist_evidence": "paper.tex lines 2470–2507 (full statement)" }, { "id": "lem:reflection-common-profile", "kind": "lemma", "label": "Lemma 5.3", "title": "Common annular kernel profile", "section": "sec:completed-reflection", "statement": "Let $V$ satisfy the hypotheses of Proposition~\\ref{prop:completed-reflection}. Fix an integer $d\\ge1$, a compact set $\\Omega\\subset\\mathbb R^d$, and real exponents $a_1,\\ldots,a_d$. Let $W$ be one smooth profile with logarithmic support in $\\Omega$, common to all rows in a given block, and let $Y>0$ be fixed within that block. Put \\[ h(\\boldsymbol y)=\\prod_{i=1}^d y_i^{a_i},\\qquad F_Y(\\boldsymbol y)=W(\\boldsymbol y)V^\\sharp(Yh(\\boldsymbol y)),\\qquad \\mathfrak m_A(Y)=\\min\\{Y^{1/4},(1+Y)^{-A}\\}. \\] For $A\\ge0$ and integers $j,J\\ge0$, p_j(F_Y)&\\ll_{A,j,d,\\Omega,\\boldsymbol a} \\mathfrak m_A(Y)p_j(W) \\mathfrak M_{A,\\lceil4A\\rceil+j+2}(V), \\\\ \\|F_Y\\|_{J,\\mathrm{sep}}&\\ll_{A,J,d,\\Omega,\\boldsymbol a} \\mathfrak m_A(Y)p_{J+d+2}(W) \\mathfrak M_{A,\\lceil4A\\rceil+J+d+4}(V). Here $p_j$ is the homogeneous seminorm of a single profile, not the inhomogeneous seminorm of a tuple. The constants are independent of $Y$ and of the actual norm labels within the block.", "argument_gist": "Represent the reflected annular kernel by one joint smooth profile common to every row in a block, with a controlled Fourier coefficient measure.", "gist_origin": "source_audit_annotation", "proof_anchor": { "source": "paper.tex", "line_start": 2568, "line_end": 2579, "char_start": 119474, "char_end": 120044 }, "proof_opening": "On the support of $W$, the monomial $h$ is bounded above and below by positive constants depending only on $\\Omega$ and the exponents. Each operator $y_i\\partial_{y_i}$ acting on $V^\\sharp(Yh(\\boldsymbol y))$ is $a_i$ times the Euler derivative of $V^\\sharp$ at $Yh(\\boldsymbol y)$. It produces no additional power of $Y$. Leibniz's rule and Equation~\\eqref{eq:reflection-kernel} prove Equation~\\eqref{eq:reflection-common-profile}. Applying Lemma~\\ref{lem:smooth-calculus} with $j=J+d+2$ gives Equation~\\eqref{eq:reflection-common-measure}.", "anchor": { "source": "paper.tex", "line_start": 2540, "line_end": 2567, "char_start": 118304, "char_end": 119472 }, "gist_evidence": "paper.tex lines 2540–2567 (full statement)" }, { "id": "lem:lattice-kernel-tail", "kind": "lemma", "label": "Lemma 5.4", "title": "Lattice kernel tails", "section": "sec:completed-reflection", "statement": "Let $\\Lambda$ be either $\\mathcal O$ or $\\lambda^{-4}\\mathcal O$. Suppose $K:(0,\\infty)\\to\\mathbb C$ satisfies $|K(x)|\\le C_Ax^{-A}$ for $x\\ge1$, where $A>1$. Then, for $a>0$ and $U\\ge1$, \\sum_{\\substack{0\\ne\\mu\\in\\Lambda\\\\a q_\\mu>U}}|K(aq_\\mu)| \\ll_A C_A(1+a^{-1})U^{1-A}. For $K=V^\\sharp$ one may take $C_A\\ll_A\\mathfrak M_{A,\\lceil4A\\rceil+2}(V)$.", "argument_gist": "Turn rapid decay of a reflected kernel into a summable power bound for the remote Eisenstein-lattice frequencies.", "gist_origin": "source_audit_annotation", "proof_anchor": { "source": "paper.tex", "line_start": 2630, "line_end": 2637, "char_start": 122906, "char_end": 123267 }, "proof_opening": "The shell $2^jU0$ and every fixed $A\\ge0$, \\sum_{R\\in\\mathcal I_H}|\\mathcal U_{v,\\ell_b,e_\\lambda}(R)|^2 &\\ll_{\\epsilon,A} Z^{E_0+\\epsilon}p_5(W)^2 \\mathfrak M_{A,\\lceil4A\\rceil+7}(V)^2,\\\\ E_0={}&\\max(H,v+\\ell_b)-S_0-B_0-\\ell_b-\\frac{2e_\\lambda}{3} -\\frac12(T_0-v-3\\ell_b-e_\\lambda)_+. The constants may depend on the fixed kernel order $A$, the fixed arithmetic data, the bounded log-length ranges, and the fixed support box, but not on the moving frozen local set $\\mathcal F$. Norm twists in $W_0$ have the polynomial cost supplied by its finite seminorms.", "argument_gist": "Separate the unmarked block into row and column factors and bound its energy while preserving the common profile and original masks.", "gist_origin": "source_audit_annotation", "proof_anchor": { "source": "paper.tex", "line_start": 3003, "line_end": 3077, "char_start": 140524, "char_end": 144089 }, "proof_opening": "For the fixed $h_0$ and the fixed class of $R$ modulo $M_{\\rm ref}^2$, the class of $F_{\\rm act}R$ is fixed. Proposition~\\ref{prop:completed-reflection} therefore gives one common $d$ and one common additive character $\\vartheta$ for the whole row set. Write $x_F=u\\lambda^{k+4}N_FB_F^3$, so that $\\lambda^4\\mu=x_Fnb^3$. The residual local factors are exactly \\[ \\prod_{p\\mid R}B_p(\\lambda^4\\mu) =\\chi_R(x_F)^3\\chi_R(nb^3)^3 =\\chi_R(x_F)^3\\chi_R(nb)^3. \\] Both equalities include zeros; no unit symbol has been divided. In particular, any collision with $x_F$ is a restriction depending only on $R$. Every remaining local factor is fixed and depends only on the dual index. The scalar $\\zeta_R$ of th", "anchor": { "source": "paper.tex", "line_start": 2956, "line_end": 3002, "char_start": 138565, "char_end": 140522 }, "gist_evidence": "paper.tex lines 2956–3002 (full statement)" }, { "id": "lem:unmarked-completed-moment", "kind": "lemma", "label": "Lemma 5.8", "title": "Unmarked completed-row moment", "section": "sec:completed-reflection", "statement": "Let $\\nu$ range over a fixed finite family of multiplicative finite-ray characters with conductors supported on $S$, all extended by zero away from the primary elements prime to $S$. Fix bounded ranges for $M\\ge0$ and $N\\in\\mathbb R$, a constant $C_m\\ge1$, and fixed dyadic support comparisons. For a nonzero element $m$, let \\[ m_{\\rm pow}=\\prod_{v_{\\mathfrak p}(m)\\ge2} \\mathfrak p^{v_{\\mathfrak p}(m)} \\] be its maximal powerful ideal factor. Use nonnegative centers $O$ for its dyadic ranges, with center zero for the unit range. For every $\\epsilon>0$ there exist a finite $A\\ge0$ and an integer $J\\ge0$ such that, for every test $V$ satisfying Proposition~\\ref{prop:completed-reflection}, every actual $O$-dyad, and all sufficiently large $Z$, \\sum_{\\substack{00$. For $Y,Q\\ge1$ and every real $v$, the polynomial $A_{m,\\sigma,v}(Y)$ satisfies \\[ \\sum_{\\substack{m\\in\\mathcal O\\\\q_m\\le CQ}} |A_{m,\\sigma,v}(Y)|^2 \\ll_{\\mathcal A,W_1,C}\\frac{Q+Y^2}{Y}. \\] The bound is uniform in $v$. In particular, when $Q=q_{b_*}Y^2$, \\[ \\sum_{q_m\\le CQ}|A_{m,\\sigma,v}(Y)|^2 \\ll_{\\mathcal A,W_1,C}\\frac QY. \\]", "argument_gist": "Reduced additive Gauss fractions have controlled spacing and coefficient mass; their mean square costs (Q+Y^2)/Y.", "gist_origin": "source_audit_annotation", "proof_anchor": { "source": "paper.tex", "line_start": 3604, "line_end": 3667, "char_start": 168062, "char_end": 170800 }, "proof_opening": "For a primary $s$ outside $S$, put $r_s=q_s/Y$ and \\[ c_\\sigma(s)=1_{s\\in\\sigma}\\frac{\\chi_s(b_*)}{\\tau\\xi(s)}. \\] The inverse is evaluated only on these elements prime to $S$. Here $|\\tau|=1$, $|\\xi(s)|=1$, and $|\\chi_s(b_*)|=1$, because $b_*$ is supported on $S$. Thus $|c_\\sigma(s)|\\le1$. The definition of $A$ and the Gauss expansion give the exact identity \\[ A_{m,\\sigma,v}(Y) =Y^{-3/2}\\sum_s c_\\sigma(s)W_1(r_s)r_s^{-1+iv} \\sum_{d\\bmod s}\\chi_s(d)e(-md/s). \\] The sum is over primary ideals outside $S$. For every such $s$, the character $\\chi_s(d)$ is zero exactly when $(d,s)>1$, and has absolute value one otherwise. This includes nonsquarefree $s$: a local exponent divisible by six is the", "anchor": { "source": "paper.tex", "line_start": 3587, "line_end": 3603, "char_start": 167506, "char_end": 168060 }, "gist_evidence": "paper.tex lines 3587–3602; paper.tex lines 3587–3666" }, { "id": "prop:balanced-low", "kind": "proposition", "label": "Proposition 6.3", "title": "Balanced low estimate", "section": "sec:probe", "statement": "Fix the arithmetic data and smooth weights used to define $I_\\eta$. For every $\\epsilon>0$, as $Z\\to\\infty$, \\[ \\bigl|I_\\eta(Z^{1/2},Z^{1/2},Z)\\bigr| \\ll_{\\mathcal A,\\epsilon} Z^{1/4+\\epsilon}. \\] The exponent is independent of the target character; the implied constant and lower threshold may depend on its fixed arithmetic data.", "argument_gist": "Combine the completed-row energy and additive mean square by Cauchy–Schwarz to obtain the balanced Z^(1/4+epsilon) bound.", "gist_origin": "source_audit_annotation", "proof_anchor": { "source": "paper.tex", "line_start": 3684, "line_end": 3722, "char_start": 171397, "char_end": 173036 }, "proof_opening": "Set $X=Y=Z^{1/2}$ and $Q=q_{b_*}XY=q_{b_*}Z$. The function $\\Omega$ in Equation~\\eqref{eq:low-separated} is bounded and supported in a fixed compact subinterval of $(0,\\infty)$. Thus all rows in that identity lie in $00$ \\[ \\sum_{00$, the correction product converges normally on a neighborhood of every point in each of the following regions, and hence defines a holomorphic function there: x_r\\ge51/100,\\quad z_r\\ge17/50,\\quad w_r\\ge-1/100,\\quad x_r+w_r\\ge1+\\epsilon_0;\\\\ x_r\\ge7/8,\\quad z_r\\ge33/200,\\quad w_r\\ge19/20. Uniformly in imaginary parts and unit phases, $\\mathcal H_{\\eta,u}\\ll_\\epsilon q_u^\\epsilon$, with the constant also depending on $\\epsilon_0$ in the first region. For $u=1$ in the second region, $\\mathcal H_{\\eta,1}=1+O(P_0^{-c_H})$ with, for example, $c_H=4/5$. Equation~\\eqref{eq:probe-high} begins in absolute convergence and supplies a meromorphic continuation through these regions.", "argument_gist": "Factor each sixth-power-free Poisson row into scalar Hecke L-functions and a holomorphic Euler correction; the principal row contains the target reciprocal.", "gist_origin": "source_audit_annotation", "anchor": { "source": "paper.tex", "line_start": 4010, "line_end": 4068, "char_start": 184708, "char_end": 186843 }, "gist_evidence": "paper.tex lines 4010–4067; paper.tex lines 4010–4067" }, { "id": "sec:detector", "kind": "section", "label": "Section 8", "title": "A zero detector with saturated witnesses", "section": "sec:detector", "argument_gist": "Assign buffered zero-free bins and produce simultaneous inverse and plain polynomial witnesses for selected zeros.", "anchor": { "source": "paper.tex", "line_start": 4201, "line_end": 4687, "char_start": 192621, "char_end": 212759 }, "gist_evidence": "paper.tex lines 4201–4687 (section scope)", "gist_origin": "source_audit_annotation" }, { "id": "lem:buffered-bins", "kind": "lemma", "label": "Lemma 8.1", "title": "Buffered zero-free bins", "section": "sec:detector", "statement": "For every retained row \\(u\\), there are an index \\(i\\in\\{1,\\ldots,I-1\\}\\) and a grid point \\[ a\\in(51/100+e\\mathbb Z_{\\ge0})\\cap[51/100,1] \\] with the following properties. For \\(j=1,\\ldots,I\\), set \\[ M_j(u)=\\max\\left(\\{51/100\\}\\cup \\left\\{\\Re\\rho: {l} \\psi\\in\\mathcal X_u,\\quad L(\\rho,\\psi^*)=0,\\\\ \\Re\\rho\\ge51/100,\\quad |\\Im\\rho|\\le3jT_1 \\right\\}\\right). \\] Then \\[ a\\le M_i(u)51/100\\), some \\(\\psi\\in\\mathcal X_u\\) has a zero \\(\\rho=\\sigma+i\\gamma\\) with \\[ a\\le\\sigma0\\), every \\(\\psi\\in\\mathcal X_u\\), and sufficiently large \\(Z\\), uniformly in the retained row, \\[ |L_{\\rm orig}(s,\\psi)|+|L_{\\rm orig}(s,\\psi)^{-1}| \\ll_{\\mathcal A,e,\\epsilon_1} U^{\\epsilon_1} \\quad \\left(\\Re s\\ge a+6e,\\ |\\Im s|\\le(3i+2)T_1\\right). \\] On the reflected line in the same height range, \\[ |L_{\\rm orig}(1-a-6e+it,\\psi)| \\ll_{\\mathcal A,e,\\epsilon_1} U^{a-1/2+12e+\\epsilon_1}(3+T_1)^C, \\] where \\(C\\) depends only on the fixed real strip and the field. There are \\(O_e(1)\\) possible pairs \\((i,a)\\), independently of the row and its conductor.", "argument_gist": "Assign buffered zero-free rectangles to each row family and obtain pointwise controls.", "gist_origin": "source_audit_annotation", "proof_anchor": { "source": "paper.tex", "line_start": 4324, "line_end": 4369, "char_start": 197473, "char_end": 199667 }, "proof_opening": "Each maximum exists: the collection is finite, the zeros of each nonprincipal primitive \\(L\\)-function are discrete in compact rectangles, and absolute Euler convergence excludes zeros with real part greater than one. The sequence \\(M_j(u)\\) is nondecreasing and lies in \\([51/100,1]\\). If every one of its \\(I-1\\) increments exceeded \\(e\\), then \\[ M_I(u)-M_1(u)>(I-1)e>1, \\] which is impossible. Choose an index with \\(M_{i+1}(u)-M_i(u)\\le e\\), and round \\(M_i(u)\\) down on the stated grid. This gives the two inequalities. When \\(a>51/100\\), the maximum \\(M_i(u)\\) is attained by an actual zero, giving \\(\\rho\\). This reasoning allows a zero on the line one. For \\(|t|\\le(3i+2)T_1\\), the closed di", "anchor": { "source": "paper.tex", "line_start": 4281, "line_end": 4323, "char_start": 196226, "char_end": 197471 }, "gist_evidence": "paper.tex lines 4281–4322; paper.tex lines 4281–4368" }, { "id": "lem:detector-dyads", "kind": "lemma", "label": "Lemma 8.2", "title": "Pointwise dyadic estimates", "section": "sec:detector", "statement": "Fix bounded nonnegative ranges for \\(r,m\\), and let \\[ M_r=M_\\psi(r;W_M),\\qquad S_m=S_\\psi(m;W_S) \\] denote the polynomials of Equations~\\eqref{old-eq:1.1a} and \\eqref{old-eq:1.1}, now at base \\(U\\). Suppose \\(\\psi\\in\\mathcal X_u\\), the untwisted profiles have uniformly bounded smooth seminorms on a fixed annulus, and any pure norm twist has height at most \\((3i+1)T_1\\). All additional Mellin frequencies entering the same \\(L\\)-argument are required to have total absolute value at most \\(T_1/2\\). For every \\(\\epsilon>0\\), there are \\(e_0>0\\), depending only on \\(\\epsilon\\) and the bounded length ranges, and a finite height order \\(A_{\\mathcal A}\\), uniform in the moving rows and profiles, such that, when \\[ 01\\) gives \\[ &U^{-r/2}\\sum_n\\mu(n)\\psi(n) W(q_n/U^r)(q_n/U^r)^{-\\sigma+i\\omega}\\\\ &\\quad=\\frac1{2\\pi i}\\int_{(c)} \\mathcal MW(s)\\, U^{r(s+\\sigma-i\\omega-1/2)} L_{\\rm orig}(s+\\sigma-i\\omega,\\psi)^{-1}\\,ds. \\] Here \\(\\mathcal MW\\) is the Mellin transform defined in Lemma~\\ref{lem:smooth-calculus}. The twist occurs in the \\(L\\)-argument, not in the transform whose tails are estimated. Shift only the portion with added frequency at most the assigned fraction", "anchor": { "source": "paper.tex", "line_start": 4385, "line_end": 4418, "char_start": 200259, "char_end": 201529 }, "gist_evidence": "paper.tex lines 4385–4417; paper.tex lines 4385–4498" }, { "id": "prop:detector-witness", "kind": "proposition", "label": "Proposition 8.3", "title": "Two saturated witnesses", "section": "sec:detector", "statement": "Let \\(u\\) have a bin \\((i,a)\\) with \\(a>51/100\\), and assume the loss and height hypotheses of Lemma~\\ref{lem:detector-dyads}. For every \\(t\\in[1,3/2]\\), and every prescribed \\(\\epsilon>0\\), after reducing its preliminary losses, there are \\(\\psi\\in\\mathcal X_u\\), a zero \\(\\rho=\\sigma+i\\gamma\\) as in Lemma~\\ref{lem:buffered-bins}, dyadic lengths \\(D=U^r\\), \\(N=U^m\\), and two polynomials \\(M_r,S_m\\) for the common row character \\(\\psi\\) such that r\\le t+O(1/\\log U),\\qquad r+m\\ge t-O(1/\\log U),\\qquad |M_rS_m|^2\\gg_{\\mathcal A,\\epsilon} U^{\\delta(r+m)-\\epsilon}. Their lengths and individual values also satisfy t-\\tfrac12-O(\\epsilon)\\le r\\le t+O(1/\\log U), \\qquad 0\\le m\\le\\tfrac12+O(\\epsilon),\\\\ |M_r|^2\\gg_{\\mathcal A,\\epsilon}U^{\\delta r-\\epsilon}, \\qquad |S_m|^2\\gg_{\\mathcal A,\\epsilon}U^{\\delta m-\\epsilon}. The constants in the \\(O(\\epsilon)\\) terms are absolute on the stated parameter ranges. Both polynomials have the same twist height \\(\\gamma-\\nu\\), where \\(|\\nu|\\le cT_1\\) for a fixed \\(c>0\\) chosen within the cumulative frequency allowance. Their untwisted profiles form a uniformly smooth annular family. For each row, both witnesses use one presentation \\(\\psi_{u,\\vartheta,\\varsigma}\\), selected by a label \\((\\vartheta,\\varsigma)\\) in the fixed finite set \\(\\Theta\\times\\{-1,1\\}\\). The presentation itself varies with \\(u\\). The dyadic pair can likewise be selected separately for each row from \\(O((\\log U)^2)\\) possibilities.", "argument_gist": "A selected zero yields two simultaneous saturated Dirichlet-polynomial witnesses, inverse and plain, sharing one presentation and twist height.", "gist_origin": "source_audit_annotation", "proof_anchor": { "source": "paper.tex", "line_start": 4549, "line_end": 4686, "char_start": 207383, "char_end": 212757 }, "proof_opening": "The truncated-inverse and Gamma-integral construction is a form of the classical zero detector; compare \\cite[Appendix C]{MP} and \\cite[Section~13.1]{GM}. The buffered row-dependent rectangle and the simultaneous lower bounds needed here are established below. Choose the zero \\(\\rho\\) supplied by the bin and set \\[ D_*=U^t,\\qquad Y_*=U^{20}. \\] Let \\(V_{\\le}\\) be a fixed smooth function equal to one on \\([0,1]\\) and zero on \\([2,\\infty)\\). Define \\[ C_\\psi(s)=\\sum_l\\mu(l)\\psi(l)V_{\\le}(q_l/D_*)q_l^{-s}, \\qquad J=\\frac1{2\\pi i}\\int_{(2)} Y_*^z\\Gamma(z)L_{\\rm orig}(\\rho+z,\\psi) C_\\psi(\\rho+z)\\,dz. \\] The row character is nonprincipal, so \\(L_{\\rm orig}\\) is entire. Moving the line to \\(\\Re z=-", "anchor": { "source": "paper.tex", "line_start": 4510, "line_end": 4548, "char_start": 205678, "char_end": 207381 }, "gist_evidence": "paper.tex lines 4510–4547; paper.tex lines 4510–4547" }, { "id": "sec:sextic-row-count", "kind": "section", "label": "Section 9", "title": "A sextic-sieve row count", "section": "sec:sextic-row-count", "argument_gist": "Prove the sextic large sieve and use it to count physical rows with large inverse witnesses.", "anchor": { "source": "paper.tex", "line_start": 4687, "line_end": 5460, "char_start": 212759, "char_end": 249371 }, "gist_evidence": "paper.tex lines 4687–5460 (section scope)", "gist_origin": "source_audit_annotation" }, { "id": "lem:sextic-large-sieve", "kind": "lemma", "label": "Lemma 9.1", "title": "Sextic large sieve", "section": "sec:sextic-row-count", "statement": "Let $K,D\\ge1$, let $\\varsigma\\in\\{-1,1\\}$, and let $(c_n)$ be any sequence of complex numbers indexed by the squarefree ideals outside $S$. The sequence is fixed independently of the row $k$. For every $\\epsilon>0$, \\sum_{\\substack{k\\ {\\rm sf}\\\\q_k\\le K}} \\left|\\sum_{\\substack{n\\ {\\rm sf}\\\\q_n\\le D}} c_n\\chi_n(k)^{\\varsigma}\\right|^2 \\ll_{S,\\epsilon}(KD)^\\epsilon \\bigl\\{K+D+(KD)^{2/3}\\bigr\\} \\sum_{\\substack{n\\ {\\rm sf}\\\\q_n\\le D}}|c_n|^2. Every symbol in this formula has its original zero value on a nonunit. A fixed restriction of the row set is allowed. A fixed restriction of the coefficient support is allowed by extending that coefficient sequence by zero. Such a restriction may depend on an object fixed before the row sum, but the coefficients may not otherwise depend on $k$.", "argument_gist": "Establish the squarefree sextic large-sieve bound K+D+(KD)^(2/3) in the primary convention with original zeros.", "gist_origin": "source_audit_annotation", "proof_anchor": { "source": "paper.tex", "line_start": 4727, "line_end": 5172, "char_start": 214781, "char_end": 235638 }, "proof_opening": "We follow the higher-order large-sieve recursion of Blomer, Goldmakher, and Louvel \\cite[Section~3]{BGL}, giving the local verification for the present primary normalization. In the finite Fourier calculation the individual residue characters act on elements; their unit-trivial quotient is the character on ideals to which we apply Poisson summation. For good ideals $a,b$, represented by their primary generators, put \\[ F_b(a)=\\chi_a(b),\\qquad J=\\{1,2,4\\}. \\] This uses the original zero-extended symbol, also when $a$ is not squarefree. It is multiplicative in both $a$ and $b$. In every power below, including a multiple of six, a nonunit still has value zero. For $M>0$ let \\[ \\mathcal I_M=\\{a\\", "anchor": { "source": "paper.tex", "line_start": 4707, "line_end": 4726, "char_start": 213826, "char_end": 214779 }, "gist_evidence": "paper.tex lines 4707–4725; paper.tex lines 4707–5171" }, { "id": "prop:sextic-row-count", "kind": "proposition", "label": "Proposition 9.2", "title": "Sextic-sieve row envelope", "section": "sec:sextic-row-count", "statement": "Fix the arithmetic data $\\mathcal A$, the bounded physical range $d_{\\min}\\le d\\le d_{\\max}$, and the height and loss hypotheses of Lemmas~\\ref{lem:buffered-bins} and~\\ref{lem:detector-dyads}. Put $U=Z^d$ and retain the notation $T_1$ of those lemmas. Let $\\mathcal B$ be any set of retained sixth-power-free element rows $u$ with $q_u\\asymp U$, all in one bin $(i,a)$ with $a>51/100$. Put $\\delta=2a-1$, so $1/50<\\delta\\le1$. For every $\\epsilon>0$, after reducing the preliminary losses in Proposition~\\ref{prop:detector-witness}, for all sufficiently large $Z$, \\#\\mathcal B\\ll_{\\mathcal A,\\epsilon} U^{R(\\delta)+\\epsilon}(1+T_1)^{A_{\\mathcal A}}, \\qquad R(\\delta)=\\min\\left\\{1, \\max\\left(1-\\frac\\delta2,\\frac43-\\delta\\right)\\right\\}. The finite order $A_{\\mathcal A}$ is uniform in the physical dyad, the bin, and all moving rows. It may be fixed before the positive exponent in $T_1=Z^\\tau$ is chosen. The estimate is valid for each fixed retained tuple of external parameters and for the original row restrictions. The implied constant may depend on the fixed comparison constants and finitely many of the shared profile seminorms. The rowwise profile choices are precisely those permitted by Proposition~\\ref{prop:detector-witness} and Lemma~\\ref{lem:smooth-calculus}; no arbitrary row-dependent coefficients are allowed. The floor bin $a=51/100$ is excluded from the witness assertion. For that bin the elementary bound $\\#\\mathcal B\\ll_{\\mathcal A}U$ applies; it agrees with the numerical value $R(1/50)=1$.", "argument_gist": "Apply the sextic sieve to frozen powerful parts and squarefree residual rows to count large inverse-witness rows.", "gist_origin": "source_audit_annotation", "proof_anchor": { "source": "paper.tex", "line_start": 5215, "line_end": 5447, "char_start": 237755, "char_end": 249037 }, "proof_opening": "Fix a small preliminary loss $\\epsilon_0>0$. Apply Proposition~\\ref{prop:detector-witness} with $t=1$ to every row in $\\mathcal B$. Subdivide by its presentation labels in $\\psi_{u,\\nu,\\varsigma}(n)=\\nu(n)\\chi_n(u)^{\\varsigma}$ and its dyadic pair of polynomial lengths. There are a fixed finite number of presentation labels and $O_{\\mathcal A}((\\log U)^2)$ pairs. Within one such subdivision, the inverse length $D=U^r$ and the pair $(\\nu,\\varsigma)$ are fixed. In this subdivision write \\[ M_u(D;W)=D^{-1/2}\\sum_n\\mu(n)\\nu(n)\\chi_n(u)^{\\varsigma}W(q_n/D) \\] for the inverse polynomial of that presentation. The witness gives \\frac12-O(\\epsilon_0)\\le r\\le1+O(1/\\log U), \\qquad |M_u(D;W_u)|^2\\gg_{\\m", "anchor": { "source": "paper.tex", "line_start": 5181, "line_end": 5214, "char_start": 236063, "char_end": 237753 }, "gist_evidence": "paper.tex lines 5181–5213; paper.tex lines 5181–5446" }, { "id": "sec:shared-analytic-estimates", "kind": "section", "label": "Section 10", "title": "Analytic estimates for the high expansion", "section": "sec:shared-analytic-estimates", "argument_gist": "Establish reusable contour estimates for central bins, the principal signal, external tails, and small and large row norms.", "anchor": { "source": "paper.tex", "line_start": 5460, "line_end": 6493, "char_start": 249371, "char_end": 296885 }, "gist_evidence": "paper.tex lines 5460–6493 (section scope)", "gist_origin": "source_audit_annotation" }, { "id": "def:stage-high-data", "kind": "definition", "label": "Definition 10.1", "title": "Data for an exact high representation", "section": "sec:shared-analytic-estimates", "statement": "Fix real numbers $l_x,l_y>0$ and $\\ell\\ge0$ such that X=Z^{l_x},\\qquad Y=Z^{l_y},\\qquad h=1-l_x+\\ell>0, \\qquad C(s)=s+\\frac{l_x}{2}-1+\\frac h6 =s-\\frac56+\\frac{l_x}{3}+\\frac\\ell6. These real numbers are fixed before the target character. For each primitive finite-order target $\\eta$, fix the data $S,\\xi,W_0,W_1$ of Section~\\ref{sec:probe}, independently of $Z$. Data for an exact high representation consist of the following objects and assertions. First, $\\mathscr I_\\eta(Z)$ is an independently specified complex-valued quantity for every sufficiently large real $Z$. In each application it is given by a finite linear combination, for that $Z$, of the completed sums in Equation~\\eqref{eq:general-probe}, allowing specified restrictions on the completed index $A$ and specified rescalings of its three positive scales. The number of terms and coefficients in this finite combination may depend on $Z$. Any direct estimate for $\\mathscr I_\\eta$ refers to this expression, not to a separately defined high integral. Second, for every nonzero sixth-power-free element $u$ with $(u,S)=1$ and every such $Z$, there is a function $\\mathfrak H_{\\eta,u,Z}(s,w,z)$. It is holomorphic on a neighborhood of each point of $\\mathcal D_2$ and of every $\\mathcal D_1(\\epsilon_0)$. For every $D>0$ and every real box $\\mathcal K$ contained in one of these regions, there are finite constants $B_{\\eta,\\mathcal K,D}$ and $J_{\\eta,\\mathcal K,D}$ such that, uniformly for $1\\le q_u\\le Z^D$, all imaginary parts, and real parts in $\\mathcal K$, |\\mathfrak H_{\\eta,u,Z}(s,w,z)| \\ll_{\\eta,\\mathcal K,D} Z^{B_{\\eta,\\mathcal K,D}} (1+|\\Im s|+|\\Im w|+|\\Im z|)^{J_{\\eta,\\mathcal K,D}}. The constants and exponents are independent of $Z,u$ and of any later order of integration by parts. No nonvanishing of $\\mathfrak H$ or of any of its local factors is assumed. Finally, the following identity is asserted for the independently given $\\mathscr I_\\eta(Z)$, with absolute convergence of the sum and integrals on the displayed lines: \\mathscr I_\\eta(Z)=\\frac1{(2\\pi i)^3} \\int_{(3)}\\int_{(3)}\\int_{(2)} &\\mathcal W(X,Y,Z;s,w,z) \\sum_u^{(6)}q_u^{-z}\\overline{\\xi(u)}\\\\ &\\cdot \\frac{\\zeta_F^S(6z)L^S(w,\\chi_\\bullet(u))} {L^S(s,\\eta\\overline{\\chi_\\bullet(u)})} \\mathfrak H_{\\eta,u,Z}(s,w,z)\\,dz\\,dw\\,ds, where the sum has the physical restrictions of Equation~\\eqref{eq:probe-poisson-identity}, and \\[ \\mathcal W(X,Y,Z;s,w,z) =X^{1/2-z}Z^{s+z-1}Y^{w-1} e^{(s+z-1)^2}M(z)\\widehat W_1(w). \\] Thus Equation~\\eqref{eq:stage-high-identity} is a hypothesis to be proved for the physical expression, not its definition.", "argument_gist": "Specify the common contract for an exact high representation: one independently defined physical probe, a row Euler correction, and the precise Mellin identity and normalization.", "gist_origin": "source_audit_annotation", "anchor": { "source": "paper.tex", "line_start": 5582, "line_end": 5651, "char_start": 255047, "char_end": 258005 }, "gist_evidence": "paper.tex lines 5582–5651 (full statement)" }, { "id": "lem:external-contour-tails", "kind": "lemma", "label": "Lemma 10.2", "title": "External integrated and trace tails", "section": "sec:shared-analytic-estimates", "statement": "Let $m\\ge2$, let $\\mathcal K$ be a compact set of real contour parameters, and, for $\\boldsymbol r\\in\\mathcal K$ and $Z\\ge2$, let $K_{\\boldsymbol r,Z}$ be a nonnegative Borel measurable kernel on $\\mathbb R^m$. Suppose it is jointly Borel measurable in $(\\boldsymbol r,\\boldsymbol t)$ and, for every integer $L\\ge0$, \\[ K_{\\boldsymbol r,Z}(\\boldsymbol t) \\ll_L(1+|\\boldsymbol t|)^{-L} \\] uniformly in $\\boldsymbol r,Z,\\boldsymbol t$. Let $\\mathcal D\\subseteq\\mathbb R^m$ be Borel measurable, and suppose a Borel measurable factor $F_Z$ satisfies on $\\mathcal D$ |F_Z(\\boldsymbol t)|\\le C Z^B(1+|\\boldsymbol t|)^J, where $B,J,C$ are fixed independently of $Z$ and of the integer $N$ below. Write $\\widehat{\\boldsymbol t}_j$ for the vector obtained by omitting the $j$th coordinate of $\\boldsymbol t$. Uniformly for $T\\ge1$ and $\\boldsymbol r\\in\\mathcal K$, \\int_{\\mathcal D\\cap\\{\\max_j|t_j|>T\\}} K_{\\boldsymbol r,Z}(\\boldsymbol t)|F_Z(\\boldsymbol t)|\\,d\\boldsymbol t &\\ll_N Z^B T^{-N}, \\\\ \\int_{\\mathcal D\\cap\\{t_j=T\\}} K_{\\boldsymbol r,Z}(\\boldsymbol t)|F_Z(\\boldsymbol t)|\\, d\\widehat{\\boldsymbol t}_j &\\ll_N Z^B(1+|T|)^{-N}. The second assertion holds for either sign of $T$ and every $j$, using coordinate Lebesgue measure on $t_j=T$. Both assertions remain valid after integration over a real contour interval of bounded length when the hypotheses are uniform there and the kernels, domain indicators and arithmetic factors are jointly Borel measurable in that real parameter and the remaining coordinates.", "argument_gist": "Integrate rapidly decaying external kernels against polynomial arithmetic bounds, including tail and trace estimates on contour joins.", "gist_origin": "source_audit_annotation", "proof_anchor": { "source": "paper.tex", "line_start": 5713, "line_end": 5729, "char_start": 260885, "char_end": 261487 }, "proof_opening": "Multiply the kernel bound by Equation~\\eqref{eq:external-arithmetic-majorant}. Integrating $(1+|\\boldsymbol t|)^{J-L}$ outside the box proves Equation~\\eqref{eq:external-integrated-tail} when $L>N+J+m$. On $t_j=T$, integration in the other $m-1$ coordinates gives \\[ \\int_{\\mathbb R^{m-1}} (1+|T|+|\\widehat{\\boldsymbol t}_j|)^{J-L} d\\widehat{\\boldsymbol t}_j \\ll(1+|T|)^{J-L+m-1}. \\] Taking $L>N+J+m-1$ proves Equation~\\eqref{eq:external-trace-tail}. Restricting to $\\mathcal D$ decreases these positive integrals, and a bounded real interval contributes only its length.", "anchor": { "source": "paper.tex", "line_start": 5672, "line_end": 5712, "char_start": 259094, "char_end": 260883 }, "gist_evidence": "paper.tex lines 5672–5712 (full statement)" }, { "id": "lem:fixed-bin-contour", "kind": "lemma", "label": "Lemma 10.3", "title": "Contour transformation for a fixed bin", "section": "sec:shared-analytic-estimates", "statement": "Assume Definition~\\ref{def:stage-high-data}. Fix $\\sigma_0\\in[7/8,1)$ with $\\beta_*>\\sigma_0$, put $z_0=17/50$, and let $U=Z^d$ with $02$, where $0<\\tau\\le d_{\\min}/100$. The real ranges and $e$ are independent of the target. Fix positive constants $c_s,c_w,c_z\\le1/2$, independently of $Z$ and the rows. Retain the original heights $|\\Im s|\\le c_sT_1$, $|\\Im w|\\le c_wT_1$ and $|\\Im z|\\le c_zT_1$. The contribution of $\\mathcal B$ on the starting lines in Equation~\\eqref{eq:stage-high-identity} equals its integral on these retained segments of \\Re s=a+16e,\\qquad \\Re w=1-a-6e,\\qquad \\Re z=z_0, up to $O_{\\eta,N}(Z^{B_\\eta}T_1^{-N})$ for every fixed $N$. The finite exponent $B_\\eta$ is fixed before $N$. An empty row set contributes zero. The transformation uses the full holomorphic correction; no decomposition of that correction or subdivision depending on a contour point is needed.", "argument_gist": "Move the Mellin contours of one fixed row bin only after its rows and zero-free buffers have been fixed; track the resulting heights and errors.", "gist_origin": "source_audit_annotation", "proof_anchor": { "source": "paper.tex", "line_start": 5836, "line_end": 5897, "char_start": 266725, "char_end": 269898 }, "proof_opening": "Section~\\ref{sec:detector} gives $a\\le\\beta_*$, including the floor bin. Isolate the finite sum over $\\mathcal B$ on the absolute lines in Equation~\\eqref{eq:stage-high-identity}. Move $z$ from $2$ to $z_0$, keep $w=3$, and move $s$ from $3$ to $\\beta_*+20e$. These moves stay in $\\mathcal D_1(\\epsilon_0)$ for a fixed positive $\\epsilon_0$, the two numerator arguments remain to the right of their poles, and the reciprocal stays in $\\Re s>\\beta_*$. Its global bound follows from Lemmas~\\ref{lem:logarithmic-control} and~\\ref{lem:deleted-euler-factors}, using the conductor and deletion radical bounds in Section~\\ref{sec:detector}. The correction obeys Equation~\\eqref{eq:stage-all-height-majorant}", "anchor": { "source": "paper.tex", "line_start": 5808, "line_end": 5835, "char_start": 265369, "char_end": 266723 }, "gist_evidence": "paper.tex lines 5808–5835 (full statement)" }, { "id": "lem:bin-contour-accounting", "kind": "lemma", "label": "Lemma 10.4", "title": "A retained integral and its exponent", "section": "sec:shared-analytic-estimates", "statement": "Assume the data of Definition~\\ref{def:stage-high-data}. Fix $\\sigma_0\\in[7/8,1)$ with $\\beta_*>\\sigma_0$, and put $z_0=17/50$. Let $02$ with $0<\\tau\\le d_{\\min}/100$. The real ranges and $e$ are chosen independently of the target. The bin $\\mathcal B$ is fixed before any Mellin variable is moved. On the retained central contours of Equation~\\eqref{eq:stage-central-contours}, suppose that there is a decomposition \\[ \\mathfrak H_{\\eta,u,Z}(s,w,z) =\\sum_{\\lambda\\in\\Lambda} \\mathfrak H^{(\\lambda)}_{\\eta,u,Z}(s,w,z), \\] where $\\Lambda$ is finite, nonempty and fixed before $Z$. This equality is required only on the retained contours; the summands need not be holomorphic away from them. For each $\\lambda$ and each retained $(s,w,z)$, suppose $\\mathcal B$ is partitioned into at most $C(1+\\log Z)^D$ sets $\\mathcal B_{\\lambda,\\nu}(s,w,z)$, with fixed $C,D$. For each set at each retained point let $R,g$ be real numbers in a fixed bounded range. Assume that, for fixed $\\varepsilon_c\\ge0$ and a finite $A_\\eta\\ge0$, \\sum_{u\\in\\mathcal B_{\\lambda,\\nu}(s,w,z)} \\left|L^S(w,\\chi_\\bullet(u)) \\mathfrak H^{(\\lambda)}_{\\eta,u,Z}(s,w,z)\\right| \\ll_\\eta U^{R+\\delta/2+\\varepsilon_c} Z^{\\ell(z_0-1/2)+g+\\varepsilon_c}(1+T_1)^{A_\\eta}. The bound is uniform in the retained point, row set and moving labels. The number $\\varepsilon_c$ and the bounded range for $R,g$ are fixed before the target; $R,g$ may depend on $d,a$, the pointwise set and the global number $\\beta_*$, but not otherwise on $\\eta$. Let $\\mathcal P_Z$ be the set of all pairs $(R,g)$ that occur at any retained point in this decomposition. Here ``retained'' has the following precise height meaning. List once all external coordinates $v_1,\\ldots,v_m$ that enter a buffered $L$-value, reciprocal, or logarithmic derivative in the verification of Equation~\\eqref{eq:stage-central-bound}, and include all three original Mellin heights, with zero coefficients when they do not occur in an argument. After pure-twist translations, every such argument has imaginary part $\\gamma_j+\\sum_{k=1}^m a_{jk}v_k$, where the finite matrix $(a_{jk})$ is fixed and $|\\gamma_j|\\le(3i+1)T_1$. Restrict |v_k|\\le c_kT_1,\\qquad c_k=\\frac1{2m(1+\\max_j|a_{jk}|)}. Coordinates with zero coefficients may be restricted by the same rule. Then every added height is at most $T_1/2$ in total. Fixed bounded enlargements are included by increasing the lower threshold for $Z$. If extra Mellin integrals are used to establish Equation~\\eqref{eq:stage-central-bound}, that inequality is required for its full left side after those integrations. Any discarded parts must have been left on their global or absolute lines, bounded by Lemma~\\ref{lem:external-contour-tails} or Lemma~\\ref{lem:smooth-calculus}, and included in the displayed bound before the hypothesis is asserted. The allowance $T_1/2$ is not renewed at successive estimates. The common ideal exponent associated with a pair $(R,g)$ is E_{\\sigma_0}(d;R,g)=a-\\sigma_0+h(z_0-1/6)-a l_y-\\ell/2+g +d(R+\\delta/2-z_0). If $\\mathcal B$ is empty its contribution is zero. Otherwise put \\[ E^{\\max}_{\\sigma_0,Z}(d) =\\sup_{(R,g)\\in\\mathcal P_Z}E_{\\sigma_0}(d;R,g). \\] For every $\\varepsilon_d,\\varepsilon_p>0$, the total contribution of $\\mathcal B$ on the central contours is, for sufficiently large $Z$, \\ll_\\eta{}& Z^{C(\\sigma_0)+E^{\\max}_{\\sigma_0,Z}(d)+(16-6l_y)e}\\\\ &\\cdot Z^{(1+d)\\varepsilon_c+d\\varepsilon_d+\\varepsilon_p} (1+T_1)^{A_\\eta}. The supremum is finite because the pairs lie in a fixed bounded range. The power $\\varepsilon_p$ absorbs the pointwise logarithmic multiplicity. No separate integral for a pointwise piece or row set is asserted. All discarded portions and horizontal joins in this contour move are $O_{\\eta,N}(Z^{B_\\eta}T_1^{-N})$ for every fixed $N$, where $B_\\eta$ is finite and is fixed before $N$.", "argument_gist": "Move only a fixed retained bin and account for its exponent, row count and height factors within a common high representation.", "gist_origin": "source_audit_annotation", "proof_anchor": { "source": "paper.tex", "line_start": 6117, "line_end": 6156, "char_start": 279450, "char_end": 281562 }, "proof_opening": "Apply Lemma~\\ref{lem:fixed-bin-contour} to the original triple integral, using for its three box constants the corresponding $c_k$ in Equation~\\eqref{eq:stage-height-allocation}. These constants are positive and at most $1/2$. The full correction and the fixed bin therefore give the retained integral and the asserted errors before any pointwise pieces are introduced. Any auxiliary integrations used for the central bound are subject to the additional hypotheses in the statement. Now use the central decomposition and form the pointwise sets $\\mathcal B_{\\lambda,\\nu}$. At each retained point apply the triangle inequality to the original row sum, and then apply Equation~\\eqref{eq:stage-central-b", "anchor": { "source": "paper.tex", "line_start": 6021, "line_end": 6116, "char_start": 274813, "char_end": 279448 }, "gist_evidence": "paper.tex lines 6021–6115; paper.tex lines 6021–6155" }, { "id": "lem:principal-residue-interface", "kind": "lemma", "label": "Lemma 10.5", "title": "Extraction of the principal signal", "section": "sec:shared-analytic-estimates", "statement": "Assume Definition~\\ref{def:stage-high-data} and Equation~\\eqref{eq:shared-principal-nonvanishing}. Fix $\\sigma_0\\in[7/8,1)$ with $\\beta_*>\\sigma_0$ and $00$, uniformly in all imaginary parts on \\[ \\Re s=\\beta_*+e,\\qquad 19/20\\le\\Re w\\le1+e, \\qquad 33/200\\le\\Re z\\le1/6+e, \\] one has |\\mathfrak H_{\\eta,1,Z}(s,w,z)| \\ll_{\\eta,e,\\epsilon} Z^{\\ell\\Re z+\\epsilon} (1+|\\Im s|+|\\Im w|+|\\Im z|)^{J_{\\eta,e,\\epsilon}}, with finite $J_{\\eta,e,\\epsilon}$ fixed before any external tail order. Suppose also that $A_\\eta(Z)\\ne0$ for every sufficiently large real $Z$, that $|A_\\eta(Z)|^{-1}\\ll_{\\eta,\\epsilon}Z^\\epsilon$ for every $\\epsilon>0$, and that on $\\Re s=\\beta_*+e$, uniformly in $\\Im s$, \\mathfrak H_{\\eta,1,Z}(s,1,1/6) =H_\\eta(s)Z^{\\ell/6}A_\\eta(Z)(1+\\mathcal R_{\\eta,Z}(s)). Here either $\\mathcal R_{\\eta,Z}$ is identically zero, or there is a number $\\mu>0$, chosen independently of $\\eta$, such that $|\\mathcal R_{\\eta,Z}(s)|\\ll_\\eta Z^{-\\mu}$ on that entire line. Let $\\mathscr P_\\eta(Z)$ be the $u=1$ term of Equation~\\eqref{eq:stage-high-identity}, and define \\[ f_\\eta(Z)=\\frac1{2\\pi i}\\int_{(2)} Z^{C(s)}e^{(s-5/6)^2}\\frac{H_\\eta(s)}{L_F^S(s,\\eta)}\\,ds. \\] Then $c_S>0$, and, for every $\\epsilon>0$, \\frac{\\mathscr P_\\eta(Z)}{c_SA_\\eta(Z)}-f_\\eta(Z) \\ll_{\\eta,e,\\epsilon}{}& Z^{C(\\beta_*)+(1+h)e-l_y/20+\\epsilon} +Z^{C(\\beta_*)+e-h/600+\\epsilon}\\notag\\\\ &+Z^{C(\\beta_*)+e-\\mu+\\epsilon}. The last term is omitted when $\\mathcal R_{\\eta,Z}$ is identically zero. The implied constants and lower thresholds may depend on the target, but the displayed real exponents do not.", "argument_gist": "Extract the target Mellin signal from the principal row and bound its unresidued contour remainders.", "gist_origin": "source_audit_annotation", "proof_anchor": { "source": "paper.tex", "line_start": 6224, "line_end": 6302, "char_start": 284458, "char_end": 288298 }, "proof_opening": "For $u=1$, the scalar factor in the high identity is \\[ \\frac{\\zeta_F^S(6z)\\zeta_F^S(w)}{L_F^S(s,\\eta)}. \\] Isolate this term on the absolute contours. Move to $\\Re s=\\beta_*+e$, $\\Re w=1+e$ and $\\Re z=1/6+e$. The reciprocal stays in $\\Re s>\\beta_*$, and both zeta arguments stay to the right of one. The paths lie in $\\mathcal D_2$, so the correction is holomorphic. The all-height majorant and Lemma~\\ref{lem:external-contour-tails} justify the horizontal limits. Move $w$ to $19/20$, crossing its simple pole at $w=1$. In that residue move $z$ to $33/200=1/6-1/600$, crossing its simple pole at $z=1/6$. The unresidued $w$ integral keeps $\\Re z=1/6+e$. All these paths lie in $\\mathcal D_2$, and $", "anchor": { "source": "paper.tex", "line_start": 6174, "line_end": 6223, "char_start": 282504, "char_end": 284456 }, "gist_evidence": "paper.tex lines 6174–6222; paper.tex lines 6174–6301" }, { "id": "lem:outer-row-tails", "kind": "lemma", "label": "Lemma 10.6", "title": "Outer row norms", "section": "sec:shared-analytic-estimates", "statement": "Assume Definition~\\ref{def:stage-high-data}. Fix $\\sigma_0\\in[7/8,1)$ with $\\beta_*>\\sigma_0$, $00$. For a dyadic number $U\\ge1$, let $\\mathscr R_\\eta(U;Z)$ denote the contribution in Equation~\\eqref{eq:stage-high-identity} of physical rows $U\\le q_u<2U$, omitting $u=1$. Suppose that for every $\\epsilon>0$, for all such rows with $q_u\\le2Z^{d_{\\min}}$, and uniformly in all imaginary parts on $(\\Re s,\\Re w,\\Re z)=(\\beta_*+e,1/2,z_0)$, |\\mathfrak H_{\\eta,u,Z}(s,w,z)| \\ll_{\\eta,e,\\epsilon} Z^{\\ell z_0+\\epsilon}q_u^\\epsilon (1+|\\Im s|+|\\Im w|+|\\Im z|)^{J_{\\eta,e,\\epsilon}}. Then, for every $\\epsilon>0$ and $1\\le U\\le Z^{d_{\\min}}$, |\\mathscr R_\\eta(U;Z)| \\ll_{\\eta,e,\\epsilon} Z^{C(\\beta_*)+h(z_0-1/6)-l_y/2+e+\\epsilon} U^{63/50+\\epsilon}. In particular the sum of these dyads is \\ll_{\\eta,e,\\epsilon} Z^{C(\\beta_*)+h(z_0-1/6)-l_y/2+e+(63/50)d_{\\min}+\\epsilon}. For the other end, suppose that for every fixed $v>2$ and $\\epsilon>0$, for every physical row, uniformly in all imaginary parts on $(\\Re s,\\Re w,\\Re z)=(2,2,v)$, |\\mathfrak H_{\\eta,u,Z}(s,w,z)| \\ll_{\\eta,v,\\epsilon} Z^{\\ell v+\\epsilon}q_u^\\epsilon (1+|\\Im s|+|\\Im w|+|\\Im z|)^{J_{\\eta,v,\\epsilon}}. Write $B_0=l_x/2+1+l_y$. Then, for every dyad $U$, |\\mathscr R_\\eta(U;Z)| \\ll_{\\eta,v,\\epsilon}Z^{B_0+hv+\\epsilon}U^{1+\\epsilon-v}. For every fixed $\\zeta>0$, if $\\epsilonZ^{h+\\zeta}$ is \\ll_{\\eta,v,\\epsilon} Z^{B_0+(h+\\zeta)(1+\\epsilon)-\\zeta v+\\epsilon}. Thus a fixed sufficiently large $v$ makes this last contribution smaller than any prescribed power of $Z$.", "argument_gist": "Handle small and large physical row norms separately using global or absolute contour lines.", "gist_origin": "source_audit_annotation", "proof_anchor": { "source": "paper.tex", "line_start": 6403, "line_end": 6464, "char_start": 292223, "char_end": 295513 }, "proof_opening": "Every row under consideration has nonprincipal numerator by the classification preceding Lemma~\\ref{lem:principal-residue-interface}. The denominator may be principal for a bounded unit row; its reciprocal is nevertheless holomorphic and bounded on $\\Re s=\\beta_*+e$ by the principal specialization of Lemma~\\ref{lem:logarithmic-control}. For all rows, the conductor and deletion radical bounds in Section~\\ref{sec:detector}, together with Lemma~\\ref{lem:deleted-euler-factors}, give an arbitrarily small power of $q_u$ and a fixed height polynomial for that reciprocal. For a small dyad move its finite sum to $(\\Re s,\\Re w,\\Re z)=(\\beta_*+e,1/2,z_0)$. The reciprocal stays global. The numerator is ", "anchor": { "source": "paper.tex", "line_start": 6342, "line_end": 6402, "char_start": 290159, "char_end": 292221 }, "gist_evidence": "paper.tex lines 6342–6401; paper.tex lines 6342–6463" }, { "id": "sec:eleven-twelfths-conclusion", "kind": "section", "label": "Section 11", "title": "The \\texorpdfstring{$11/12$}{11/12} conclusion", "section": "sec:eleven-twelfths-conclusion", "argument_gist": "Choose heights after retained orders, close the first-stage 11/12 bound, and transfer a Hecke half-plane to Dirichlet L-functions.", "anchor": { "source": "paper.tex", "line_start": 6493, "line_end": 6810, "char_start": 296885, "char_end": 311100 }, "gist_evidence": "paper.tex lines 6493–6810 (section scope)", "gist_origin": "source_audit_annotation" }, { "id": "lem:late-height-closure", "kind": "lemma", "label": "Lemma 11.1", "title": "Late choice of height and external order", "section": "sec:eleven-twelfths-conclusion", "statement": "Fix $\\sigma_0\\in(1/2,1)$ with $\\Delta_0=\\beta_*-\\sigma_0>0$ and an affine function $C(s)=s+c$. Suppose that all real parameters and finite structural choices in an application have been fixed independently of $\\eta$. Suppose there are common numbers $m>0$ and $0<\\omega<\\Delta_0$ such that, for every primitive finite-order target, functions $J_\\eta,f_\\eta$ independent of $T_1$ satisfy \\[ |J_\\eta(Z)|\\ll_\\eta Z^{C(\\sigma_0)+\\omega}. \\] Assume that after a finite sum of estimates, for every integer $N\\ge0$, |J_\\eta(Z)-f_\\eta(Z)| \\ll_{\\eta,N} Z^{C(\\beta_*)-m}(1+T_1)^{A_\\eta} +Z^{B_\\eta}T_1^{-N}, where $A_\\eta\\ge0$ and $B_\\eta$ are finite and independent of $N$. Suppose there is a number $\\tau_{0,\\eta}>0$, fixed after the real choices and the finite profile orders for the target but before $N$ and $Z$, such that the estimate is valid for sufficiently large $Z$ whenever $T_1=Z^\\tau$ and $0<\\tau\\le\\min\\{d_{\\min}/100,\\tau_{0,\\eta}\\}$, with a fixed $d_{\\min}>0$. This additional ceiling may encode a height condition in a preceding estimate, such as the one in Lemma~\\ref{lem:detector-dyads}. The lower threshold may depend on $\\eta,N,\\tau$. Increasing $N$ is assumed to change only external test seminorms, not $A_\\eta$ or the already fixed real powers in Equation~\\eqref{eq:late-height-hypothesis}. Then $\\tau$ and $N$ can be chosen after $\\eta$ so that \\[ |J_\\eta(Z)-f_\\eta(Z)|\\ll_\\eta Z^{C(\\beta_*)-m/2}. \\] In particular the common saving $\\sigma=m/2$, together with the displayed low estimate, has the target quantifier required by Proposition~\\ref{lem:continuation-criterion}.", "argument_gist": "Fix all retained orders first; choose a target-dependent height power and external tail order while preserving a common final saving.", "gist_origin": "source_audit_annotation", "proof_anchor": { "source": "paper.tex", "line_start": 6560, "line_end": 6581, "char_start": 299709, "char_end": 300549 }, "proof_opening": "For the fixed target choose \\[ 0<\\tau_\\eta\\le \\min\\left\\{\\frac{d_{\\min}}{100},\\tau_{0,\\eta}, \\frac{m}{4(A_\\eta+1)}\\right\\}, \\qquad T_1=Z^{\\tau_\\eta}. \\] Then $(1+T_1)^{A_\\eta}\\ll_\\eta Z^{m/4}$, so the first term in Equation~\\eqref{eq:late-height-hypothesis} is $O_\\eta(Z^{C(\\beta_*)-3m/4})$. Next choose a fixed integer $N$ such that \\[ B_\\eta-N\\tau_\\eta0$ and $B_\\eta$ is finite. The second term is then bounded by the required power. Increase the lower threshold for $Z$ after these choices. Both $m$ and $\\omega$ were fixed before the target, whereas $\\tau_\\eta,N$ and the threshold may depend on it. Since $J_\\eta$ and $f_\\eta$ do not conta", "anchor": { "source": "paper.tex", "line_start": 6520, "line_end": 6559, "char_start": 297938, "char_end": 299707 }, "gist_evidence": "paper.tex lines 6520–6558; paper.tex lines 6520–6580" }, { "id": "prop:balanced-high", "kind": "proposition", "label": "Proposition 11.2", "title": "Balanced high estimate", "section": "sec:eleven-twelfths-conclusion", "statement": "Under the supposition $\\Delta_1>0$, for every primitive finite-order target $\\eta$ and all sufficiently large $Z$, \\[ |J_{\\mathrm I,\\eta}(Z)-f_{\\mathrm I,\\eta}(Z)| \\ll_\\eta Z^{C_{\\mathrm I}(\\beta_*)-1/4800}. \\] The displayed saving is independent of the target; the implied constant and lower threshold may depend on it.", "argument_gist": "Sum central, principal and outer contributions to approximate the balanced signal with common saving 1/4800.", "gist_origin": "source_audit_annotation", "proof_anchor": { "source": "paper.tex", "line_start": 6594, "line_end": 6699, "char_start": 300964, "char_end": 305993 }, "proof_opening": "Use $l_x=l_y=h=1/2$, $\\ell=0$ and $C=C_{\\mathrm I}$. The exact identity and local correction were verified at the start of Section~\\ref{sec:shared-analytic-estimates}; Equation~\\eqref{eq:balanced-central-contribution} estimates its intermediate rows, and the principal and outer specializations there give the margins listed above. It remains to choose their losses. \\paragraph{Order of choices.} We give a common loss budget. Put \\[ m_0=\\frac1{1200},\\qquad \\epsilon_*=10^{-6},\\qquad m=\\frac1{2400},\\qquad \\omega=\\frac{\\Delta_1}{2}. \\] The error-free central, small, principal, and large savings just proved are all at least $m_0$. Fix the displayed geometry, row ranges, and $v=4000$ before the targ", "anchor": { "source": "paper.tex", "line_start": 6582, "line_end": 6593, "char_start": 300551, "char_end": 300962 }, "gist_evidence": "paper.tex lines 6582–6592; paper.tex lines 6582–6698" }, { "id": "prop:hecke-dirichlet-transfer", "kind": "proposition", "label": "Proposition 11.3", "title": "Quadratic transfer", "section": "sec:eleven-twelfths-conclusion", "statement": "Let $\\sigma_0\\in[1/2,1)$. Suppose every primitive finite-order Hecke $L$-function over $F=\\mathbb Q(\\sqrt{-3})$ is zero-free on $\\Re s>\\sigma_0$, with its principal pole at one allowed. Then every finite-order Hecke $L$-function over $F$ and every Dirichlet $L$-function is zero-free on that same strict half-plane, again allowing the principal pole at one.", "argument_gist": "Transfer a Hecke family half-plane through the quadratic norm Euler-product factorization, checking poles at one separately.", "gist_origin": "source_audit_annotation", "proof_anchor": { "source": "paper.tex", "line_start": 6715, "line_end": 6782, "char_start": 306707, "char_end": 309768 }, "proof_opening": "Passing from a primitive Hecke character to one that it induces changes only finitely many factors $1-\\eta(\\mathfrak p)N\\mathfrak p^{-s}$. They are nonzero for $\\Re s>0$, so the Hecke assertion extends to all finite-order characters. The same observation for factors $1-\\chi(p)p^{-s}$ reduces the Dirichlet assertion to a primitive Dirichlet character $\\chi$ of conductor $q$. Let $\\chi_{-3}$ be the quadratic character of conductor three, and let $\\eta$ be the finite-order Hecke character given by $\\mathfrak a\\mapsto\\chi(N\\mathfrak a)$ on ideals coprime to $3q$. It is a ray character: if $\\alpha\\equiv1\\pmod{q\\mathcal O}$, then $N\\alpha\\equiv1\\pmod q$, so the norm character is trivial on the cor", "anchor": { "source": "paper.tex", "line_start": 6705, "line_end": 6714, "char_start": 306250, "char_end": 306705 }, "gist_evidence": "paper.tex lines 6705–6713; paper.tex lines 6705–6781" }, { "id": "sec:compensation", "kind": "section", "label": "Section 12", "title": "The compensated probe", "section": "sec:compensation", "argument_gist": "Bootstrap from 11/12 and define the compensated probe with asymmetric scales and selected prime factors.", "anchor": { "source": "paper.tex", "line_start": 6810, "line_end": 6918, "char_start": 311100, "char_end": 316222 }, "gist_evidence": "paper.tex lines 6810–6918 (section scope)", "gist_origin": "source_audit_annotation" }, { "id": "sec:additional-arithmetic", "kind": "section", "label": "Section 13", "title": "Coefficient conventions and finite correlations", "section": "sec:additional-arithmetic", "argument_gist": "Specify plain and prime polynomial conventions, fixed-ray prime normalization, and exact finite Fourier correlations.", "anchor": { "source": "paper.tex", "line_start": 6918, "line_end": 7249, "char_start": 316222, "char_end": 331919 }, "gist_evidence": "paper.tex lines 6918–7249 (section scope)", "gist_origin": "source_audit_annotation" }, { "id": "lem:ray-prime-normalizer", "kind": "lemma", "label": "Lemma 13.1", "title": "Fixed-ray prime normalizer", "section": "sec:additional-arithmetic", "statement": "Let $T$ be a fixed quotient of a ray class group of $F$, and write $p\\in1_T$ when the image of an unramified prime ideal is the identity. For a fixed nonnegative, nonzero smooth annular weight $W$, \\sum_{p\\in1_T}W(q_p/P)q_p^{-5/6} \\sim\\frac{P^{1/6}}{|T|\\log P} \\int_0^\\infty W(y)y^{-5/6}\\,dy. Deleting any further fixed finite set of primes does not change this asymptotic. Its lower threshold may depend on all the fixed data.", "argument_gist": "Fixed-ray-class prime counting makes each prime normalizer asymptotic and positive, without needing an error exponent uniform in the conductor.", "gist_origin": "source_audit_annotation", "proof_anchor": { "source": "paper.tex", "line_start": 7019, "line_end": 7041, "char_start": 322027, "char_end": 323025 }, "proof_opening": "For the fixed abelian extension of $F$ corresponding to $T$, the prime ideal theorem in a fixed Frobenius class gives \\[ \\pi_{1_T}(x):=\\#\\{p\\in1_T:q_p\\le x\\} \\sim\\frac{\\operatorname{Li}(x)}{|T|}. \\] This is the fixed-extension consequence of Chebotarev in \\cite[Theorem 1.1]{TZ}; the extension and its conductor are fixed as $x\\to\\infty$. If $\\operatorname{supp}W\\subset[a,b]\\subset(0,\\infty)$, the error $o(x/\\log x)$ is uniform for $aP\\le x\\le bP$ as $P\\to\\infty$. Stieltjes integration by parts therefore changes the left side of Equation~\\eqref{eq:ray-prime-normalizer} by $o(P^{1/6}/\\log P)$ when $d\\pi_{1_T}(x)$ is replaced by $dx/(|T|\\log x)$. The resulting integral is \\[ \\frac{P^{1/6}}{|T|\\l", "anchor": { "source": "paper.tex", "line_start": 7006, "line_end": 7018, "char_start": 321440, "char_end": 322025 }, "gist_evidence": "paper.tex lines 7006–7017; paper.tex lines 7006–7040" }, { "id": "lem:prime-power-fourier", "kind": "lemma", "label": "Lemma 13.2", "title": "Prime-power Fourier sums", "section": "sec:additional-arithmetic", "statement": "For $P=q_p$ and every integer $a\\ge1$, |G(p^a,k)|&=P^{(a-1)/2}1_{v_p(k)=a-1},&&6\\nmid a,\\\\ G(p^a,k)&=P^{a/2}1_{p^a\\mid k} -P^{a/2-1}1_{p^{a-1}\\mid k},&&6\\mid a.", "argument_gist": "Evaluate local prime-power Fourier sums, retaining the nonunit zeros and the Ramanujan-sum branch when the exponent is divisible by six.", "gist_origin": "source_audit_annotation", "proof_anchor": { "source": "paper.tex", "line_start": 7067, "line_end": 7080, "char_start": 323931, "char_end": 324513 }, "proof_opening": "Write a residue modulo $p^a$ as $x_0+py$, with $x_0\\bmod p$ and $y\\bmod p^{a-1}$. The sum over $y$ is zero unless $p^{a-1}\\mid k$, and equals $P^{a-1}$ otherwise. In the latter case write $k=p^{a-1}k_0$. The remaining normalized sum is \\[ P^{a/2-1}\\sum_{x_0\\bmod p}\\chi_p(x_0)^a e(k_0x_0/p). \\] If $6\\nmid a$, its inner sum vanishes when $p\\mid k_0$ and otherwise has absolute value $\\sqrt P$. If $6\\mid a$, the inner sum is the sum of the additive character over the units, namely $P1_{p\\mid k_0}-1$. These are exactly the two cases in the statement.", "anchor": { "source": "paper.tex", "line_start": 7055, "line_end": 7066, "char_start": 323583, "char_end": 323929 }, "gist_evidence": "paper.tex lines 7055–7066 (full statement)" }, { "id": "lem:full-correlation", "kind": "lemma", "label": "Lemma 13.3", "title": "Full correlation and common factors", "section": "sec:additional-arithmetic", "statement": "For primary moduli $u,v$ outside $S$ and $j\\in\\mathcal O$, put F(u,v;j) =\\sum_{\\substack{x\\bmod u,\\ y\\bmod v\\\\ vx-uy\\equiv j\\pmod{uv}}} \\chi_u(x)\\overline{\\chi_v(y)}. Then \\[ F(u,v;j)=\\frac1{\\sqrt{q_uq_v}}\\sum_{h\\bmod uv} G(u,h)\\overline{G(v,h)}e(-jh/(uv)). \\] At zero frequency, $F(u,v;0)=0$ unless $u=v$, and $F(u,u;0)=\\varphi(u)$, where $\\varphi(u)$ is the number of units modulo $u$. Let $C=(u,v)$ and write $u=Cn_1$, $v=Cn_2$, where $(n_1,n_2)=1$. Then $F(u,v;j)=0$ unless $C\\mid j$. For $j=Ck$, F(Cn_1,Cn_2;Ck) &=\\chi_{n_1}(k)\\overline{\\chi_{n_2}(-k)} \\mathcal R(n_1,n_2)L_C(n_1,n_2;k),\\\\ L_C(n_1,n_2;k) &=\\sum_{\\substack{x,y\\bmod C\\\\n_2x-n_1y\\equiv k\\pmod C}} \\chi_C(x)\\overline{\\chi_C(y)} =\\prod_{p^c\\parallel C}L_{p^c}. For $P=q_p$ and $p\\nmid n_1n_2$, the local factor is L_{p^c}=P^{c-1}\\chi_p(n_1/n_2)^c P-1,&p\\mid k,\\\\ -1,&p\\nmid k,\\ 6\\nmid c,\\\\ P-2,&p\\nmid k,\\ 6\\mid c. If $p$ divides exactly one of $n_1,n_2$, the local factor is $P^{c-1}(P-1)1_{6\\mid c}1_{p\\nmid k}$. On the genuine residual locus $(n_1,n_2)=1$, $L_C$ means the congruence sum in Equation~\\eqref{eq:correlation-lift}. When $L_C$ is used on all residual pairs, it instead denotes the artificial product extension of these local formulas, with the local value defined to be zero if $p\\mid(n_1,n_2)$. Outside the genuine residual locus this is not the original congruence sum. The genuine function and this artificial extension both satisfy $|L_C|\\le q_C$, and the extension is periodic modulo $\\operatorname{rad}C$ in each residual column.", "argument_gist": "Compute finite Fourier correlations with full local supports and common factors retained.", "gist_origin": "source_audit_annotation", "proof_anchor": { "source": "paper.tex", "line_start": 7134, "line_end": 7191, "char_start": 326422, "char_end": 329338 }, "proof_opening": "Expanding both Gauss sums in the displayed Fourier transform leaves \\[ \\frac1{q_uq_v}\\sum_{x\\bmod u,y\\bmod v}\\chi_u(x)\\overline{\\chi_v(y)} \\sum_{h\\bmod uv}e\\bigl(h(vx-uy-j)/(uv)\\bigr). \\] Additive orthogonality makes the inner sum $q_uq_v$ when the congruence holds and zero otherwise. If $j=0$ and a summand is nonzero, $x$ and $y$ are units modulo $u$ and $v$. Reducing the congruence modulo $u$ gives $u\\mid v$, and reducing modulo $v$ gives $v\\mid u$. Since the generators are primary, $u=v$. The congruence then says $x=y\\bmod u$ and gives $\\varphi(u)$. This argument uses the zero masks also when a local character power is principal. The divisibility by $C$ is immediate. After division by $C$", "anchor": { "source": "paper.tex", "line_start": 7081, "line_end": 7133, "char_start": 324515, "char_end": 326420 }, "gist_evidence": "paper.tex lines 7081–7132; paper.tex lines 7081–7190" }, { "id": "lem:complete-support-correlation", "kind": "lemma", "label": "Lemma 13.4", "title": "Complete-common-support correlation", "section": "sec:additional-arithmetic", "statement": "Suppose $u=Da$, $v=Eb$ are primary and outside $S$, with $(a,b)=1$ and $(ab,DE)=1$. Then, for every $j\\in\\mathcal O$, F(Da,Eb;j)={}&F(D,E;j)\\mathcal R(a,E) \\overline{\\mathcal R(b,D)}\\mathcal R(a,b)\\\\ &\\hspace{8mm}\\cdot\\chi_a(j)\\overline{\\chi_b(-j)}. The moduli $D,E$ may contain any prime powers, including shared auxiliary prime factors.", "argument_gist": "Keep a common nonprimitive modulus during complete support extraction and identify the induced child character.", "gist_origin": "source_audit_annotation", "proof_anchor": { "source": "paper.tex", "line_start": 7212, "line_end": 7230, "char_start": 330094, "char_end": 330916 }, "proof_opening": "At primes of $a$ and $b$, solving the congruence in Equation~\\eqref{old-eq:2.11} gives respectively $\\chi_a(j)\\overline{\\chi_a(Eb)}$ and $\\overline{\\chi_b(-j)}\\chi_b(Da)$. These statements remain valid when a character of $j$ is zero; only the displayed unit factors are inverted. At primes of $DE$ change variables $x'=bx\\bmod D$, $y'=ay\\bmod E$. The congruence becomes $Ex'-Dy'\\equiv j\\pmod{DE}$ and its character factor changes by $\\overline{\\chi_D(b)}\\chi_E(a)$. The remaining unit factor is \\[ \\frac{\\chi_E(a)}{\\chi_a(E)} \\frac{\\chi_b(D)}{\\chi_D(b)} \\frac{\\chi_b(a)}{\\chi_a(b)} =\\mathcal R(a,E)\\overline{\\mathcal R(b,D)}\\mathcal R(a,b). \\] All denominators here are symbols of units by the hypot", "anchor": { "source": "paper.tex", "line_start": 7196, "line_end": 7211, "char_start": 329540, "char_end": 330092 }, "gist_evidence": "paper.tex lines 7196–7210; paper.tex lines 7196–7229" }, { "id": "sec:marked", "kind": "section", "label": "Section 14", "title": "Marked completion and reflected row energy", "section": "sec:marked", "argument_gist": "Reflect whole-index product-form marks and derive the hybrid quadratic–cubic energy bound for the resulting rows.", "anchor": { "source": "paper.tex", "line_start": 7249, "line_end": 8061, "char_start": 331919, "char_end": 370072 }, "gist_evidence": "paper.tex lines 7249–8061 (section scope)", "gist_origin": "source_audit_annotation" }, { "id": "cor:marked-reflection", "kind": "corollary", "label": "Corollary 14.1", "title": "Completed reflection with whole-index marks", "section": "sec:marked", "statement": "Fix an invocation of Proposition~\\ref{prop:completed-reflection}, with base local prime set \\(\\mathcal P_0\\), powers \\(j_p\\), fixed multiplier \\(\\phi\\), and test \\(V\\). Put \\(M_{\\rm ref}=\\lambda^{12}L^4\\) for this invocation. Let \\(\\mathcal Q\\) be a finite set of distinct good primes disjoint from \\(\\mathcal P_0\\). In the direct completed coefficient sum, insert the factor \\(\\prod_{q\\in\\mathcal Q}1_{q\\mid nb^3}\\); denote this sum by \\(\\mathcal T_{\\mathcal Q}(X;\\Psi,V)\\). It has the same normalization as Equation~\\eqref{eq:marked-completion}, with \\(\\Psi_k,V_*\\) replaced by \\(\\Psi,V\\). The primal squarefree and cube ideals may share primes. There is an exact finite expansion \\mathcal T_{\\mathcal Q}(X;\\Psi,V) =\\sum_{\\mathcal B}\\zeta_{\\mathcal Q} \\sum_{0\\ne\\mu\\in\\lambda^{-4}\\mathcal O} &\\frac{d(\\mu)\\alpha(\\mu)\\vartheta(\\lambda^4\\mu)}{\\sqrt{q_\\mu}} \\prod_{p\\in\\mathcal A_0}B_p(\\lambda^4\\mu)\\\\ &\\cdot\\prod_{q\\in\\mathcal A_{\\mathcal Q}} q_q^{-1/2}\\chi_q(\\lambda^4\\mu)^{-2} V^\\sharp\\!\\left(\\frac{q_\\mu X}{q_c^2}\\right). Here a branch \\(\\mathcal B\\) specifies \\(h_0\\bmod L\\), the base active and inactive sets \\(\\mathcal A_0,\\mathcal I_0\\), and a partition \\(\\mathcal Q=\\mathcal A_{\\mathcal Q}\\sqcup\\mathcal I_{\\mathcal Q}\\). All base primes with \\(j_p\\ne0\\) are active. Put \\[ r=\\prod_{p\\in\\mathcal A_0\\cup\\mathcal A_{\\mathcal Q}}p, \\qquad c=c_F(h_0)r. \\] The base factors \\(B_p\\) and the canonical functions \\(d,\\vartheta\\) are those of Proposition~\\ref{prop:completed-reflection} for this active radical. With every \\(\\sigma_p,\\epsilon_p,\\omega_{p,j}\\) computed using this branch's whole \\(c\\), the scalar is \\zeta_{\\mathcal Q}={}&-\\frac{i}{81}\\bar\\alpha(c)^2 \\widehat\\phi(h_0)\\bar\\kappa_F \\prod_{p\\in\\mathcal I_0}(1-q_p^{-1}) \\prod_{q\\in\\mathcal I_{\\mathcal Q}}q_q^{-1}\\\\ &\\cdot\\prod_{p\\in\\mathcal A_0}\\chi_p(\\sigma_p)^{-2}\\omega_{p,j_p} \\prod_{q\\in\\mathcal A_{\\mathcal Q}} \\chi_q(\\sigma_q)^{-2}(-\\omega_{q,0}). In particular \\(|\\zeta_{\\mathcal Q}|\\le1/81\\). The functions \\(d,\\vartheta\\) and the scalar \\(\\kappa_F\\) have the canonical dependence on \\(h_0\\) and \\(r\\bmod M_{\\rm ref}^2\\) from Part I. Every dual sum is absolutely convergent and has the same full source support as there. If a marked prime instead belongs to the base local set, the corresponding primal term is identically zero and is removed before this formula is used.", "argument_gist": "Apply the completed reflection to product-form marks on the whole index, preserving branch compatibility and row-slot phases.", "gist_origin": "source_audit_annotation", "proof_anchor": { "source": "paper.tex", "line_start": 7417, "line_end": 7442, "char_start": 339995, "char_end": 341506 }, "proof_opening": "For every good prime and every \\(A\\in\\mathcal O\\), the zero convention gives the pointwise identity \\(1_{q\\mid A}=1-\\chi_q(A)^0\\). Expand its product over \\(\\mathcal Q\\), and apply Proposition~\\ref{prop:completed-reflection} to each of the resulting finitely many completed sums. For a fixed \\(h_0\\) and active radical, the Proposition uses the same \\(c_F,c,d,\\vartheta,\\kappa_F\\), other local phases, and kernel whether an absent prime is omitted from the local set or included with exponent zero in its inactive branch. Thus the two inactive contributions at \\(q\\) combine with coefficient \\(1-(1-q_q^{-1})=q_q^{-1}\\). The active contribution comes only from the subtracted zero-mask transform and ", "anchor": { "source": "paper.tex", "line_start": 7361, "line_end": 7416, "char_start": 337290, "char_end": 339993 }, "gist_evidence": "paper.tex lines 7361–7415; paper.tex lines 7361–7441" }, { "id": "lem:hybrid-energy", "kind": "lemma", "label": "Lemma 14.2", "title": "A quadratic--cubic norm bound", "section": "sec:marked", "statement": "Let \\(K,N,B,L\\ge1\\). Let \\(k,n,P\\) be squarefree primary ideals with \\(q_k\\ll K\\), \\(q_n\\asymp N\\), and \\(q_P\\asymp L\\); let \\(b\\) be any primary ideal with \\(q_b\\asymp B\\). The ideals \\(k,P\\) avoid the current excluded set \\(S\\); additional fixed exclusions are allowed. The source ideals \\(n,b\\) may contain primes of \\(S\\) other than \\(\\lambda\\). Let \\(a(P)\\) and \\(\\beta(n,b)\\) satisfy \\[ |a(P)|\\le1,\\qquad |\\beta(n,b)|\\le1. \\] Here \\(a\\) is independent of \\(k,n,b\\), and \\(\\beta\\) is independent of \\(k,P\\). Then &\\sum_k\\left| \\sum_P\\frac{a(P)}{\\sqrt{q_P}}1_{(P,k)=1} \\sum_{n,b}\\beta(n,b)\\chi_k(nb)^3 \\chi_P(n)^{-2}1_{(P,b)=1} \\right|^2\\\\ &\\qquad\\ll_\\epsilon (KNBL)^\\epsilon(K+NB)B\\{N+L+(NL)^{2/3}\\}. Fixed restrictions on the \\(k\\)-set and fixed restrictions on the \\((n,b)\\)- or \\(P\\)-supports are allowed, as are fixed ray sectors. All such restrictions must preserve the two coefficient-independence conditions above. Divisor-bounded multiplicities may be included if they preserve those conditions; otherwise they must first be removed by the divisor Cauchy inequalities in the proof.", "argument_gist": "Combine quadratic and cubic character norms while explicitly resolving the coupled masks.", "gist_origin": "source_audit_annotation", "proof_anchor": { "source": "paper.tex", "line_start": 7542, "line_end": 7656, "char_start": 346460, "char_end": 351561 }, "proof_opening": "Write \\(\\mathcal H\\) for the left side of Equation~\\eqref{old-eq:4.6b}. First remove the complete moving row-mark mask: \\[ 1_{(P,k)=1}=\\sum_{D\\mid(P,k)}\\mu(D). \\] For each row the number of such divisors is at most \\(d_{\\mathcal O}(k)\\). Rowwise divisor Cauchy, followed by the positive sum over rows, therefore costs a permitted factor \\((KNBL)^\\epsilon\\). In a fixed \\(D\\)-summand write \\(P=DP'\\) and \\(k=Dk'\\). Squarefreeness gives the fixed restrictions \\((D,P')=(D,k')=1\\). Define \\[ a_D(P')&=a(DP')1_{(D,P')=1},\\\\ \\beta_D(n,b)&=\\beta(n,b)\\chi_D(nb)^3\\chi_D(n)^{-2}1_{(D,b)=1}. \\] All these factors retain their zeros and \\(|\\beta_D|\\le1\\). Factoring \\(q_P^{-1/2}=q_D^{-1/2}q_{P'}^{-1/2}\\) gives", "anchor": { "source": "paper.tex", "line_start": 7510, "line_end": 7541, "char_start": 345174, "char_end": 346458 }, "gist_evidence": "paper.tex lines 7510–7540; paper.tex lines 7510–7655" }, { "id": "lem:reflected-energy", "kind": "lemma", "label": "Lemma 14.3", "title": "Reflected energy", "section": "sec:marked", "statement": "Retain the \\hyperref[par:reflected-energy-data]{reflected block data} above. Let \\(Z\\ge2\\), and suppose their log-lengths range over fixed bounded sets. In the completed sum of Equation~\\eqref{eq:marked-completion}, the fixed character, puncture, and slot coefficients are independent of \\(k,f\\), the slot supports are disjoint, and the slot coefficients are products as in Equation~\\eqref{old-eq:4.2}. The fixed row restrictions after freezing \\(k_{\\rm pow},k_{\\rm sup}\\) must be independent of the active slots and dual variables, apart from the explicit zero mask \\(1_{(P,k_{\\rm res})=1}\\). Fix a small \\(\\tau_{\\rm ref}>0\\), put \\(X=Z^{N_*}\\). Define \\(T_d\\) and call a dyad retained by the following formula: T_d=2H+2A_0+2z_a-N_*-N_0-3B_0,\\\\ v+3\\ell_b+e_\\lambda\\le T_d+\\tau_{\\rm ref} \\quad\\text{for a retained dyad}. Terms with \\[ v+3\\ell_b+e_\\lambda>T_d+\\tau_{\\rm ref} \\] have arbitrarily small total size, with a fixed polynomial height cost, after a sufficiently far kernel contour shift and a threshold for \\(Z\\) depending only on the fixed support intervals and \\(\\tau_{\\rm ref}\\). Every retained dyad has coefficient exponent, per active tuple, -v/2-\\ell_b-e_\\lambda/3-(S_0+B_0+z_a)/2. Define u=\\min\\{v,z_a,(v+z_a)/3\\}, and E_{\\rm ref}={}&O/2+\\max(H,v+\\ell_b)-S_0-B_0+z_a-u-\\ell_b -2e_\\lambda/3\\\\ &-\\tfrac12(T_d-v-3\\ell_b-e_\\lambda)_+. For every \\(\\epsilon>0\\), the contribution with fixed \\(k_{\\rm pow},k_{\\rm sup}\\) satisfies \\[ \\sum_{k_{\\rm res}\\ {\\rm sf}} |\\mathcal U_{v,\\ell_b,e_\\lambda}(k_{\\rm res})|^2 \\ll Z^{E_{\\rm ref}-O/2+\\epsilon}. \\] Summing the fixed parts in their \\(O\\)-dyad and the local choices changes this to \\(O(Z^{E_{\\rm ref}+\\epsilon})\\). The bounds are uniform in the moving ideals and punctures in the stated ranges, with finitely many smooth seminorms and a fixed polynomial cost for norm-twist heights. The negative value of \\(e_\\lambda\\), when present, is \\(O(1/\\log Z)\\). In particular, the formula is used only on actual residual-row dyads satisfying Equation~\\eqref{eq:actual-residual-row-dyad}.", "argument_gist": "Estimate the marked reflected row energy uniformly over admissible moving supports and frozen labels.", "gist_origin": "source_audit_annotation", "proof_anchor": { "source": "paper.tex", "line_start": 7806, "line_end": 7994, "char_start": 358171, "char_end": 367330 }, "proof_opening": "First remove the inactive slots by triangle inequality in the row Hilbert space. At such a slot the absolute coefficient mass is \\[ \\sum_{p\\in\\mathcal P_i}q_p^{-1}\\ll_\\epsilon Z^\\epsilon, \\] uniformly under any fixed restriction, including for a bounded or zero-length list. It therefore suffices to prove a uniform bound with those primes fixed. Each such branch has its own conductor, cusp sector, and kernel scale, with the inactive prime absent from the conductor. Its earlier collision exclusion is now a fixed row restriction. Fix \\(f,\\rho,k_{\\rm pow},k_{\\rm sup}\\), the non-slot local choices, \\(h_0\\), and separate classes of the residual row product and active slot product modulo the full \\", "anchor": { "source": "paper.tex", "line_start": 7747, "line_end": 7805, "char_start": 355757, "char_end": 358169 }, "gist_evidence": "paper.tex lines 7747–7804; paper.tex lines 7747–7993" }, { "id": "sec:compensated-low", "kind": "section", "label": "Section 15", "title": "The compensated low estimate", "section": "sec:compensated-low", "argument_gist": "Combine the compensated completed-row norm with a quantitative additive Gram estimate to bound the modified probe directly.", "anchor": { "source": "paper.tex", "line_start": 8061, "line_end": 8651, "char_start": 370072, "char_end": 395751 }, "gist_evidence": "paper.tex lines 8061–8651 (section scope)", "gist_origin": "source_audit_annotation" }, { "id": "lem:probe-row-norm", "kind": "lemma", "label": "Lemma 15.1", "title": "The compensated completed-row norm", "section": "sec:compensated-low", "statement": "For every $\\epsilon>0$, every fixed rescaled subset $J$, and every $\\sigma\\in T$, with $Q=q_{b_*}X'Y'\\asymp Z^{M'}$, \\sum_{00$, one has \\sum_{q_m\\ll Q}|A_m|^2 \\ll (1+|\\nu|)^{J_{\\mathrm{Gr}}}\\frac Q{Y'} \\left(1+P_a^{1/6}+\\frac{P_a^2}{Y'}\\right)Z^\\epsilon. Here $J_{\\mathrm{Gr}}$ is a fixed seminorm order, independent of the moving scales and of $\\nu$. The same assertion holds for any fixed finite linear combination of the ray coefficients $1_{s\\in\\sigma}\\chi_s(b_*)/\\xi(s)$. No arbitrary row-dependent arithmetic coefficient is asserted.", "argument_gist": "Obtain the quantitative additive Gram bound used with the asymmetric compensated scales.", "gist_origin": "source_audit_annotation", "proof_anchor": { "source": "paper.tex", "line_start": 8363, "line_end": 8558, "char_start": 383109, "char_end": 392491 }, "proof_opening": "Majorize the row ball by a fixed nonnegative radial Schwartz function. Put $r_s=q_s/Y'$ and, on the primary elements prime to $S$, $c_\\sigma(s)=1_{s\\in\\sigma}\\chi_s(b_*)/(\\tau\\xi(s))$, extended by zero elsewhere before the inverse character is evaluated. The exact annular rewriting of the polynomial is \\[ A_m=Y'^{-3/2}\\sum_s c_\\sigma(s)W_1(r_s)r_s^{-1+i\\nu} g_{\\chi_s}(s,-m). \\] Thus the expanded square has the common normalization $Y'^{-3}$ and the joint annular profile contains the factors $W_1(r_{s_1})r_{s_1}^{-1+i\\nu}W_1(r_{s_2})r_{s_2}^{-1-i\\nu}$. Write $s_1=Cn_1$, $s_2=Cn_2$, with $(n_1,n_2)=1$. Poisson in the row variable has factor $Q/(q_{s_1}q_{s_2})$, whereas the complete transform ", "anchor": { "source": "paper.tex", "line_start": 8340, "line_end": 8362, "char_start": 382185, "char_end": 383107 }, "gist_evidence": "paper.tex lines 8340–8361; paper.tex lines 8340–8557" }, { "id": "prop:probe-low", "kind": "proposition", "label": "Proposition 15.3", "title": "The compensated low estimate", "section": "sec:compensated-low", "statement": "For the probe in Equation~\\eqref{eq:compensated-probe-definition}, the geometry in Equation~\\eqref{old-eq:5.1}, and every $\\epsilon>0$, |I_{\\eta,\\mathrm{modified}}(Z)| \\ll Z^{l_x/2+b/12+\\epsilon}=Z^{3/16+\\epsilon}. The exponent is independent of the target and the slot mesh; the constant and lower threshold may depend on the fixed arithmetic data, slot system, and smooth tests.", "argument_gist": "Combine row energy and additive Gram estimates to bound the compensated physical probe directly.", "gist_origin": "source_audit_annotation", "proof_anchor": { "source": "paper.tex", "line_start": 8578, "line_end": 8650, "char_start": 393184, "char_end": 395749 }, "proof_opening": "For a fixed rescaled subset and tuple, put $Q=q_{b_*}X'Y'$. Its separation is Equation~\\eqref{eq:low-separated} with $A_{m,\\sigma,v}(Y')$ and $B^J_{m,\\sigma}(Z)$. Put $r_J=q_{p_J}/Z^d$; the fixed annular supports give $02$, on \\[ x_r=w_r=2,\\qquad z_r=z_\\infty, \\] one has $G_p=O_{z_\\infty}(1)$ at every selected prime. In either case the positive sum over the full selected tuples satisfies \\sum_{(p_i)\\in\\prod_i\\mathcal P_i(Z)} \\prod_i\\bigl[|W_i(q_{p_i}/P_i)|q_{p_i}^{z_r-1}|G_{p_i}|\\bigr] \\prod_{\\substack{p\\notin S\\\\p\\notin\\{p_1,\\ldots,p_K\\}}}|H_p| \\ll_\\epsilon U^\\epsilon\\prod_iP_i^{z_r}. The constants may depend on the fixed data, $e$, and, in the second case, $z_\\infty$, but not on $Z,u$ or the imaginary parts. These estimates remain valid at zeros of $P_p$ or $H_p$.", "argument_gist": "Retain full selected factors on the small/absolute lines where the central main/error split is unavailable.", "gist_origin": "source_audit_annotation", "proof_anchor": { "source": "paper.tex", "line_start": 9133, "line_end": 9197, "char_start": 418020, "char_end": 420768 }, "proof_opening": "Both sets of lines lie in the first Euler region with $\\epsilon_0=3/8$. Its primewise defect estimates give $H_p=O(1)$, while Equation~\\eqref{eq:unselected-local-product} bounds the positive product of all unselected factors by $O_\\epsilon(U^\\epsilon)$. Consider first $x_r=\\beta_*+e$, $w_r=1/2$, and $z_r=17/50$. For a selected $p\\nmid u$, Equation~\\eqref{old-eq:5.13b} gives \\[ |G_p+\\overline{\\chi_p(u)}H_p| \\ll Q^{-x_r}+Q^{-6z_r} +Q^{4-5x_r-6z_r}+Q^{1-w_r-6z_r}. \\] The four exponents are at most $-7/8-e,-51/25,-483/200-5e,-77/50$, respectively. Consequently $G_p=O(1)$ off $u$, without division by $H_p$. For a selected $p\\mid u$, one has $W=D=0$ and $\\mathcal E_p=P_p^*$. Equation~\\eqref{old-eq", "anchor": { "source": "paper.tex", "line_start": 9101, "line_end": 9132, "char_start": 416744, "char_end": 418018 }, "gist_evidence": "paper.tex lines 9101–9131; paper.tex lines 9101–9196" }, { "id": "sec:inverse", "kind": "section", "label": "Section 17", "title": "The inverse moment with prime factors", "section": "sec:inverse", "argument_gist": "Prove the marked inverse moment through a finite canonical induction, verify its initialization, and amplify its no-slot consequence.", "anchor": { "source": "paper.tex", "line_start": 9209, "line_end": 12477, "char_start": 421375, "char_end": 578490 }, "gist_evidence": "paper.tex lines 9209–12477 (section scope)", "gist_origin": "source_audit_annotation" }, { "id": "lem:marked", "kind": "lemma", "label": "Lemma 17.1", "title": "Marked inverse moment", "section": "sec:inverse", "statement": "Fix bounded ranges for the nonnegative parameters \\(m,r,z_i\\), a bound for \\(|I|\\), and positive constants \\(c_1,c_2\\). Suppose \\[ r+2z\\le m-c_1,\\qquad 2r+8z\\le3m-c_2. \\] Then, for every \\(\\epsilon>0\\), \\sum_{\\substack{u\\in\\mathcal O\\\\00\\). Let \\(f\\) range over squarefree ideals outside \\(S\\) with \\(q_f\\asymp Z^V\\). Let \\(w\\) be a nonnegative function of \\(f\\) alone satisfying \\(w(f)\\le C d_{\\mathcal O}(f)^C\\) for one fixed \\(C\\ge1\\). Neither \\(C\\) nor the choice of \\(w\\) depends on \\(Z\\), the current rows, or any other averaged variable. The implied constant below may depend on \\(C\\), but is uniform over all \\(w\\) satisfying this fixed bound. Let \\(\\nu\\) be a fixed multiplicative finite ray character whose full zero-extended defining modulus has prime support in \\(S\\), and let \\(\\rho(n)=1_{(n,\\mathfrak r_\\rho)=1}\\) be a fixed puncture. Both are independent of \\(k,f\\). Let \\(\\mathfrak d\\) have the product form in Equation~\\eqref{old-eq:4.2}, with disjoint fixed prime lists of total length at most \\(z_0\\) and bounded coefficients independent of \\(k,f\\). No other row-dependent column coefficient or puncture is allowed. There is no additional residual coefficient \\(b(n)\\), even one independent of \\(k,f\\): arbitrary bounded weights are permitted only as the stated product of individual prime-slot coefficients. Put \\(F_0=N+V\\). Suppose q_{\\mathfrak r_\\rho}\\le Z^{F_0-M-z_0-c_*},\\\\ 3M+6z_0+c_*\\le4F_0. For every smooth annular \\(W\\) and every \\(\\epsilon>0\\), define \\mathcal E={}&Z^{-V}\\sum_f w(f) \\sum_{\\substack{k\\in\\mathcal O\\\\00\\), &\\sum_{k\\in\\mathcal O}\\psi(k)1_{(k,R)=1}\\Phi(q_k/K)\\\\ &\\quad=\\frac{K\\gamma(\\psi;m)}{\\sqrt{q_m}} \\sum_{d\\mid\\operatorname{rad}R}\\frac{\\mu(d)\\psi(d)}{q_d} \\sum_{h\\in\\mathcal O}\\bar\\psi(h) \\mathcal F\\Phi\\!\\left(\\frac{Kq_h}{q_dq_m}\\right), where \\(\\gamma(\\psi;m)=q_m^{-1/2}\\sum_{x\\bmod m}\\psi(x)e(x/m)\\). For a nonprincipal primitive character, \\(|\\gamma(\\psi;m)|=1\\) and the \\(h=0\\) term is zero. For the primitive principal character the conventions are \\(m=1,\\ \\gamma(1;1)=1,\\ \\psi(h)=1\\), including at \\(h=0\\).", "argument_gist": "Give exact finite Poisson summation for a primitive character with a fixed mask, including principal restoration.", "gist_origin": "source_audit_annotation", "proof_anchor": { "source": "paper.tex", "line_start": 9745, "line_end": 9782, "char_start": 447964, "char_end": 449750 }, "proof_opening": "Expand the mask as \\(\\sum_{d\\mid\\operatorname{rad}R,\\ d\\mid k}\\mu(d)\\) and write \\(k=dk'\\). The scale becomes \\(K/q_d\\) and the character contributes \\(\\psi(d)\\). Poisson summation in residue classes modulo \\(m\\) has prefactor \\(K/(q_dq_m)\\). Its finite transform is \\[ \\sum_{x\\bmod m}\\psi(x)e(hx/m) =\\sqrt{q_m}\\gamma(\\psi;m)\\bar\\psi(h). \\] For unit \\(h\\) this follows by changing variables. For nonunit \\(h\\) the transform is zero by primitivity; for \\(m=1\\) the stated principal convention applies. This proves Equation~\\eqref{eq:masked-poisson}. For a nonprincipal primitive character, finite Parseval gives \\(\\sum_h|\\sum_x\\psi(x)e(hx/m)|^2=q_m\\sum_x|\\psi(x)|^2\\). Writing \\(\\phi(m)=|(\\mathcal O/m", "anchor": { "source": "paper.tex", "line_start": 9722, "line_end": 9744, "char_start": 446956, "char_end": 447962 }, "gist_evidence": "paper.tex lines 9722–9743; paper.tex lines 9722–9781" }, { "id": "lem:inverse-amplification", "kind": "lemma", "label": "Lemma 17.6", "title": "Sixth-power amplification", "section": "sec:inverse", "statement": "Let \\(\\nu,W,\\varepsilon_\\chi\\) and the zero extensions be as in Lemma~\\ref{lem:marked}, with no prime slots. Let \\(U,D\\ge1\\), and sum over elements \\(u\\) with \\(q_u\\asymp U\\) such that every prime valuation of the ideal \\((u)\\) is at most five. For every fixed \\(c>0\\) and \\(\\epsilon>0\\), put \\[ H=\\max(2U,D^{1+c}),\\qquad P=(H/U)^{1/6}. \\] Then \\sum_u|M_u(D;W)|^2 \\ll \\frac HP(UD)^\\epsilon \\ll \\max\\{U,U^{1/6}D^{5(1+c)/6}\\}(UD)^\\epsilon. The implied constant depends only on \\(c,\\epsilon\\), the fixed arithmetic data, and finitely many seminorms of \\(W\\). In particular, for \\(D=U^r\\) with \\(r\\) in a fixed bounded nonnegative range, for every \\(\\epsilon>0\\) the exponent may be written \\[ \\sum_u|M_u(U^r;W)|^2\\ll U^{e(r)+\\epsilon}, \\qquad e(r)=\\max\\{1,(1+5r)/6\\}. \\] Both assertions permit a separate test \\(W_{\\sigma_u,t_u}(y)=W(y)y^{-\\sigma_u+it_u}\\) in each row, with \\(\\sigma_u\\) in a fixed compact interval and \\(|t_u|\\le T_1\\), at a cost \\((1+T_1)^A\\) for a fixed \\(A\\).", "argument_gist": "Amplify an unmarked inverse polynomial on sixth-power-free rows to longer column lengths.", "gist_origin": "source_audit_annotation", "proof_anchor": { "source": "paper.tex", "line_start": 12391, "line_end": 12470, "char_start": 574653, "char_end": 578206 }, "proof_opening": "First, Equation~\\eqref{eq:raw-moment} and Lemma~\\ref{lem:smooth-calculus} imply the rowwise scale bound \\sum_{\\substack{v\\in\\mathcal O\\\\00\\), there is a slot mesh \\(\\eta>0\\) such that, if \\(z_i\\le\\eta\\) for every \\(i\\), then \\sum_{k\\in\\mathcal R_z} \\left|S_{\\psi_k}(n_1;W_1)S_{\\psi_k}(n_2;W_2)Q\\right|^2 \\ll Z^{M+\\epsilon} in either of the following cases: \\item \\(z=0\\). There is no restriction on the bounded nonnegative lengths \\(n_1,n_2\\), and no lower bound on the conductor. \\item \\(z>0\\) and n_1+n_2+6\\kappa z=A+(6\\kappa-1)z\\le M. If \\(\\kappa<1\\), assume in addition that \\(\\beta_*\\le(1+\\kappa)/2\\). If \\(\\kappa=1\\), no zero-free hypothesis is required. For an empty slot list, \\(Q=1\\). If \\(z=0\\) and the list contains zero-length slots, their scales are bounded and absolute counting absorbs them into the constant; the zero-slot assertion therefore has the same strength. All real length parameters range over prescribed bounded sets. The mesh depends only on those sets and \\(\\epsilon\\), uniformly for \\(\\kappa\\in[3/4,1]\\). For each fixed \\(Z\\)-independent arithmetic datum \\(\\mathcal A\\) and fixed slot count, the bound uses finitely many smooth seminorms and a fixed polynomial in the separated norm-twist heights. Their orders, and the bound itself, are uniform over all moving moduli, admissible masks, and frozen outer labels in the stated ranges.", "argument_gist": "Control a fourth moment of two plain witnesses and short prime slots, with a dynamic zero-free parameter and exclusion of a finite inducing-character family.", "gist_origin": "source_audit_annotation", "anchor": { "source": "paper.tex", "line_start": 12531, "line_end": 12580, "char_start": 581433, "char_end": 583670 }, "gist_evidence": "paper.tex lines 12531–12579; paper.tex lines 12531–12600" }, { "id": "lem:centered-coefficient-invariant", "kind": "lemma", "label": "Lemma 18.2", "title": "Common coefficient under complete extraction", "section": "sec:plain", "statement": "For a centered input, the two transforms and the intervening Gauss-row enlargement just constructed preserve the following coefficient data. Each Gauss polynomial and each amplifier error retains Equation~\\eqref{eq:centered-coefficient-form}, multiplied by \\(G(u,h)\\), apart from bounded frozen scalars and row scalars of modulus at most one. For an uncentered input, omit the second rectangle throughout. After the second transform and the full M\\\"obius factor allocation, each separated child side has that coefficient form without \\(G\\) and without \\(s\\mid u\\). Every surviving slot has the child's one whole-product character and its original coefficient \\(\\nu_i\\), with no Gauss coefficient. Within a side the plain variables have the same character, puncture mask, and norm power in both rectangles. The two sides of one squared Gauss norm may have different norm powers; their inducing characters differ by a member of \\(\\Theta\\). If the new inducing character belongs to \\(\\Theta\\), fixing the live slot labels leaves the plain coefficient \\[ \\vartheta(l_1)\\vartheta(l_2) 1_{(l_1l_2,\\mathfrak R_*)=1}q_{l_1l_2}^{it}D_{\\mathbf b}(l_1,l_2) \\] for one \\(\\vartheta\\in\\Theta\\), one squarefree mask \\(\\mathfrak R_*\\) of polynomial norm, and one real \\(t\\). The mask may depend on the frozen row, but is common to both variables and both rectangles. All coefficients are understood with their retained frozen scalars, including zeros, and the exact normalizations recorded below.", "argument_gist": "Preserve one common centered coefficient under complete divisor/support extraction.", "gist_origin": "source_audit_annotation", "proof_anchor": { "source": "paper.tex", "line_start": 13983, "line_end": 13997, "char_start": 648533, "char_end": 649404 }, "proof_opening": "The first-transform calculation proved the Gauss coefficient form; the local \\(p\\)-power calculation proved it for each amplifier error. The genuine complete-support identity and the full M\\\"obius allocation above proved the child form, preserving one character and norm power on each whole residual product. Thus ``common'' concerns one separated side; the displayed character ratio is the relation between the two sides. When the inducing character belongs to \\(\\Theta\\), it equals some \\(\\vartheta\\in\\Theta\\) on units. Its redundant natural zeros and the fixed punctures combine into one \\(\\mathfrak R_*\\) of polynomial norm. For each fixed \\(\\Pi\\), multiplicativity factors its value, mask, and n", "anchor": { "source": "paper.tex", "line_start": 13953, "line_end": 13982, "char_start": 646930, "char_end": 648531 }, "gist_evidence": "paper.tex lines 13953–13981; paper.tex lines 13953–13996" }, { "id": "lem:centered-lattice-cancellation", "kind": "lemma", "label": "Lemma 18.3", "title": "Masked rectangle cancellation", "section": "sec:plain", "statement": "Let \\(\\vartheta\\) range over a fixed finite set of finite-order ray characters with conductor primes in \\(\\mathcal S\\). Let \\(\\mathfrak R_*\\) be squarefree with \\(q_{\\mathfrak R_*}\\le Z^{B_*}\\) for a fixed \\(B_*\\), and let \\(W_1,W_2\\) be smooth profiles on fixed annuli. For \\(X>0\\) and \\(t\\in\\mathbb R\\), define \\[ \\mathcal L_i(X,t):= \\sum_l\\vartheta(l)1_{(l,\\mathfrak R_*)=1} q_l^{it}W_i(q_l/X). \\] Then \\mathcal L_i(X,t) &=c_{\\vartheta,\\mathfrak R_*}X^{1+it}I_i(t) +O\\left(Z^{\\epsilon_1}(1+|t|)^J\\right), \\\\ |\\mathcal L_i(X,t)|&\\ll X. Here \\(J\\) is fixed, the constants use finitely many seminorms, \\(c_{\\vartheta,\\mathfrak R_*}\\) is independent of \\(X,t\\), and \\(I_i(t)\\) is a fixed measure constant times \\(\\int_0^\\infty W_i(y)y^{it}\\,dy\\). Both estimates are uniform for \\(X>0\\), for the stated masks, and for the finite character set. Let \\(D_{\\mathbf b}\\) be as in Equation~\\eqref{old-eq:2.18a}, and put \\[ U_i=X_i/q_{\\mathfrak b_i},\\qquad V_i=Y_i/q_{\\mathfrak b_i},\\qquad T=U_1U_2=V_1V_2. \\] If \\(U_i,V_i\\ge Z^r\\) for \\(i=1,2\\) and some \\(r\\ge0\\), then T^{-1/2}\\left| \\sum_{l_1,l_2} \\vartheta(l_1)\\vartheta(l_2) 1_{(l_1l_2,\\mathfrak R_*)=1} q_{l_1l_2}^{it}D_{\\mathbf b}(l_1,l_2) \\right| \\ll Z^{\\epsilon_1}(1+|t|)^{2J}T^{1/2}Z^{-r}. Without a nonnegative lower length, the same expression is \\(O(T^{1/2})\\).", "argument_gist": "Cancel the equal-product-scale centered rectangles on the exceptional inducing-character rows.", "gist_origin": "source_audit_annotation", "proof_anchor": { "source": "paper.tex", "line_start": 14598, "line_end": 14681, "char_start": 675782, "char_end": 679074 }, "proof_opening": "First omit the mask. The primary generators representing ideals outside \\(\\mathcal S\\), with the fixed character weight \\(\\vartheta\\), are a finite weighted collection of residue classes in a fixed lattice. Apply Poisson on that lattice to \\[ f_{X,t}(z)=q_z^{it}W_i(q_z/X) =X^{it}f_t(z/\\sqrt X),\\qquad f_t(z)=q_z^{it}W_i(q_z). \\] The Fourier transform is \\(X^{1+it}\\widehat f_t(\\sqrt X\\,\\xi)\\). Because \\(f_t\\) is supported on a fixed annulus, integration by parts for any fixed \\(J_0>2\\) gives \\[ |\\widehat f_t(\\xi)| \\ll_{J_0} (1+|t|)^{J_0}(1+|\\xi|)^{-J_0}, \\] using finitely many seminorms of \\(W_i\\). For \\(X\\ge1\\), the sum over nonzero points of the fixed dual lattice is therefore \\[ \\ll (1+|t|)", "anchor": { "source": "paper.tex", "line_start": 14545, "line_end": 14597, "char_start": 674183, "char_end": 675780 }, "gist_evidence": "paper.tex lines 14545–14596; paper.tex lines 14545–14680" }, { "id": "sec:refined-row-counts", "kind": "section", "label": "Section 19", "title": "Prime amplitudes and refined row counts", "section": "sec:refined-row-counts", "argument_gist": "Control prime amplitudes and combine the simultaneous witnesses with inverse and plain slot capacities to refine the row counts.", "anchor": { "source": "paper.tex", "line_start": 14985, "line_end": 15495, "char_start": 694835, "char_end": 716739 }, "gist_evidence": "paper.tex lines 14985–15495 (section scope)", "gist_origin": "source_audit_annotation" }, { "id": "lem:bin-prime-bound", "kind": "lemma", "label": "Lemma 19.1", "title": "Prime bound in a bin", "section": "sec:refined-row-counts", "statement": "Under the hypotheses of Lemma~\\ref{lem:detector-dyads}, with the prime-annulus Mellin frequency included in its cumulative allowance, for every \\(\\epsilon_1>0\\) \\[ |Q_i(u;z)|\\ll_{\\mathcal A,e,\\epsilon_1} U^{\\epsilon_1}P_i^{a-1/2+O(e)}. \\] The implied \\(O(e)\\) is uniform for \\(51/100\\le a\\le1\\) and the fixed real ranges. It holds for every main slot in the row.", "argument_gist": "Control each prime amplitude inside a buffered row bin with finite height/profile costs.", "gist_origin": "source_audit_annotation", "proof_anchor": { "source": "paper.tex", "line_start": 15027, "line_end": 15063, "char_start": 696714, "char_end": 698576 }, "proof_opening": "Expand \\(1_{p\\in1_T}=|T|^{-1}\\sum_{\\theta\\in\\widehat T}\\theta(p)\\). Every resulting prime character belongs to \\(\\mathcal X_u\\). Lemma~\\ref{lem:logarithmic-control}, applied to the buffered disks, bounds its logarithmic derivative on \\(\\Re s=a+8e\\) by \\(O_{\\mathcal A,e}(\\log U)\\); the logarithmic derivative of the deleted product has the same bound. Mellin inversion of the corresponding smooth von Mangoldt annulus, shifted only in the retained central range, gives \\(U^{\\epsilon_1}P_i^{a-1/2+O(e)}\\) after normalization. The pure twist \\((q_p/P_i)^{i\\Im z}\\) is translated into the logarithmic-derivative argument before the added Mellin frequency is truncated. The joins have bounded real length", "anchor": { "source": "paper.tex", "line_start": 15015, "line_end": 15026, "char_start": 696272, "char_end": 696712 }, "gist_evidence": "paper.tex lines 15015–15025; paper.tex lines 15015–15062" }, { "id": "prop:detector-counts", "kind": "proposition", "label": "Proposition 19.2", "title": "Row counts from the witnesses", "section": "sec:refined-row-counts", "statement": "Let \\(\\mathcal B\\) be a fixed dynamic and amplitude bin of retained sixth-power-free physical rows \\(q_u\\asymp U=Z^d\\), with \\(a>51/100\\), \\(0<\\delta=2a-1\\le\\alpha\\), and mean amplitude \\(q\\in[0,\\delta/2]\\). Put \\(x=q/\\delta\\), so \\(0\\le x\\le1/2\\), and define \\[ D_x=3-\\frac{17x}{9},\\qquad P_x=\\left(2-\\frac{8x}{9}\\right)(1-x). \\] Assume the total available prime length exceeds \\(7/37\\) by a fixed positive amount. For every \\(t\\in[1,3/2]\\) and every \\(\\epsilon>0\\), the capacity decrements and slot mesh can be chosen using only \\(\\epsilon,\\Delta\\) and the bounded real ranges so that \\[ \\#\\mathcal B\\ll_{\\mathcal A,\\epsilon} U^{\\max\\{R_{\\rm short}(t),L(t)\\}+\\Delta/4+\\epsilon} (1+T_1)^{A_{\\mathcal A}}, \\] where \\(A_{\\mathcal A}<\\infty\\) is uniform over moving rows and \\[ R_{\\rm short}(t)=1-\\delta+\\frac{\\delta P_x}{D_x}(3/2-t), \\qquad L(t)=1-\\delta+(\\alpha-\\delta)(t-1). \\] The estimate is valid for the rowwise witnesses of Proposition~\\ref{prop:detector-witness}, with its height condition. If no prime slots are selected, then \\(t=1\\) instead gives \\[ \\#\\mathcal B\\ll_{\\mathcal A,\\epsilon} U^{1-2\\delta/3+\\epsilon}(1+T_1)^{A_{\\mathcal A}}, \\] without a prime-supply hypothesis. At zero inverse capacity use Lemma~\\ref{lem:inverse-amplification}; at zero plain capacity use the zero-slot case of Lemma~\\ref{lem:plain}.", "argument_gist": "Combine saturated witnesses with inverse/plain capacities and available short prime slots to get refined row counts.", "gist_origin": "source_audit_annotation", "proof_anchor": { "source": "paper.tex", "line_start": 15222, "line_end": 15447, "char_start": 705746, "char_end": 714391 }, "proof_opening": "Subdivide by the witness presentation and dyadic pair. The number of choices is \\(O_{\\mathcal A}((\\log U)^2)\\); the remaining rowwise smooth parameters are handled by the preceding Sobolev argument. Work in one subdivision, denoting it again by \\(\\mathcal B\\), and let \\(r,m\\) be the actual lengths from Proposition~\\ref{prop:detector-witness}. Whenever the selected slots satisfy the corresponding moment hypotheses and have total requested capacity \\(z\\), their spike and the individual witness spikes imply, after reducing preliminary losses, \\[ {ll} \\#\\mathcal B \\ll U^{1-\\delta r-2qz+\\epsilon}(1+T_1)^{A_{\\mathcal A}}, &\\text{from Lemma~\\ref{lem:marked}},\\\\[2mm] \\#\\mathcal B \\ll U^{1-2\\delta m-", "anchor": { "source": "paper.tex", "line_start": 15185, "line_end": 15221, "char_start": 704309, "char_end": 705744 }, "gist_evidence": "paper.tex lines 15185–15220; paper.tex lines 15185–15446" }, { "id": "remark:line:15448", "kind": "remark", "label": "Remark 19.3", "title": "Unselected inverse witnesses beyond the Part II bin ceiling", "section": "sec:refined-row-counts", "statement": "The derivation of Equation~\\eqref{eq:no-slot-inverse-count} uses only the inverse spike from Proposition~\\ref{prop:detector-witness} and Lemma~\\ref{lem:inverse-amplification}. It can therefore be repeated for any instance of the shared detector satisfying those hypotheses, independently of the current Part~II bin ceiling. Consequently, for each fixed \\(t\\in[1,3/2]\\), the bound in Equation~\\eqref{eq:no-slot-inverse-count} holds on each fixed presentation/dyadic subdivision of a fixed dynamic bin of retained rows with actual witnesses from that proposition whenever \\[ a>51/100,\\qquad 1/50<\\delta=2a-1\\le1, \\] and \\(r\\) lies in the prescribed bounded nonnegative range. It uses the common inverse profile \\(W_{\\rm base}\\), rowwise parameter bounds, and witness loss and height hypotheses used in the preceding proof, but requires neither \\(\\delta\\le\\alpha\\) nor a prime-supply hypothesis. This conclusion does not include all rows in the floor bin \\(a=51/100\\), which need not have an actual witness.", "argument_gist": "Explain that the unselected inverse-witness count can be reused beyond the Part II bin ceiling when the same detector and amplification hypotheses hold.", "gist_origin": "source_audit_annotation", "anchor": { "source": "paper.tex", "line_start": 15448, "line_end": 15468, "char_start": 714393, "char_end": 715491 }, "gist_evidence": "paper.tex lines 15448–15468 (full statement)" }, { "id": "sec:conclusion", "kind": "section", "label": "Section 20", "title": "The seven-eighths bound", "section": "sec:conclusion", "argument_gist": "Balance the refined counts, certify the endpoint exponent, choose all losses in the required order, and close the claimed 7/8 bound.", "anchor": { "source": "paper.tex", "line_start": 15495, "line_end": 16678, "char_start": 716739, "char_end": 766316 }, "gist_evidence": "paper.tex lines 15495–16678 (section scope)", "gist_origin": "source_audit_annotation" }, { "id": "lem:high-bin", "kind": "lemma", "label": "Lemma 20.1", "title": "A high exponent for one row bin", "section": "sec:conclusion", "statement": "Let \\(02\\), \\(0<\\tau\\le d_{\\min}/100\\), and the cumulative allocation in Equation~\\eqref{eq:stage-height-allocation}. The bounded real ranges, the positive losses \\(e,\\vartheta,\\epsilon\\), and the slot system are fixed before the target. For each retained tuple of external parameters and each subset of error slots, partition \\(\\mathcal B\\) pointwise into the amplitude sets of Section~\\ref{sec:amplitudes}, and into the witness subdivisions of Section~\\ref{sec:selection} when a witness count is used. Suppose each such set \\(\\mathcal C\\) has main-slot mean \\(q\\in[0,\\delta/2]\\), as in Equation~\\eqref{old-eq:6.6}, and \\[ \\#\\mathcal C\\ll_{\\mathcal A,\\epsilon} U^{R+\\epsilon}(1+T_1)^{A_{\\mathcal A}}, \\] uniformly in the retained tuple and moving labels. The occurring \\(R,q\\) range over a fixed bounded set and may depend on the pointwise set, \\(d,a\\), and \\(\\beta_*\\), but not otherwise on the target. For each occurring pair define E(d) &=a-\\frac78+h\\left(\\frac{17}{50}-\\frac16\\right) -a l_y-(1-a)\\ell-(\\delta/2-q)\\ell +d\\left(R+\\frac\\delta2-\\frac{17}{50}\\right)\\\\ &=C_0+\\frac23\\delta+\\frac q6-h(1-R) +(d-h)\\left(R+\\frac\\delta2-\\frac{17}{50}\\right), \\qquad C_0=-\\frac1{48}. If \\(\\mathcal B\\ne\\varnothing\\), let \\(E_{\\max,Z}(d)\\) be the supremum of these values over all pointwise sets at all retained tuples. On the central contours \\[ \\Re s=a+16e,\\qquad \\Re w=1-a-6e,\\qquad \\Re z=\\frac{17}{50}, \\] the contribution of the original fixed dynamic-bin sum is bounded by \\[ Z^{C(7/8)+E_{\\max,Z}(d)+O(e+\\vartheta+\\epsilon)} (1+T_1)^{A'_{\\mathcal A}}, \\] where \\(\\vartheta\\) is the amplitude-bin width and \\(A'_{\\mathcal A}<\\infty\\) is uniform in moving labels. An empty bin contributes zero. The constant in the \\(O(e+\\vartheta+\\epsilon)\\) term depends only on the bounded real ranges and the fixed slot system, not on the target. No separate integral of an individual pointwise set is asserted. After fixing \\(\\tau>0\\), all discarded external pieces and joins are \\(O_{\\mathcal A,N}(Z^B T_1^{-N})\\), for a fixed \\(B\\) and every fixed \\(N\\).", "argument_gist": "Translate a refined row bin count and local correction decomposition into the compensated high exponent.", "gist_origin": "source_audit_annotation", "proof_anchor": { "source": "paper.tex", "line_start": 15759, "line_end": 15842, "char_start": 727499, "char_end": 731547 }, "proof_opening": "We verify the central hypothesis of Lemma~\\ref{lem:bin-contour-accounting}, which supplies the contour move and its tails for the full correction just specified. Its bin ceiling holds by Equation~\\eqref{eq:part-II-bin-ceiling}. The physical \\(\\Im z\\) coordinate is included in its height allocation because it enters the prime profiles. The other witness, dyadic, and prime-annulus coordinates are the fixed finite list used in the estimates of Section~\\ref{sec:selection}. Their complete pointwise bounds include the discarded auxiliary integrals on global or absolute lines. Thus the single cumulative allocation in Equation~\\eqref{eq:stage-height-allocation} applies, without renewing an allowance", "anchor": { "source": "paper.tex", "line_start": 15699, "line_end": 15758, "char_start": 724843, "char_end": 727497 }, "gist_evidence": "paper.tex lines 15699–15757; paper.tex lines 15699–15841" }, { "id": "lem:balanced-endpoint", "kind": "lemma", "label": "Lemma 20.2", "title": "Compensated endpoint certificate", "section": "sec:conclusion", "statement": "For \\(0\\le\\delta\\le5/6\\) and \\(0\\le x\\le1/2\\), define \\(\\mathcal J,t,R_*\\) by Equation~\\eqref{eq:target-cutoff} where \\(\\mathcal J>0\\), and let \\(E_*\\) be \\(E(h)\\) with \\(q=x\\delta\\) and \\(R=R_*\\). Then \\[ -E_*\\ge\\frac{49}{440640}>\\frac1{10000}. \\] In particular the bound is uniform on the range \\(1/50<\\delta\\le5/6\\), including the closed endpoint.", "argument_gist": "Certify a strictly positive numerical margin at the balanced endpoint by an explicit completed-square polynomial identity.", "gist_origin": "source_audit_annotation", "proof_anchor": { "source": "paper.tex", "line_start": 16006, "line_end": 16073, "char_start": 737810, "char_end": 739519 }, "proof_opening": "Set \\(y=1/2-x\\in[0,1/2]\\) and \\(v=51+41y\\). To verify the calculation, write \\[ p_y=7+18y+8y^2,\\qquad j_y=185+170y+(-138+12y+96y^2)\\delta. \\] Then \\[ D_x=\\frac{37+34y}{18},\\qquad P_x=\\frac{p_y}{9},\\qquad \\mathcal J=\\frac{j_y}{108}. \\] From \\(R_*=1-\\delta+(\\alpha-\\delta)\\delta P_x/(2\\mathcal J)\\) and Equation~\\eqref{eq:common-high-exponent}, \\[ -E_*= \\frac{1+(3+8y)\\delta}{48} -\\frac{13}{32}\\, \\frac{(5/6-\\delta)\\delta P_x}{\\mathcal J}. \\] Multiplying out gives the explicit quadratic \\[ 10368\\mathcal J(-E_*) &=2j_y[1+(3+8y)\\delta] -468(5/6-\\delta)\\delta p_y\\\\ &=10(37+34y) -8(237+377y+26y^2)\\delta\\\\ &\\hspace{12mm} +48(51+131y+94y^2+32y^3)\\delta^2. \\] Multiplication by \\(v\\) and completion of the", "anchor": { "source": "paper.tex", "line_start": 15994, "line_end": 16005, "char_start": 737367, "char_end": 737808 }, "gist_evidence": "paper.tex lines 15994–16004; paper.tex lines 15994–16072" }, { "id": "prop:parameter-order", "kind": "proposition", "label": "Proposition 20.3", "title": "Order of choices", "section": "sec:conclusion", "statement": "Under the contradiction in Equation~\\eqref{eq:part-II-bootstrap}, there are numbers \\[ 0<\\omega<\\Delta,\\qquad \\sigma>0 \\] and a fixed geometry, moment losses, capacity decrements, slot system, amplitude width, bin width \\(e>0\\), and extension \\(0<\\zeta<1/48\\), all chosen before the target character, with the following property. For every primitive finite-order target \\(\\eta\\), one can then choose its fixed arithmetic data, a positive height exponent \\(\\tau_\\eta\\), a finite external tail order \\(N_\\eta\\), and a lower threshold \\(Z_{0,\\eta}\\) such that, for \\(Z\\ge Z_{0,\\eta}\\), \\[ |J_{\\mathrm{II},\\eta}(Z)|\\ll_\\eta Z^{C(7/8)+\\omega},\\qquad |J_{\\mathrm{II},\\eta}(Z)-f_{\\mathrm{II},\\eta}(Z)| \\ll_\\eta Z^{C(\\beta_*)-\\sigma}. \\] The exponents \\(\\omega,\\sigma\\) do not depend on \\(\\eta\\). The cutoff \\(T_1=Z^{\\tau_\\eta}\\) occurs only in the estimates, not in either function.", "argument_gist": "Choose geometry, slots, losses and margins before the target; retained orders before height/tail orders; conclude a family-common positive high saving.", "gist_origin": "source_audit_annotation", "proof_anchor": { "source": "paper.tex", "line_start": 16219, "line_end": 16454, "char_start": 745562, "char_end": 757121 }, "proof_opening": "The ideal margins are \\[ m_{\\rm lo}=\\Delta,\\qquad m_{\\rm hi}=\\frac{51}{64}\\Delta,\\qquad m_{\\rm high}:=\\min\\{m_{\\rm hi},m_{\\rm small}\\}>0. \\] The principal contour margins are \\(m_w,m_z\\) in Equation~\\eqref{old-eq:5.14}. Reserve one half of each of \\(m_{\\rm lo},m_{\\rm high},m_w,m_z\\). On the compact real parameter ranges, the coefficients multiplying all requested detector, moment, capacity, and rounding losses are bounded independently of the target. Indeed, \\(\\delta\\ge1/50\\), \\(D_x\\ge37/18\\), and \\(\\mathcal J\\ge35/54\\); the dyadic lengths are bounded, and Proposition~\\ref{prop:detector-counts} gives fixed inverse width and supply margins. The equality \\(\\delta=\\alpha\\) uses the separate no-", "anchor": { "source": "paper.tex", "line_start": 16196, "line_end": 16218, "char_start": 744597, "char_end": 745560 }, "gist_evidence": "paper.tex lines 16196–16217; paper.tex lines 16196–16453" }, { "id": "manuscript", "kind": "paper", "label": "The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane Re(s)>7/8", "argument_gist": "The paper tries to contradict the existence of a rightmost possible zero in an entire Hecke L-function family. It constructs a completed cubic-theta sum with two exact representations: reflection gives a small direct bound, while Poisson summation exposes a principal Mellin signal containing the reciprocal of the target L-function. Bounds for the other rows would continue that reciprocal across the possible zeros. A balanced first stage claims a zero-free half-plane to the right of 11/12; prime compensation and two additional moment inductions claim 7/8. The full Hecke family is needed because Poisson introduces twists, even when the final target is zeta. These are the manuscript’s proposed steps, not a completed verification of its difficult inputs.", "anchor": { "source": "paper.tex", "line_start": 1, "line_end": 16678, "char_start": 0, "char_end": 766316 }, "gist_origin": "source_audit_annotation" }, { "id": "old-eq:1.1b", "kind": "anchor", "label": "old-eq:1.1b", "owner": "sec:analytic-reduction", "section": "sec:analytic-reduction", "anchor": { "source": "paper.tex", "line_start": 384, "line_end": 385, "char_start": 19906, "char_end": 19925 }, "argument_gist": "", "excerpt": "ta)=0\\\\ \\text{for some primitive finite-order Hecke character }\\eta \\right\\}\\right). Poles are not included. Absolute convergence of the Euler product excludes zeros in $\\Re s>1$, so $1/2\\le\\beta_*\\le1$. For a finite set $\\mathcal S$ of prime ideals, a superscript $\\mathcal S$ means that the corresponding Euler factors have been deleted: \\[ L_F^{\\mathcal S}(s,\\eta)" }, { "id": "eq:signal-euler-contract", "kind": "anchor", "label": "eq:signal-euler-contract", "owner": "lem:continuation-criterion", "section": "sec:analytic-reduction", "anchor": { "source": "paper.tex", "line_start": 414, "line_end": 415, "char_start": 21144, "char_end": 21176 }, "argument_gist": "", "excerpt": "iciently large real $Z$, such that \\sup_{\\Re s>\\sigma_0}|H_\\eta(s)-1|\\le\\frac12. For $Z>0$, define f_\\eta(Z)=\\frac1{2\\pi i}\\int_{\\Re s=2} Z^{C(s)}e^{(s-5/6)^2} \\frac{H_\\eta(s)}{L_F^{\\mathcal S}(s,\\eta)}\\,ds. Assume that, as $Z\\to\\infty$, |J_\\eta(Z)|&\\ll_\\eta Z^{C(\\sigma_0)+\\omega}," }, { "id": "eq:common-residue", "kind": "anchor", "label": "eq:common-residue", "owner": "lem:continuation-criterion", "section": "sec:analytic-reduction", "anchor": { "source": "paper.tex", "line_start": 421, "line_end": 422, "char_start": 21340, "char_end": 21365 }, "argument_gist": "", "excerpt": "rac1{2\\pi i}\\int_{\\Re s=2} Z^{C(s)}e^{(s-5/6)^2} \\frac{H_\\eta(s)}{L_F^{\\mathcal S}(s,\\eta)}\\,ds. Assume that, as $Z\\to\\infty$, |J_\\eta(Z)|&\\ll_\\eta Z^{C(\\sigma_0)+\\omega}, \\\\ |J_\\eta(Z)-f_\\eta(Z)|&\\ll_\\eta Z^{C(\\beta_*)-\\sigma}. The implied constants, lower thresholds, excluded" }, { "id": "eq:low-probe-contract", "kind": "anchor", "label": "eq:low-probe-contract", "owner": "lem:continuation-criterion", "section": "sec:analytic-reduction", "anchor": { "source": "paper.tex", "line_start": 426, "line_end": 426, "char_start": 21478, "char_end": 21507 }, "argument_gist": "", "excerpt": "n} Assume that, as $Z\\to\\infty$, |J_\\eta(Z)|&\\ll_\\eta Z^{C(\\sigma_0)+\\omega}, \\\\ |J_\\eta(Z)-f_\\eta(Z)|&\\ll_\\eta Z^{C(\\beta_*)-\\sigma}. The implied constants, lower thresholds, excluded set, and function $H_\\eta$ may depend on $\\eta$. Then these assumptions contradict $\\beta_*>\\sigma_0$. Put \\[" }, { "id": "eq:high-probe-contract", "kind": "anchor", "label": "eq:high-probe-contract", "owner": "lem:continuation-criterion", "section": "sec:analytic-reduction", "anchor": { "source": "paper.tex", "line_start": 428, "line_end": 429, "char_start": 21572, "char_end": 21602 }, "argument_gist": "", "excerpt": "\\\\ |J_\\eta(Z)-f_\\eta(Z)|&\\ll_\\eta Z^{C(\\beta_*)-\\sigma}. The implied constants, lower thresholds, excluded set, and function $H_\\eta$ may depend on $\\eta$. Then these assumptions contradict $\\beta_*>\\sigma_0$. Put \\[ \\epsilon_*:=\\min\\{\\Delta_0-\\omega,\\sigma\\}>0. \\] Because $C$ has slope one, the triangle ineq" }, { "id": "eq:uniform-saving", "kind": "anchor", "label": "eq:uniform-saving", "owner": "lem:continuation-criterion", "section": "sec:analytic-reduction", "anchor": { "source": "paper.tex", "line_start": 443, "line_end": 444, "char_start": 22007, "char_end": 22032 }, "argument_gist": "", "excerpt": "y gives |f_\\eta(Z)|\\ll_\\eta Z^{C(\\beta_*)-\\epsilon_*} \\qquad (Z\\ge Z_{0,\\eta}). Moreover $\\epsilon_*\\le\\Delta_0-\\omega<\\Delta_0$, so $\\beta_*-\\epsilon_*>\\sigma_0$. We also need control as $Z\\downarrow0$. By Equation~\\eqref{eq:signal-euler-contract}, $H_\\eta$ is bounded on $\\Re s>\\sigma_0$. The reciprocal Euler product is absolutely and uniformly bounded on $\\Re" }, { "id": "part:eleven-twelfths", "kind": "anchor", "label": "part:eleven-twelfths", "owner": "sec:analytic-reduction", "section": "sec:analytic-reduction", "anchor": { "source": "paper.tex", "line_start": 523, "line_end": 524, "char_start": 25124, "char_end": 25152 }, "argument_gist": "", "excerpt": "r thresholds may depend on it in both applications. \\clearpage \\part{The quasi-Riemann hypothesis} \\section{A first zero-free half-plane} We first prove a fixed zero-free half-plane using the basic completed cubic-theta sum. This isolates the mechanism that excludes zeros before the additional estimates needed for the sharper boundary are introduced. [Th" }, { "id": "eq:first-stage-gap", "kind": "anchor", "label": "eq:first-stage-gap", "owner": "sec:first-stage", "section": "sec:first-stage", "anchor": { "source": "paper.tex", "line_start": 542, "line_end": 543, "char_start": 25850, "char_end": 25876 }, "argument_gist": "", "excerpt": "cke assertion, suppose for contradiction that \\Delta_1:=\\beta_*-\\frac{11}{12}>0. The proof will construct one character sum with two exact representations. Cubic-theta reflection bounds the sum after an elementary additive large-sieve estimate. Poisson summation expresses the same sum as a principal Mellin signal and a family of remaining character rows. A zero d" }, { "id": "old-eq:1.1", "kind": "anchor", "label": "old-eq:1.1", "owner": "sec:arithmetic", "section": "sec:arithmetic", "anchor": { "source": "paper.tex", "line_start": 708, "line_end": 709, "char_start": 34478, "char_end": 34496 }, "argument_gist": "", "excerpt": "olynomial of real log-length $n$ is S_\\psi(n;W)=Z^{-n/2}\\sum_l\\psi(l)W(q_l/Z^n), and the centrally normalized inverse polynomial of log-length $r$ is M_\\psi(r;W)=Z^{-r/2}\\sum_n\\mu(n)\\psi(n)W(q_n/Z^r). Every mask in $\\psi$ is retained in both definitions. At base $U$, when $\\psi(n)=\\nu(n)\\chi_n(u)$, we also de" }, { "id": "old-eq:1.1a", "kind": "anchor", "label": "old-eq:1.1a", "owner": "sec:arithmetic", "section": "sec:arithmetic", "anchor": { "source": "paper.tex", "line_start": 713, "line_end": 714, "char_start": 34651, "char_end": 34670 }, "argument_gist": "", "excerpt": "lynomial of log-length $r$ is M_\\psi(r;W)=Z^{-r/2}\\sum_n\\mu(n)\\psi(n)W(q_n/Z^r). Every mask in $\\psi$ is retained in both definitions. At base $U$, when $\\psi(n)=\\nu(n)\\chi_n(u)$, we also denote the latter polynomial by $M_u(U^r;W)$, or by $M_u(D;W)$ when $D=U^r$. An annular test is a smooth function with support in a fixed compact subinterval of $(0,\\infty)$." }, { "id": "eq:quadratic-four-term", "kind": "anchor", "label": "eq:quadratic-four-term", "owner": "lem:quadratic-four-term", "section": "sec:arithmetic", "anchor": { "source": "paper.tex", "line_start": 847, "line_end": 848, "char_start": 40614, "char_end": 40644 }, "argument_gist": "", "excerpt": "odd $c\\in\\mathcal O$, define \\[ \\Gamma(c)=|c|^{-1}\\sum_{x\\bmod c}e(x^2/c). \\] Then \\Gamma(c)=\\frac12\\sum_{y\\bmod2\\mathcal O}e(-cy^2/4). For $c=a+b\\omega$ the right side is $(1+i^{-b}+i^a+i^{b-a})/2$. In particular $\\Gamma$ depends only on $c\\bmod4\\mathcal O$ and $\\Gamma(cv^2)=\\Gamma(c)$ for every odd $v$. The units modulo $4$ have square subgroup $\\{1,\\omega,\\ome" }, { "id": "eq:reciprocity-four-class", "kind": "anchor", "label": "eq:reciprocity-four-class", "owner": "lem:quadratic-four-term", "section": "sec:arithmetic", "anchor": { "source": "paper.tex", "line_start": 857, "line_end": 858, "char_start": 41145, "char_end": 41178 }, "argument_gist": "", "excerpt": "es on these units, and $\\mathfrak r(a,b)=\\Gamma(ab)/(\\Gamma(a)\\Gamma(b))$ satisfies {c|rrrr} c&1&-1&\\lambda&-\\lambda\\\\ \\hline \\Gamma(c)&1&1&i&-i \\\\ \\mathfrak r((-1)^e\\lambda^f,(-1)^g\\lambda^h) =(-1)^{eh+fg+fh},\\qquad e,f,g,h\\in\\{0,1\\}. Thus $|\\Gamma(c)|=1$ and $\\mathfrak r$ is a symmetric $\\{\\pm" }, { "id": "eq:G-quadratic-refinement", "kind": "anchor", "label": "eq:G-quadratic-refinement", "owner": "lem:fixed-gauss-phase", "section": "sec:arithmetic", "anchor": { "source": "paper.tex", "line_start": 950, "line_end": 951, "char_start": 45014, "char_end": 45047 }, "argument_gist": "", "excerpt": "s one and factors through a fixed ray group supported at $2,3$. For all such $v,w$, G(vw)=G(v)G(w)\\mathcal R(v,w),\\qquad G(1)=1. Moreover $G(v^2)=\\chi_v(4)$, and at good primes G(p^3)=\\gamma_3(p),\\qquad \\mathcal R(p,p)=\\gamma_3(p)^2=\\chi_p(-1). On every primary $n$ outside $S$, one has $\\chi_n(-1)=\\math" }, { "id": "eq:fixed-gauss-phases", "kind": "anchor", "label": "eq:fixed-gauss-phases", "owner": "lem:fixed-gauss-phase", "section": "sec:arithmetic", "anchor": { "source": "paper.tex", "line_start": 954, "line_end": 955, "char_start": 45173, "char_end": 45202 }, "argument_gist": "", "excerpt": "(v,w),\\qquad G(1)=1. Moreover $G(v^2)=\\chi_v(4)$, and at good primes G(p^3)=\\gamma_3(p),\\qquad \\mathcal R(p,p)=\\gamma_3(p)^2=\\chi_p(-1). On every primary $n$ outside $S$, one has $\\chi_n(-1)=\\mathcal R(n,n)$. The diagonal $t\\mapsto\\mathcal R(t,t)$ is a character of the fixed ray group. In the square-class notation of Equation~\\eqref{eq:reciprocity" }, { "id": "eq:signal", "kind": "anchor", "label": "eq:signal", "owner": "lem:fixed-gauss-phase", "section": "sec:arithmetic", "anchor": { "source": "paper.tex", "line_start": 965, "line_end": 966, "char_start": 45718, "char_end": 45735 }, "argument_gist": "", "excerpt": "lass of the unit ideal $(-1)$. For squarefree $c$ the complete Gauss identities are \\gamma_2(c)^3=\\mu(c)\\alpha(c),\\qquad \\gamma_1(c)\\gamma_2(c)=\\mu(c)\\alpha(c)G(c),\\qquad G(c)=\\overline{\\chi_c(4)}\\gamma_3(c),\\quad |G(c)|=1. All occurrences of $G$ on nonsquarefree indices mean the finite-ray function just defined, not a nonsquarefree Gauss sum. \\begin" }, { "id": "eq:row-fixed-ray-reduction", "kind": "anchor", "label": "eq:row-fixed-ray-reduction", "owner": "sec:arithmetic", "section": "sec:arithmetic", "anchor": { "source": "paper.tex", "line_start": 1064, "line_end": 1065, "char_start": 50073, "char_end": 50107 }, "argument_gist": "", "excerpt": "side $S$, multiplicativity and sextic reciprocity give the zero-preserving identity \\chi_A(m)=\\chi_A(u m_S)\\mathcal R(A,m_{\\mathrm{good}}) \\prod_{p\\mid m_{\\mathrm{good}}}\\chi_p(A)^{v_p(m)}. For $(A,m_{\\mathrm{good}})=1$ this is the reciprocity formula just proved; if they meet, both sides are zero and $\\mathcal R$ is evaluated as its separately defined bicha" }, { "id": "eq:parameter-sobolev", "kind": "anchor", "label": "eq:parameter-sobolev", "owner": "lem:smooth-calculus", "section": "sec:arithmetic", "anchor": { "source": "paper.tex", "line_start": 1143, "line_end": 1144, "char_start": 53692, "char_end": 53720 }, "argument_gist": "", "excerpt": "a unit box $I\\subset\\mathbb R^d$ and a smooth scalar function $F$ on $I+[-1,1]^d$, \\sup_I|F|^2\\ll_d \\sum_{\\alpha\\in\\{0,1\\}^d} \\int_{I+[-1,1]^d}|\\partial^\\alpha F|^2. Consequently, under the corresponding derivative bounds, rowwise choices of scales in a polynomial range cost powers of $\\log Z$, and rowwise choices of norm-twist heights of absolute value at most" }, { "id": "eq:late-fourier-tail", "kind": "anchor", "label": "eq:late-fourier-tail", "owner": "lem:smooth-calculus", "section": "sec:arithmetic", "anchor": { "source": "paper.tex", "line_start": 1154, "line_end": 1155, "char_start": 54112, "char_end": 54140 }, "argument_gist": "", "excerpt": "at most $T_1$ cost a fixed power of $1+T_1$. For $T_1\\ge1$ and integers $J,N\\ge0$, \\int_{|\\boldsymbol t|>T_1}|\\widehat w(\\boldsymbol t)| (1+|\\boldsymbol t|)^J\\,d\\boldsymbol t \\le T_1^{-N}\\|w\\|_{J+N,\\mathrm{sep}}. With the Mellin convention $\\mathcal MW(s)=\\int_0^\\infty W(y)y^s\\,dy/y$, a pure twist obeys \\[ \\mathcal M[W(y)y^{i\\omega}](s)=\\mathcal MW(" }, { "id": "eq:pointwise-mellin-tail", "kind": "anchor", "label": "eq:pointwise-mellin-tail", "owner": "lem:smooth-calculus", "section": "sec:arithmetic", "anchor": { "source": "paper.tex", "line_start": 1168, "line_end": 1169, "char_start": 54729, "char_end": 54761 }, "argument_gist": "", "excerpt": "tegrations by parts vanish. Then, uniformly for $\\sigma\\in I$ and $T\\in\\mathbb R$, |\\mathcal MW(\\sigma+iT)| \\ll_{I,N}(1+|T|)^{-N} \\sum_{j=0}^N\\int_0^\\infty y^\\sigma|D_y^jW(y)|\\,\\frac{dy}{y}. These hypotheses hold for annular $W$ on every such $I$. They also hold when $W$ is smooth at zero and Schwartz at infinity and $I$ is a compact subset of $(0,\\infty)$." }, { "id": "eq:joint-gaussian-slice", "kind": "anchor", "label": "eq:joint-gaussian-slice", "owner": "lem:smooth-calculus", "section": "sec:arithmetic", "anchor": { "source": "paper.tex", "line_start": 1181, "line_end": 1182, "char_start": 55419, "char_end": 55450 }, "argument_gist": "", "excerpt": "e functions $a(v),b(w)$ for which the right side is finite, and integers $J,N\\ge0$, &\\int_{\\mathbb R^2}e^{-(T+v)^2}|a(v)b(w)| (1+|T|+|v|+|w|)^J\\,dv\\,dw\\\\ &\\quad\\ll_{J,N}(1+|T|)^{-N} \\sup_v(1+|v|)^{J+N}|a(v)| \\int_{\\mathbb R}(1+|w|)^J|b(w)|\\,dw. Thus the product of a Gaussian in the sum of two heights and rapidly dec" }, { "id": "eq:gaussian-annular-sum", "kind": "anchor", "label": "eq:gaussian-annular-sum", "owner": "lem:gaussian-annular", "section": "sec:arithmetic", "anchor": { "source": "paper.tex", "line_start": 1302, "line_end": 1303, "char_start": 61349, "char_end": 61380 }, "argument_gist": "", "excerpt": "fixed compact logarithmic support and, for every fixed $A\\ge0$ and integer $j\\ge0$, \\sum_{k\\in\\mathbb Z}e^{A|k|}p_j(w_k)<\\infty. Consequently Lemma~\\ref{lem:smooth-calculus} may be applied on each annulus after $y=e^k x$, and the resulting weighted separation norms are summable. For fixed $\\kappa>0$, $A,B,C_0\\ge0$, and $M>0$ one also has \\[ Z^B\\sum_{|k|>\\kappa\\log" }, { "id": "eq:hecke-growth", "kind": "anchor", "label": "eq:hecke-growth", "owner": "lem:hecke-strip-growth", "section": "sec:arithmetic", "anchor": { "source": "paper.tex", "line_start": 1428, "line_end": 1429, "char_start": 67383, "char_end": 67406 }, "argument_gist": "", "excerpt": "ve nonprincipal finite-order Hecke character of $F$, with conductor norm $Q$. Then |L(\\sigma+it,\\psi)|\\ll Q^{3/5}(3+|t|)^2, \\qquad-1/10\\le\\sigma\\le11/10. For the principal character, the same bound with $Q=1$ holds for $(s-1)\\zeta_F(s)/(s+1)$, with the removable value used at $s=1$. A finite-order character has trivial infinite type, si" }, { "id": "eq:disk-control", "kind": "anchor", "label": "eq:disk-control", "owner": "lem:logarithmic-control", "section": "sec:arithmetic", "anchor": { "source": "paper.tex", "line_start": 1539, "line_end": 1540, "char_start": 72585, "char_end": 72608 }, "argument_gist": "", "excerpt": "e$ centered at $2+it$. Uniformly on the closed concentric disk of radius $2-a-6e$, |\\mathcal L(s)|+|\\mathcal L(s)^{-1}| \\ll_{e,\\epsilon}\\{2Q(3+|t|)^2\\}^{\\epsilon}. On the disk of radius $2-a-8e$ one also has \\[ |\\mathcal L'(s)/\\mathcal L(s)|\\ll_e\\log\\{2Q(3+|t|)^2\\}. \\] In particular, for every fixed $b\\ge\\beta_*$ and $v>0$, these bounds hold on $\\Re s\\ge b+v$, w" }, { "id": "eq:reflection", "kind": "anchor", "label": "eq:reflection", "owner": "prop:completed-reflection", "section": "sec:completed-reflection", "anchor": { "source": "paper.tex", "line_start": 1763, "line_end": 1764, "char_start": 82614, "char_end": 82635 }, "argument_gist": "", "excerpt": "ing data have properties (1)--(3). For these terms and every $a>1$, the identity is &\\frac1{2\\pi i}\\int_{(a)}\\widehat V(s-1/2)X^{s-1/2}T(s,\\Psi)\\,ds\\\\ &\\quad=\\sum_{\\text{terms}}\\zeta \\sum_{0\\ne\\mu\\in\\lambda^{-4}\\mathcal O} \\frac{d(\\mu)\\alpha(\\mu)\\vartheta(\\lambda^4\\mu)}{\\sqrt{q_\\mu}} \\prod_{p\\ {\\rm active}}B_p(\\lambda^4\\mu) V^\\sharp\\!\\left(\\frac{q_\\mu X}{q_c" }, { "id": "old-eq:4.3", "kind": "anchor", "label": "old-eq:4.3", "owner": "prop:completed-reflection", "section": "sec:completed-reflection", "anchor": { "source": "paper.tex", "line_start": 1801, "line_end": 1802, "char_start": 84136, "char_end": 84154 }, "argument_gist": "", "excerpt": "\\{u\\lambda^k n b^3:k\\ge-4,\\ n,b\\equiv1\\pmod3,\\ n\\ {\\rm sf}\\}, \\qquad |d(\\mu)|\\le27\\,3^{k/6}|b|. The bound refers to the displayed representation of a supported index. It permits common primes of $n$ and $b$, and it imposes no exclusion at the primes of $L$ other than the displayed restriction at $\\lambda$. The character $\\vartheta$ is an additive character of a quotient of $\\ma" }, { "id": "eq:reflection-row-phase", "kind": "anchor", "label": "eq:reflection-row-phase", "owner": "prop:completed-reflection", "section": "sec:completed-reflection", "anchor": { "source": "paper.tex", "line_start": 1832, "line_end": 1833, "char_start": 85312, "char_end": 85343 }, "argument_gist": "", "excerpt": "ute value one, depending only on $h_0$ and the class of $r$ modulo $M^2$, such that \\zeta=-\\frac i{81}\\bar\\alpha(c)^2\\widehat\\phi(h_0)\\bar\\kappa_F \\prod_{p\\ {\\rm inactive}}(1-q_p^{-1}) \\prod_{p\\ {\\rm active}}\\chi_p(\\sigma_p)^{-2}\\omega_{p,j_p}. In each of the finitely many classes of $h_0$ and $r\\bmod M^2$, both $d$ and $\\vartheta$ are fixed. Consequently these" }, { "id": "eq:reflection-kernel", "kind": "anchor", "label": "eq:reflection-kernel", "owner": "prop:completed-reflection", "section": "sec:completed-reflection", "anchor": { "source": "paper.tex", "line_start": 1860, "line_end": 1861, "char_start": 86594, "char_end": 86622 }, "argument_gist": "", "excerpt": "\\mathbb R} (1+|u|)^B|\\widehat V(\\eta+iu)|\\,du. \\] For every integer $j\\ge0$, |(x\\partial_x)^jV^\\sharp(x)| \\ll_{A,j}\\min\\{x^{1/4},(1+x)^{-A}\\} \\mathfrak M_{A,\\lceil4A\\rceil+j+2}(V). If $\\chi\\in C_c^\\infty(0,\\infty)$ is fixed, $Y>0$, and $J\\ge0$ is an integer, then \\int_{\\mathbb R}(1+|u|)^J \\left|" }, { "id": "eq:reflection-annular-fourier", "kind": "anchor", "label": "eq:reflection-annular-fourier", "owner": "prop:completed-reflection", "section": "sec:completed-reflection", "anchor": { "source": "paper.tex", "line_start": 1867, "line_end": 1868, "char_start": 86850, "char_end": 86887 }, "argument_gist": "", "excerpt": "If $\\chi\\in C_c^\\infty(0,\\infty)$ is fixed, $Y>0$, and $J\\ge0$ is an integer, then \\int_{\\mathbb R}(1+|u|)^J \\left|\\int_0^\\infty\\chi(y)V^\\sharp(Yy)y^{-iu}\\frac{dy}{y}\\right|du \\ll_{A,J,\\chi}\\min\\{Y^{1/4},(1+Y)^{-A}\\} \\mathfrak M_{A,\\lceil4A\\rceil+J+4}(V). These are finite smooth seminorms. More precisely, \\label{eq:reflection-test-seminor" }, { "id": "eq:reflection-test-seminorm", "kind": "anchor", "label": "eq:reflection-test-seminorm", "owner": "prop:completed-reflection", "section": "sec:completed-reflection", "anchor": { "source": "paper.tex", "line_start": 1874, "line_end": 1875, "char_start": 87154, "char_end": 87189 }, "argument_gist": "", "excerpt": "A\\rceil+J+4}(V). These are finite smooth seminorms. More precisely, \\mathfrak M_{A,B}(V)\\ll_{A,B} \\int_0^\\infty(1+x^{-A}+x^{1/4}) \\bigl(|V(x)|+|(x\\partial_x)^{B+2}V(x)|\\bigr)\\frac{dx}{x}. Replacing $V(x)$ by $x^{iu_0}V(x)$ multiplies $\\mathfrak M_{A,B}$ by at most $(1+|u_0|)^B$. We first express the character" }, { "id": "eq:reflection-cusp-expansion", "kind": "anchor", "label": "eq:reflection-cusp-expansion", "owner": "prop:completed-reflection", "section": "sec:completed-reflection", "anchor": { "source": "paper.tex", "line_start": 1936, "line_end": 1937, "char_start": 89676, "char_end": 89712 }, "argument_gist": "", "excerpt": "e are the only cusp representatives needed below. For any of them, define $d_H$ by \\bar\\theta(H(z,v))=C_H(v)+ \\sum_{\\mu\\ne0}d_H(\\mu)vK_{1/3}(4\\pi|\\mu|v)\\check e(\\mu z). Here $C_H(v)$ is independent of $z$. The three functions $\\Theta_0,\\Theta_1,\\Theta_2$ are respectively $F_1,F_{19},F_{10}$ in~\\cite{DR}. Its Appendix~A, rows $1,19,10$, expresses their coefficien" }, { "id": "eq:reflection-cubic-source-gauss", "kind": "anchor", "label": "eq:reflection-cubic-source-gauss", "owner": "prop:completed-reflection", "section": "sec:completed-reflection", "anchor": { "source": "paper.tex", "line_start": 1956, "line_end": 1957, "char_start": 90661, "char_end": 90701 }, "argument_gist": "", "excerpt": "explicit, for every primary squarefree $n$ define, as in~\\cite[Equation (1.7)]{DR}, \\widetilde g_3(n)=q_n^{-1/2}\\sum_{v\\bmod n} \\left(\\frac vn\\right)_3\\check e(v/n), \\qquad \\widetilde g_3(1)=1. The ordinary cubic symbol is defined at every prime other than $\\lambda$ and is extended by zero on nonunits and multiplicatively in the denominator. Every primary $" }, { "id": "eq:reflection-infinity-coefficient", "kind": "anchor", "label": "eq:reflection-infinity-coefficient", "owner": "prop:completed-reflection", "section": "sec:completed-reflection", "anchor": { "source": "paper.tex", "line_start": 1992, "line_end": 1993, "char_start": 92672, "char_end": 92714 }, "argument_gist": "", "excerpt": "gives $x\\equiv-1\\pmod3$. At the isolated index the coefficient in $\\bar\\theta$ is d_I(\\lambda^{-3}nb^3)=3^{5/2}|b|\\widetilde g_3(n). This is the unmasked source formula, valid for every primary squarefree $n$ and every primary $b$, with no coprimality condition between them. Only when $(n,S)=1$ may it be expressed in the global sextic notation: substituting $y=\\" }, { "id": "eq:reflection-cubic-sextic-conversion", "kind": "anchor", "label": "eq:reflection-cubic-sextic-conversion", "owner": "prop:completed-reflection", "section": "sec:completed-reflection", "anchor": { "source": "paper.tex", "line_start": 1999, "line_end": 2000, "char_start": 93063, "char_end": 93108 }, "argument_gist": "", "excerpt": "ed in the global sextic notation: substituting $y=\\lambda v$ in $\\gamma_2(n)$ gives \\widetilde g_3(n)=\\left(\\frac\\lambda n\\right)_3^{-1}\\gamma_2(n) =\\chi_n(\\lambda)^{-2}\\gamma_2(n)\\qquad((n,S)=1). The cubic supplementary laws in~\\cite[Equation (1.5)]{DR} make $(\\lambda/x)_3$ periodic on all primary $x$ modulo $9$. On primary elements prime to $S$ it equals" }, { "id": "eq:reflection-local-fourier", "kind": "anchor", "label": "eq:reflection-local-fourier", "owner": "prop:completed-reflection", "section": "sec:completed-reflection", "anchor": { "source": "paper.tex", "line_start": 2020, "line_end": 2021, "char_start": 94152, "char_end": 94187 }, "argument_gist": "", "excerpt": "ss sum, and using the orthogonality of a nontrivial multiplicative character, gives C_{p,j}(h)= q_p^{-1/2}\\tau_{p,j}^{-}\\chi_p(h)^{-j},&j\\ne0,\\ h\\ne0,\\\\ 0,&j\\ne0,\\ h=0,\\\\ -q_p^{-1},&j=0,\\ h\\ne0,\\\\ 1-q_p^{-1},&j=0,\\ h=0. Here and below $h=0$ means $h\\equiv0\\pmod p$. Each nontrivial power of $\\chi_p$ is primitive modulo the prime $p$, s" }, { "id": "eq:reflection-translated-theta", "kind": "anchor", "label": "eq:reflection-translated-theta", "owner": "prop:completed-reflection", "section": "sec:completed-reflection", "anchor": { "source": "paper.tex", "line_start": 2044, "line_end": 2045, "char_start": 95164, "char_end": 95202 }, "argument_gist": "", "excerpt": "tiplier on its infinity coefficient $x=\\lambda^3\\mu\\in\\mathcal O$ is the finite sum \\mathcal F(z,v)=\\sum_hC(h)\\bar\\theta(z+z_h,v),\\qquad z_h=\\lambda^2\\left(h_0/L+\\sum_p h_p/p\\right). For an isolated infinity index $x=nb^3$, the piecewise definition of $\\phi$ first discards every term for which $(nb,\\mathfrak E)\\ne1$. This is a mask on the whole completed index: $" }, { "id": "eq:reflection-cusp-fraction", "kind": "anchor", "label": "eq:reflection-cusp-fraction", "owner": "prop:completed-reflection", "section": "sec:completed-reflection", "anchor": { "source": "paper.tex", "line_start": 2078, "line_end": 2079, "char_start": 96991, "char_end": 97026 }, "argument_gist": "", "excerpt": "rod_{p\\ {\\rm active}}p$, Equation~\\eqref{eq:reflection-translated-theta} then gives z_h=\\frac ac,\\qquad c=c_Fr,\\qquad a=a_Fr+\\lambda^2c_F\\sum_{p\\mid r}(r/p)h_p. At a prime dividing $c_F$, the numerator is congruent to $a_Fr$ and is a unit. At $p\\mid r$, it is congruent to $\\lambda^2c_F(r/p)h_p$ and is again a unit. Thus this fraction is reduced, even when $h_0$" }, { "id": "eq:reflection-cusp-case-zero", "kind": "anchor", "label": "eq:reflection-cusp-case-zero", "owner": "prop:completed-reflection", "section": "sec:completed-reflection", "anchor": { "source": "paper.tex", "line_start": 2132, "line_end": 2133, "char_start": 99605, "char_end": 99641 }, "argument_gist": "", "excerpt": "(3)$, and cubic reciprocity for the primary $a$ and the primary primes of $r$ gives \\kappa(G)=\\left(\\frac{c_F}{a}\\right)_3 \\left(\\frac ar\\right)_3. If $v_\\lambda(c)=1$, take the unique $u_0\\in\\{\\lambda,-\\lambda\\}$ with $u_0\\equiv c\\pmod3$ and put $H=\\mathsf L(u_0)$; these are the two possibilities because $\\mathcal O/(\\lambda)\\simeq\\mathbb F_3$. The co" }, { "id": "eq:reflection-multiplier", "kind": "anchor", "label": "eq:reflection-multiplier", "owner": "prop:completed-reflection", "section": "sec:completed-reflection", "anchor": { "source": "paper.tex", "line_start": 2185, "line_end": 2186, "char_start": 101653, "char_end": 101685 }, "argument_gist": "", "excerpt": "The last inverse uses $a\\delta'\\equiv1\\pmod c$. In all three cases we have proved \\kappa(G)=\\kappa_F\\prod_{p\\mid r}\\chi_p(a)^2, \\qquad |\\kappa_F|=1. The factor $\\kappa_F$ is constant as the active $h_p$ vary in the fixed sector. If $3\\mid c$, it is $(c_F/a)_3$: the unit and $\\lambda$-power factors are determined by $a\\bmod9$, and reciprocity determines the rema" }, { "id": "eq:reflection-additive-phase", "kind": "anchor", "label": "eq:reflection-additive-phase", "owner": "prop:completed-reflection", "section": "sec:completed-reflection", "anchor": { "source": "paper.tex", "line_start": 2207, "line_end": 2208, "char_start": 102827, "char_end": 102863 }, "argument_gist": "", "excerpt": "ive Chinese remaindering for the coprime factors of $D_Fr$ gives the exact identity \\check e(-\\delta'\\mu/c) &=e(-\\delta'x/(D_Fr))\\\\ &=\\vartheta(x)\\prod_{p\\mid r}e(\\epsilon_ph_p^{-1}x/p), \\qquad \\vartheta(x)=e(-\\delta'_F r^{-1}x/D_F). Here $\\delta'_F$ is the class of $\\delta'$ modulo $D_F$, and $r^{-1}$ in the formula for $\\vartheta$" }, { "id": "eq:reflection-cross-prime-phase", "kind": "anchor", "label": "eq:reflection-cross-prime-phase", "owner": "prop:completed-reflection", "section": "sec:completed-reflection", "anchor": { "source": "paper.tex", "line_start": 2271, "line_end": 2272, "char_start": 105714, "char_end": 105753 }, "argument_gist": "", "excerpt": "}$ of absolute value one, depending on $p,j$ but on no other active prime such that \\chi_p(\\sigma_p)^{-2}\\omega_{p,j} =\\xi_{p,j}\\chi_p(c/p)^{2j+2}. For $j=4$ this uses $2j+2\\equiv-2\\pmod6$; for the other exponents it follows immediately by inserting the expression for $\\epsilon_p$. Thus the contribution of another active prime to the phase at $p$ has exponen" }, { "id": "eq:reflection-direct-mellin", "kind": "anchor", "label": "eq:reflection-direct-mellin", "owner": "prop:completed-reflection", "section": "sec:completed-reflection", "anchor": { "source": "paper.tex", "line_start": 2308, "line_end": 2309, "char_start": 107219, "char_end": 107254 }, "argument_gist": "", "excerpt": "version}. Using $q_{\\lambda^{-3}}=27^{-1}$ and $\\bar\\alpha(\\lambda^{-3})=-i$ gives J(s)=\\frac{3^{5/2}}4\\left(\\frac{27}{(2\\pi)^2}\\right)^s \\Gamma(s+1/3)\\Gamma(s+2/3)T(s,\\Psi). For one translated term $z_h=a/c$, use the matrix $g=GH$ constructed above. The coordinate action of $g^{-1}$ on hyperbolic space, from \\cite[Equation (5.1)]{DR}, gives \\[ g^{-1}(z+" }, { "id": "eq:reflection-reflected-mellin", "kind": "anchor", "label": "eq:reflection-reflected-mellin", "owner": "prop:completed-reflection", "section": "sec:completed-reflection", "anchor": { "source": "paper.tex", "line_start": 2335, "line_end": 2336, "char_start": 108360, "char_end": 108398 }, "argument_gist": "", "excerpt": "and then $v\\mapsto1/(q_cv)$ in the integral gives, for $\\Re s<0$, the contribution -\\frac i4(2\\pi)^{2s-2}\\bar\\alpha(c)^2q_c^{1-2s}\\bar\\kappa(G) \\prod_{\\pm}\\Gamma(3/2-s\\pm1/6)\\\\ {} \\times\\sum_{\\mu\\ne0}d_H(\\mu)\\alpha(\\mu) \\check e(-\\delta'\\mu/c)q_\\mu^{-(1-s)}. To verify the scalar directly, before applying the Bessel integral th" }, { "id": "eq:reflection-row-reduction", "kind": "anchor", "label": "eq:reflection-row-reduction", "owner": "lem:reflection-row-sectors", "section": "sec:completed-reflection", "anchor": { "source": "paper.tex", "line_start": 2484, "line_end": 2485, "char_start": 115495, "char_end": 115530 }, "argument_gist": "", "excerpt": "s are primary and $u$ is a unit. On every primary element $A$ prime to $S$ one has \\chi_A(m)=\\chi_A(u m_S)\\mathcal R(A,m_{\\rm good}) \\prod_{p\\mid m_{\\rm good}}\\chi_p(A)^{v_p(m)}. All powers on the right retain their zero values, including when $v_p(m)\\equiv0\\pmod6$. Fix $u$, the valuations $v_{\\mathfrak l}(m)$ modulo six, and the class of $m_{\\rm good}$ in" }, { "id": "eq:reflection-sector-character", "kind": "anchor", "label": "eq:reflection-sector-character", "owner": "lem:reflection-row-sectors", "section": "sec:completed-reflection", "anchor": { "source": "paper.tex", "line_start": 2492, "line_end": 2493, "char_start": 115928, "char_end": 115966 }, "argument_gist": "", "excerpt": "in the fixed ray group through which $\\mathcal R$ factors. In such a sector define \\Psi_0^{[m]}(A)= \\Psi_{\\rm base}(A)\\chi_A(u m_S)\\mathcal R(A,m_{\\rm good}), &A\\equiv1\\pmod3,\\ (A,S)=1,\\\\ 0,&\\text{otherwise}. The nonzero branch is a member of a fixed finite family of finite-ray characters with conductors supported on $S$. Ther" }, { "id": "eq:reflection-common-profile", "kind": "anchor", "label": "eq:reflection-common-profile", "owner": "lem:reflection-common-profile", "section": "sec:completed-reflection", "anchor": { "source": "paper.tex", "line_start": 2557, "line_end": 2557, "char_start": 119052, "char_end": 119088 }, "argument_gist": "", "excerpt": ")&\\ll_{A,j,d,\\Omega,\\boldsymbol a} \\mathfrak m_A(Y)p_j(W) \\mathfrak M_{A,\\lceil4A\\rceil+j+2}(V), \\\\ \\|F_Y\\|_{J,\\mathrm{sep}}&\\ll_{A,J,d,\\Omega,\\boldsymbol a} \\mathfrak m_A(Y)p_{J+d+2}(W) \\mathfrak M_{A,\\lceil4A\\rceil+J+d+4}(V). Here $p_j$ is the homogeneous seminorm of a single profile, not the inhomogeneous seminorm of a tuple. The constant" }, { "id": "eq:reflection-common-measure", "kind": "anchor", "label": "eq:reflection-common-measure", "owner": "lem:reflection-common-profile", "section": "sec:completed-reflection", "anchor": { "source": "paper.tex", "line_start": 2561, "line_end": 2562, "char_start": 119223, "char_end": 119259 }, "argument_gist": "", "excerpt": ",J,d,\\Omega,\\boldsymbol a} \\mathfrak m_A(Y)p_{J+d+2}(W) \\mathfrak M_{A,\\lceil4A\\rceil+J+d+4}(V). Here $p_j$ is the homogeneous seminorm of a single profile, not the inhomogeneous seminorm of a tuple. The constants are independent of $Y$ and of the actual norm labels within the block. On the support of $W$, the monomial $h$ is bounded above and below by p" }, { "id": "eq:reflection-common-minkowski", "kind": "anchor", "label": "eq:reflection-common-minkowski", "owner": "sec:completed-reflection", "section": "sec:completed-reflection", "anchor": { "source": "paper.tex", "line_start": 2588, "line_end": 2589, "char_start": 120608, "char_end": 120646 }, "argument_gist": "", "excerpt": "es have been verified, write $F_Y=\\mathfrak m_A(Y)\\widetilde F_Y$. Minkowski gives \\|\\mathcal U\\|_{\\ell^2(\\mathcal I)} \\le \\frac{\\mathfrak m_A(Y)}{(2\\pi)^d} \\int_{\\mathbb R^d}|\\widehat{\\widetilde F_Y}(\\boldsymbol t)| \\|B(\\boldsymbol t)\\|_{\\ell^2(\\mathcal I)}\\,d\\boldsymbol t. Here $B(\\boldsymbol t)$ is the separated row vector, whose norm powers all" }, { "id": "eq:lattice-kernel-tail", "kind": "anchor", "label": "eq:lattice-kernel-tail", "owner": "lem:lattice-kernel-tail", "section": "sec:completed-reflection", "anchor": { "source": "paper.tex", "line_start": 2622, "line_end": 2623, "char_start": 122682, "char_end": 122712 }, "argument_gist": "", "excerpt": "sfies $|K(x)|\\le C_Ax^{-A}$ for $x\\ge1$, where $A>1$. Then, for $a>0$ and $U\\ge1$, \\sum_{\\substack{0\\ne\\mu\\in\\Lambda\\\\a q_\\mu>U}}|K(aq_\\mu)| \\ll_A C_A(1+a^{-1})U^{1-A}. For $K=V^\\sharp$ one may take $C_A\\ll_A\\mathfrak M_{A,\\lceil4A\\rceil+2}(V)$. The shell $2^jU1$, \\mathcal C_V(X;m,\\nu)=\\frac1{2\\pi i}\\int_{(a)} \\widehat V(s-1/2)X^{s-1/2}T(s,\\nu\\chi_\\bullet(m))\\,ds. Indeed, absolute convergence permits termwise Mellin inversion on that line. We bound the mean square of this completed sum over nonzero element rows. At equal completed and row" }, { "id": "eq:unmarked-structural-block", "kind": "anchor", "label": "eq:unmarked-structural-block", "owner": "lem:unmarked-reflected-block", "section": "sec:completed-reflection", "anchor": { "source": "paper.tex", "line_start": 2972, "line_end": 2973, "char_start": 139209, "char_end": 139245 }, "argument_gist": "", "excerpt": ",b)$, with absolute values at most one, such that $\\beta$ is independent of $R$ and \\mathcal U_{v,\\ell_b,e_\\lambda}(R) =\\frac13 Z^{-\\nu_0}\\rho(R) \\sum_{n,b}\\beta(n,b)\\chi_R(nb)^3F_Y(y_R,y_n,y_b). The sum retains all the fixed dual restrictions from the definition. The only arithmetic factor here that depends simultaneously on $R$ and the dual variables is" }, { "id": "eq:unmarked-actual-annuli", "kind": "anchor", "label": "eq:unmarked-actual-annuli", "owner": "lem:unmarked-reflected-block", "section": "sec:completed-reflection", "anchor": { "source": "paper.tex", "line_start": 2980, "line_end": 2981, "char_start": 139618, "char_end": 139651 }, "argument_gist": "", "excerpt": "and the dual variables is the displayed zero-extended quadratic symbol. Moreover, \\frac{q_\\mu X}{q_c^2}=Yy_ny_b^3y_R^{-2},\\qquad c=c_FF_{\\rm act}R. This is an exact identity on the original support, including when $N_F$ shares primes with $b$. For every $\\epsilon>0$ and every fixed $A\\ge0$, \\s" }, { "id": "eq:unmarked-reflected-energy", "kind": "anchor", "label": "eq:unmarked-reflected-energy", "owner": "lem:unmarked-reflected-block", "section": "sec:completed-reflection", "anchor": { "source": "paper.tex", "line_start": 2988, "line_end": 2989, "char_start": 139896, "char_end": 139932 }, "argument_gist": "", "excerpt": "when $N_F$ shares primes with $b$. For every $\\epsilon>0$ and every fixed $A\\ge0$, \\sum_{R\\in\\mathcal I_H}|\\mathcal U_{v,\\ell_b,e_\\lambda}(R)|^2 &\\ll_{\\epsilon,A} Z^{E_0+\\epsilon}p_5(W)^2 \\mathfrak M_{A,\\lceil4A\\rceil+7}(V)^2,\\\\ E_0={}&\\max(H,v+\\ell_b)-S_0-B_0-\\ell_b-\\frac{2e_\\lambda}{3} -\\frac12(T_0-v-3\\ell_b-e_\\lambda)_+. \\end{equati" }, { "id": "eq:actual-residual-row-dyad", "kind": "anchor", "label": "eq:actual-residual-row-dyad", "owner": "sec:completed-reflection", "section": "sec:completed-reflection", "anchor": { "source": "paper.tex", "line_start": 3130, "line_end": 3131, "char_start": 146624, "char_end": 146659 }, "argument_gist": "", "excerpt": "C_O, \\qquad q_R\\ge Z^H/C_H. \\] Since $q_{m_{\\rm sup}}\\ge1$, multiplicativity gives H\\le M-O+\\frac{\\log C_{\\rm res}}{\\log Z}, \\qquad C_{\\rm res}=C_HC_mC_O. This inequality does not require $m_{\\rm sup}$ to have bounded norm, and it need not be an equality. We will use it with the three factors on disjoint prime supports and with the unit residual ideal in the bou" }, { "id": "eq:unmarked-completed-moment", "kind": "anchor", "label": "eq:unmarked-completed-moment", "owner": "lem:unmarked-completed-moment", "section": "sec:completed-reflection", "anchor": { "source": "paper.tex", "line_start": 3156, "line_end": 3157, "char_start": 147848, "char_end": 147884 }, "argument_gist": "", "excerpt": "{prop:completed-reflection}, every actual $O$-dyad, and all sufficiently large $Z$, \\sum_{\\substack{00. Here is a proof that does not require the Fourier transform itself to be nonnegative. For $0<\\sigma<1$, \\[ |y|^{2\\sigma-2}=\\frac1{\\Gamma(1-\\sigma)} \\int_0^\\infty t^{-\\sigma}e^{-t|y|^2}\\,dt. \\] The pairing of the left side with $\\widetilde W_0(|y|^2)$ is abso" }, { "id": "eq:local-full-series", "kind": "anchor", "label": "eq:local-full-series", "owner": "sec:poisson-representation", "section": "sec:poisson-representation", "anchor": { "source": "paper.tex", "line_start": 3807, "line_end": 3808, "char_start": 176863, "char_end": 176891 }, "argument_gist": "", "excerpt": "sion of the two weights. For a nonzero sixth-power-free $u$ with $(u,S)=1$, define \\mathcal F_{\\eta,u}(x,w,z)= \\sum_{\\substack{c\\ {\\rm sf}\\\\n,s,a}} K_\\eta(c,n,s,a;u)\\, q_c^{-1/2-x}q_n^{-1-3x}q_s^{-w}q_a^{-6z}, where $A=cn^3$ and the coefficient, with all normalizations retained, is K_\\eta(c,n,s,a;u)= \\frac{F(s" }, { "id": "eq:local-full-coefficient", "kind": "anchor", "label": "eq:local-full-coefficient", "owner": "sec:poisson-representation", "section": "sec:poisson-representation", "anchor": { "source": "paper.tex", "line_start": 3815, "line_end": 3816, "char_start": 177127, "char_end": 177160 }, "argument_gist": "", "excerpt": "{equation} where $A=cn^3$ and the coefficient, with all normalizations retained, is K_\\eta(c,n,s,a;u)= \\frac{F(s,A,ua^6)\\chi_s(b_*)} {\\sqrt{q_{b_*}}\\tau\\xi(s)\\overline{\\xi(u)}} \\gamma_2(c)\\overline{\\alpha(A)G(A)} \\eta(A)\\Xi(A)^{-1}\\mathcal R(A,s). For convenience define the common Mellin weight \\[ \\mathcal W(X,Y,Z;x,w,z)= X^{1/2-z}Z^{x+z-1}Y^{w-1} \\Phi(x+z-1" }, { "id": "eq:probe-poisson-identity", "kind": "anchor", "label": "eq:probe-poisson-identity", "owner": "sec:poisson-representation", "section": "sec:poisson-representation", "anchor": { "source": "paper.tex", "line_start": 3831, "line_end": 3832, "char_start": 177661, "char_end": 177694 }, "argument_gist": "", "excerpt": "ginal line $\\Re t=4$ and lie in absolute convergence. The complete high identity is I_\\eta(X,Y,Z)=\\frac1{(2\\pi i)^3} \\int_{(3)}\\int_{(3)}\\int_{(2)} &\\mathcal W(X,Y,Z;x,w,z)\\\\ &\\cdot\\sum_u^{(6)}q_u^{-z}\\overline{\\xi(u)} \\mathcal F_{\\eta,u}(x,w,z)\\,dz\\,dw\\,dx. One may justify all interchanges on these lines by the elementar" }, { "id": "sec:local-euler", "kind": "anchor", "label": "sec:local-euler", "owner": "sec:poisson-representation", "section": "sec:poisson-representation", "anchor": { "source": "paper.tex", "line_start": 3848, "line_end": 3849, "char_start": 178276, "char_end": 178299 }, "argument_gist": "", "excerpt": "obe, not estimates for separately chosen test expressions. \\subsection{The scalar Euler identity} We next evaluate the coefficient in Equation~\\eqref{eq:local-full-coefficient}. Our objective is to prove that the high series is a scalar Euler product for the target $\\eta$. The calculation also records separately the terms with positive valuation of the completed index $A$ at a given prime. W" }, { "id": "eq:local-parameters", "kind": "anchor", "label": "eq:local-parameters", "owner": "sec:poisson-representation", "section": "sec:poisson-representation", "anchor": { "source": "paper.tex", "line_start": 3859, "line_end": 3860, "char_start": 178782, "char_end": 178809 }, "argument_gist": "", "excerpt": "Fix a nonzero sixth-power-free row $u$ with $(u,S)=1$ and a prime $p\\notin S$. Set Q=q_p,\\qquad j=v_p(u)\\in\\{0,\\ldots,5\\},\\qquad \\rho=\\chi_p(u/p^j),\\qquad \\omega_p=\\chi_p(-1),\\\\ a_p=\\overline{\\alpha(p)}^3\\eta(p)^3,\\qquad b_p=\\overline{\\alpha(p)}^2\\eta(p)^2\\overline{\\chi_p(4)}, \\qquad b_p^3=a_p^2,\\\\ V=Q^{-6z},\\qquad R=a_p^2Q^{4-6x-6z},\\qquad W_{\\rm loc}=\\r" }, { "id": "eq:local-scalar-definition", "kind": "anchor", "label": "eq:local-scalar-definition", "owner": "sec:poisson-representation", "section": "sec:poisson-representation", "anchor": { "source": "paper.tex", "line_start": 3892, "line_end": 3893, "char_start": 179975, "char_end": 180009 }, "argument_gist": "", "excerpt": "art of Equation~\\eqref{eq:local-F} at $p$, before its unit factors are restored, is C_p(t,k,j')= \\sum_{\\substack{d\\bmod p^k\\\\p^{j'}\\equiv p^td\\pmod{p^k}}} \\chi_p^k(d) g_{\\chi_p^t}\\!\\left(p^t,\\frac{p^{j'}-p^td}{p^k}\\right). At modulus one both factors in this formula mean one. Its complete evaluation is C_p(t,k,j')" }, { "id": "eq:local-scalar-values", "kind": "anchor", "label": "eq:local-scalar-values", "owner": "sec:poisson-representation", "section": "sec:poisson-representation", "anchor": { "source": "paper.tex", "line_start": 3900, "line_end": 3901, "char_start": 180266, "char_end": 180296 }, "argument_gist": "", "excerpt": "} At modulus one both factors in this formula mean one. Its complete evaluation is C_p(t,k,j')= 1,&t=k=0,\\\\ 1_{j'=0},&t=0,\\ k>0,\\\\ g_{\\chi_p^t}(p^t,p^{j'}),&t>0,\\ k=0,\\\\ \\omega_p^{-1}G_1g_{\\chi_p^{t-1}}(p^t,p^{j'-1}), &t>0,\\ k=1,\\ j'\\ge1,\\\\ 0,&\\text{otherwise}. For $t=0$ and $k>0$, the congruence fixes $d=p^{j" }, { "id": "eq:local-unit-factor", "kind": "anchor", "label": "eq:local-unit-factor", "owner": "sec:poisson-representation", "section": "sec:poisson-representation", "anchor": { "source": "paper.tex", "line_start": 3930, "line_end": 3931, "char_start": 181481, "char_end": 181509 }, "argument_gist": "", "excerpt": "g and the substitution $d=h_p(b_*A_p^\\circ)^{-1}d'$ give the additional unit factor U_p=\\chi_p^{k-t}(h_p)\\chi_p^t(s_p) \\chi_p^{t-k}(A_p^\\circ)\\chi_p^{t-k}(b_*). More explicitly, the outer residue character gives $\\chi_p^k(h_p)\\chi_p^{-k}(b_*A_p^\\circ)$, while the Chinese remainder factor and the rescaling of the Gauss argument give $\\chi_p^t(b_*A_p^\\circ)\\ch" }, { "id": "eq:local-P", "kind": "anchor", "label": "eq:local-P", "owner": "sec:poisson-representation", "section": "sec:poisson-representation", "anchor": { "source": "paper.tex", "line_start": 3997, "line_end": 3998, "char_start": 184294, "char_end": 184312 }, "argument_gist": "", "excerpt": "rated into prime factors. In absolute convergence, its factor at $p$ is the series P_p={}&\\sum_{\\substack{e_0=0,1\\\\l,k,m\\ge0}} (-1)^{e_0}\\gamma_1^{-e_0}\\eta(p)^{e_0} (a_p\\overline{\\gamma_3})^l\\rho^{k-t} \\omega_p^{tk-e_0l-l(l-1)/2}C_p(t,k,j+6m)\\\\ &\\hspace{33mm}\\cdot Q^{-(x+1/2)e_0-(1+3x)l-wk-6zm},\\qquad t=e_0+3l. Let $P_p^*$ be th" }, { "id": "old-eq:5.13a", "kind": "anchor", "label": "old-eq:5.13a", "owner": "lem:local-euler", "section": "sec:poisson-representation", "anchor": { "source": "paper.tex", "line_start": 4013, "line_end": 4014, "char_start": 184902, "char_end": 184922 }, "argument_gist": "", "excerpt": "r-free $u$ with $(u,S)=1$ and every prime $p\\notin S$, the factors just defined are P_p^*&=\\frac1{1-R}\\left\\{ \\frac{R(1-Q^{-1})-\\eta(p)(Q-1) Q^{-x-w}V^{1_{j\\le1}}}{1-V}+J_j\\right\\},\\\\ P_p&=\\frac1{1-V}+1_{j=0}\\frac{W_{\\rm loc}}{1-W_{\\rm loc}}+P_p^*, where the six values of $J_j$ are \\[ {c|l} j&J_j\\\\\\hline 0&-\\eta(p)\\rh" }, { "id": "eq:local-H-definition", "kind": "anchor", "label": "eq:local-H-definition", "owner": "lem:local-euler", "section": "sec:poisson-representation", "anchor": { "source": "paper.tex", "line_start": 4037, "line_end": 4038, "char_start": 185579, "char_end": 185608 }, "argument_gist": "", "excerpt": "erline{\\chi_p(u)}Q^{-x}$, with the original zero extension at $p\\mid u$, and define H_p=P_p\\frac{(1-V)(1-W)}{1-D},\\qquad \\mathcal H_{\\eta,u}=\\prod_{p\\notin S}H_p. Then the series in Equation~\\eqref{eq:local-full-series} has the scalar factorization \\mathcal F_{\\eta,u}(x,w,z)= \\frac{\\zeta_F^S(6z)L^S(w,\\chi_\\bullet(u))} {L^S(" }, { "id": "eq:probe-high", "kind": "anchor", "label": "eq:probe-high", "owner": "lem:local-euler", "section": "sec:poisson-representation", "anchor": { "source": "paper.tex", "line_start": 4043, "line_end": 4044, "char_start": 185807, "char_end": 185828 }, "argument_gist": "", "excerpt": "en the series in Equation~\\eqref{eq:local-full-series} has the scalar factorization \\mathcal F_{\\eta,u}(x,w,z)= \\frac{\\zeta_F^S(6z)L^S(w,\\chi_\\bullet(u))} {L^S(x,\\eta\\overline{\\chi_\\bullet(u)})} \\mathcal H_{\\eta,u}(x,w,z). Write $x_r=\\Re x$, $w_r=\\Re w$, and $z_r=\\Re z$. For every fixed $\\epsilon_0>0$, the correction product converges normally on a neighborhood" }, { "id": "eq:local-region-one", "kind": "anchor", "label": "eq:local-region-one", "owner": "lem:local-euler", "section": "sec:poisson-representation", "anchor": { "source": "paper.tex", "line_start": 4055, "line_end": 4055, "char_start": 186321, "char_end": 186348 }, "argument_gist": "", "excerpt": "ere: x_r\\ge51/100,\\quad z_r\\ge17/50,\\quad w_r\\ge-1/100,\\quad x_r+w_r\\ge1+\\epsilon_0;\\\\ x_r\\ge7/8,\\quad z_r\\ge33/200,\\quad w_r\\ge19/20. Uniformly in imaginary parts and unit phases, $\\mathcal H_{\\eta,u}\\ll_\\epsilon q_u^\\epsilon$, with the constant also depending on $\\epsilon_0$ in the first region. For $u=1$ in the second region, $\\mathcal" }, { "id": "eq:local-region-two", "kind": "anchor", "label": "eq:local-region-two", "owner": "lem:local-euler", "section": "sec:poisson-representation", "anchor": { "source": "paper.tex", "line_start": 4057, "line_end": 4058, "char_start": 186401, "char_end": 186428 }, "argument_gist": "", "excerpt": "+w_r\\ge1+\\epsilon_0;\\\\ x_r\\ge7/8,\\quad z_r\\ge33/200,\\quad w_r\\ge19/20. Uniformly in imaginary parts and unit phases, $\\mathcal H_{\\eta,u}\\ll_\\epsilon q_u^\\epsilon$, with the constant also depending on $\\epsilon_0$ in the first region. For $u=1$ in the second region, $\\mathcal H_{\\eta,1}=1+O(P_0^{-c_H})$ with, for example, $c_H=4/5$. Equation~\\eqref{eq:pr" }, { "id": "eq:local-family-table", "kind": "anchor", "label": "eq:local-family-table", "owner": "sec:poisson-representation", "section": "sec:poisson-representation", "anchor": { "source": "paper.tex", "line_start": 4086, "line_end": 4087, "char_start": 187738, "char_end": 187767 }, "argument_gist": "", "excerpt": "displayed. Every omitted case is zero by Equation~\\eqref{eq:local-scalar-values}. {c|c|l|l} (e_0,l)&k&\\text{condition on }(j,m)&R^{-r}\\text{ times summand}\\\\\\hline (0,2r+2)&0&m\\ge r+1&R(1-Q^{-1})V^{m-r-1}\\\\ (0,2r+2)&0&j=5,\\ m=r&-a_p^2Q^{3-6x}\\\\ (0,2r+2)&1&j=0,\\ m=r+1&W_{\\rm loc}R\\\\ (1,2r)&0&j=0,\\ m=r&-\\eta(p)\\rho^{-1}Q^{-x}\\\\ (1,2r)&1&j=1,\\ m=r&\\eta(p)Q^{-x-w}" }, { "id": "eq:local-H-defect", "kind": "anchor", "label": "eq:local-H-defect", "owner": "sec:poisson-representation", "section": "sec:poisson-representation", "anchor": { "source": "paper.tex", "line_start": 4139, "line_end": 4140, "char_start": 190099, "char_end": 190124 }, "argument_gist": "", "excerpt": "e contribution from $t=0$ is $(1-V)^{-1}+W/(1-W)$, so an exact simplification gives H_p-1= \\frac{D(V+W-VW)-VW+(1-V)(1-W)\\mathcal E_p}{1-D}. This expression has no $1-W$ denominator. At $p\\mid u$, where $D=W=0$, it reduces to $(1-V)\\mathcal E_p$. In both stated regions $|R|<1$, $|V|<1$, and $|D|<1$, uniformly away from one, so these are holomorphic local expressi" }, { "id": "par:physical-row-ramification", "kind": "anchor", "label": "par:physical-row-ramification", "owner": "sec:detector", "section": "sec:detector", "anchor": { "source": "paper.tex", "line_start": 4259, "line_end": 4260, "char_start": 195193, "char_end": 195230 }, "argument_gist": "", "excerpt": "rticular, they do not replace a row-dependent mask by an independently chosen mask. \\phantomsection If \\(v_p(u)=j\\in\\{1,\\ldots,5\\}\\) at a prime \\(p\\notin S\\), the local character of \\(\\chi_\\bullet(u)\\) on units at \\(p\\) has order \\(6/\\gcd(6,j)>1\\). A character in \\(\\Theta\\) is unramified there, so it cannot cancel this local character. Consequently a presentation in \\(\\mathcal X_u\\) can induce" }, { "id": "old-eq:6.2", "kind": "anchor", "label": "old-eq:6.2", "owner": "lem:detector-dyads", "section": "sec:detector", "anchor": { "source": "paper.tex", "line_start": 4408, "line_end": 4409, "char_start": 201245, "char_end": 201263 }, "argument_gist": "", "excerpt": "sufficiently large \\(Z\\): |M_r|^2\\ll_{\\mathcal A,\\epsilon}U^{\\delta r+\\epsilon}, |S_m|^2\\ll_{\\mathcal A,\\epsilon} U^{\\delta\\min(m,1-m)+\\epsilon}. The external tail order may be chosen after the positive number \\(\\tau\\). It does not change \\(A_{\\mathcal A}\\). We spell out the h" }, { "id": "old-eq:6.3", "kind": "anchor", "label": "old-eq:6.3", "owner": "lem:detector-dyads", "section": "sec:detector", "anchor": { "source": "paper.tex", "line_start": 4413, "line_end": 4414, "char_start": 201370, "char_end": 201388 }, "argument_gist": "", "excerpt": "quation} |S_m|^2\\ll_{\\mathcal A,\\epsilon} U^{\\delta\\min(m,1-m)+\\epsilon}. The external tail order may be chosen after the positive number \\(\\tau\\). It does not change \\(A_{\\mathcal A}\\). We spell out the height restriction because a full infinite contour shift would not be justified by the bin. For a fixed annular \\(W\\), a boun" }, { "id": "eq:detector-note", "kind": "anchor", "label": "eq:detector-note", "owner": "prop:detector-witness", "section": "sec:detector", "anchor": { "source": "paper.tex", "line_start": 4520, "line_end": 4521, "char_start": 206218, "char_end": 206242 }, "argument_gist": "", "excerpt": "\\), and two polynomials \\(M_r,S_m\\) for the common row character \\(\\psi\\) such that r\\le t+O(1/\\log U),\\qquad r+m\\ge t-O(1/\\log U),\\qquad |M_rS_m|^2\\gg_{\\mathcal A,\\epsilon} U^{\\delta(r+m)-\\epsilon}. Their lengths and individual values also satisfy t-\\tfrac12-O(\\epsilon)\\le r\\le t+O(1/\\log U), \\qqu" }, { "id": "eq:saturated-witness", "kind": "anchor", "label": "eq:saturated-witness", "owner": "prop:detector-witness", "section": "sec:detector", "anchor": { "source": "paper.tex", "line_start": 4527, "line_end": 4528, "char_start": 206443, "char_end": 206471 }, "argument_gist": "", "excerpt": "lta(r+m)-\\epsilon}. Their lengths and individual values also satisfy t-\\tfrac12-O(\\epsilon)\\le r\\le t+O(1/\\log U), \\qquad 0\\le m\\le\\tfrac12+O(\\epsilon),\\\\ |M_r|^2\\gg_{\\mathcal A,\\epsilon}U^{\\delta r-\\epsilon}, \\qquad |S_m|^2\\gg_{\\mathcal A,\\epsilon}U^{\\delta m-\\epsilon}. The constants in the \\(O(\\epsilon)\\) terms" }, { "id": "eq:sextic-large-sieve", "kind": "anchor", "label": "eq:sextic-large-sieve", "owner": "lem:sextic-large-sieve", "section": "sec:sextic-row-count", "anchor": { "source": "paper.tex", "line_start": 4712, "line_end": 4713, "char_start": 214123, "char_end": 214152 }, "argument_gist": "", "excerpt": "e $S$. The sequence is fixed independently of the row $k$. For every $\\epsilon>0$, \\sum_{\\substack{k\\ {\\rm sf}\\\\q_k\\le K}} \\left|\\sum_{\\substack{n\\ {\\rm sf}\\\\q_n\\le D}} c_n\\chi_n(k)^{\\varsigma}\\right|^2 \\ll_{S,\\epsilon}(KD)^\\epsilon \\bigl\\{K+D+(KD)^{2/3}\\bigr\\} \\sum_{\\substack{n\\ {\\rm sf}\\\\q_n\\le D}}|c_n|^2. Every symbol in this formula has" }, { "id": "eq:sextic-sieve-unit-class", "kind": "anchor", "label": "eq:sextic-sieve-unit-class", "owner": "lem:sextic-large-sieve", "section": "sec:sextic-row-count", "anchor": { "source": "paper.tex", "line_start": 4784, "line_end": 4785, "char_start": 217471, "char_end": 217505 }, "argument_gist": "", "excerpt": "/6}$. Since $q_p\\equiv1\\pmod6$, multiplication over the prime factors of $b$ gives \\chi_b(u)=u^{(q_b-1)/6}. The exponent is read modulo six: expanding a product of integers $1+6h$ proves the equality of exponents. Thus equal $t$-classes have the same restriction to the six global units. They also have the same residue modulo $4$, which controls the reciprocity f" }, { "id": "eq:sextic-sieve-pair-values", "kind": "anchor", "label": "eq:sextic-sieve-pair-values", "owner": "lem:sextic-large-sieve", "section": "sec:sextic-row-count", "anchor": { "source": "paper.tex", "line_start": 4816, "line_end": 4817, "char_start": 219000, "char_end": 219035 }, "argument_gist": "", "excerpt": "c reciprocity, with its factor fixed by $t(b)=t(c)$, gives for every good ideal $a$ \\Psi_{b,c}^{(j)}(a)=F_b(a)^j\\overline{F_c(a)^j}. For coprime inputs the two reciprocity factors cancel; on the other inputs both sides are zero. With the Gauss sums already defined in the arithmetic preliminaries, CRT using $z=cv+bw$ gives \\label{eq:sextic-sieve-pai" }, { "id": "eq:sextic-sieve-pair-gauss", "kind": "anchor", "label": "eq:sextic-sieve-pair-gauss", "owner": "lem:sextic-large-sieve", "section": "sec:sextic-row-count", "anchor": { "source": "paper.tex", "line_start": 4822, "line_end": 4823, "char_start": 219309, "char_end": 219343 }, "argument_gist": "", "excerpt": "uss sums already defined in the arithmetic preliminaries, CRT using $z=cv+bw$ gives \\Gamma_{b,c}^{(j)} &:=q_{bc}^{-1/2}\\sum_{z\\bmod bc}\\Theta_{b,c}^{(j)}(z)e(z/(bc))\\\\ &=\\chi_b(c)^j\\overline{\\chi_c(b)^j}\\gamma_j(b)\\gamma_{-j}(c) =\\eta_{t(b),j}\\gamma_j(b)\\gamma_{-j}(c), where $\\eta_{t(b),j}=\\mathcal R(b,c)^j$ depends only on the commo" }, { "id": "eq:sextic-sieve-transform-bounds", "kind": "anchor", "label": "eq:sextic-sieve-transform-bounds", "owner": "lem:sextic-large-sieve", "section": "sec:sextic-row-count", "anchor": { "source": "paper.tex", "line_start": 4842, "line_end": 4843, "char_start": 220194, "char_end": 220234 }, "argument_gist": "", "excerpt": "W(|y|^2). \\] The transform is radial, smooth at zero, and rapidly decreasing. Thus \\mathscr W(x)\\ll_A(1+x)^{-A},\\qquad \\int_{\\mathbb R}|\\mathcal M\\mathscr W(\\sigma+it)|\\,dt<\\infty \\quad(\\sigma>0). Here $\\mathcal M$ denotes the Mellin transform. The second assertion follows by integration by parts in the Mellin integral; all logarithmic derivatives are bou" }, { "id": "eq:sextic-sieve-pair-poisson", "kind": "anchor", "label": "eq:sextic-sieve-pair-poisson", "owner": "lem:sextic-large-sieve", "section": "sec:sextic-row-count", "anchor": { "source": "paper.tex", "line_start": 4868, "line_end": 4869, "char_start": 221448, "char_end": 221484 }, "argument_gist": "", "excerpt": "ummation, followed by division by the six generators of each ideal, therefore gives \\sum_{\\mathfrak a\\ne0}W(q_{\\mathfrak a}/M)\\Psi_{b,c}^{(j)}(\\mathfrak a) =\\frac{M\\Gamma_{b,c}^{(j)}}{\\sqrt{q_{bc}}} \\sum_{\\mathfrak a\\ne0}\\mathscr W(Mq_{\\mathfrak a}/q_{bc}) \\overline{\\Psi_{b,c}^{(j)}(\\mathfrak a)}. Both sums here are over all nonzero integral id" }, { "id": "eq:sextic-sieve-pair-norm", "kind": "anchor", "label": "eq:sextic-sieve-pair-norm", "owner": "lem:sextic-large-sieve", "section": "sec:sextic-row-count", "anchor": { "source": "paper.tex", "line_start": 4888, "line_end": 4889, "char_start": 222342, "char_end": 222375 }, "argument_gist": "", "excerpt": "\\Psi_{b,c}^{(j)}(\\mathfrak a). \\] We claim, for every $\\varepsilon>0$, |Q_j(M,N;\\lambda)| &\\ll_{S,\\varepsilon}(2+MN)^\\varepsilon\\|\\lambda\\|_2^2\\\\ &\\quad\\times \\left\\{M+\\frac MN \\max_{\\substack{1\\le H\\\\ H\\ll_{S,\\varepsilon}(2+MN)^\\varepsilon N^2/M}} E_j(H,N)\\right\\}. The maximum runs over dyadic shells and" }, { "id": "eq:sextic-sieve-norm-reflection", "kind": "anchor", "label": "eq:sextic-sieve-norm-reflection", "owner": "lem:sextic-large-sieve", "section": "sec:sextic-row-count", "anchor": { "source": "paper.tex", "line_start": 4991, "line_end": 4992, "char_start": 227121, "char_end": 227160 }, "argument_gist": "", "excerpt": "athrm r}\\le q_{\\mathfrak s_S}\\Delta_{\\mathrm d}$. We obtain the recursive interface E_j(M,N) &\\ll_{S,\\varepsilon}(2+MN)^\\varepsilon \\max_{\\substack{1\\le\\Delta_{\\mathrm d}\\le2N\\\\ 1\\le\\Delta_{\\mathrm r}\\le q_{\\mathfrak s_S}\\Delta_{\\mathrm d}}} \\left\\{\\frac M{\\Delta_{\\mathrm r}}\\right.\\\\ &\\qquad\\left.+ \\frac{M\\Delta_{\\mathrm d}}{N\\Delta_{\\mathrm" }, { "id": "eq:sextic-sieve-norm-symmetry", "kind": "anchor", "label": "eq:sextic-sieve-norm-symmetry", "owner": "lem:sextic-large-sieve", "section": "sec:sextic-row-count", "anchor": { "source": "paper.tex", "line_start": 5013, "line_end": 5014, "char_start": 228096, "char_end": 228133 }, "argument_gist": "", "excerpt": "and sextic reciprocity, splitting one matrix variable by its finite $t$-class, give D_j(M,N)\\ll_S D_j(N,M). The phases are unit-valued and the noncoprime entries are zero in both orientations. We also have the initial estimate D_j(M,N)\\ll_S M^2+N\\qquad(M,N\\ge1). Indeed, for the squarefree row $a$" }, { "id": "eq:sextic-sieve-norm-initial", "kind": "anchor", "label": "eq:sextic-sieve-norm-initial", "owner": "lem:sextic-large-sieve", "section": "sec:sextic-row-count", "anchor": { "source": "paper.tex", "line_start": 5018, "line_end": 5019, "char_start": 228310, "char_end": 228346 }, "argument_gist": "", "excerpt": "oncoprime entries are zero in both orientations. We also have the initial estimate D_j(M,N)\\ll_S M^2+N\\qquad(M,N\\ge1). Indeed, for the squarefree row $a$, finite Fourier inversion gives on every $b$, including nonunits, \\[ \\chi_a(b)^j=\\frac1{\\sqrt{q_a}\\gamma_{-j}(a)} \\sum_{x\\bmod a}\\overline{\\chi_a(x)^j}e(xb/a). \\] Cauchy costs at most $\\#(\\mathcal O/(a))^" }, { "id": "eq:sextic-sieve-norm-monotone", "kind": "anchor", "label": "eq:sextic-sieve-norm-monotone", "owner": "lem:sextic-large-sieve", "section": "sec:sextic-row-count", "anchor": { "source": "paper.tex", "line_start": 5038, "line_end": 5039, "char_start": 229274, "char_end": 229311 }, "argument_gist": "", "excerpt": "ge-sieve recursions of \\cite[Lemma~4.4 and Remark~5]{GL} and \\cite[Lemma~3.1]{BGL}: D_j(M_1,N)\\ll_S D_j(M_2,N) \\quad\\text{if }M_2\\ge C M_1\\log(2M_1N),\\quad M_1,M_2,N\\ge1. Here is a direct verification. The assertion is immediate for a zero norm. Otherwise choose a maximizing unit coefficient vector for $D_j(M_1,N)$, and split the row shell at $3M_1/2$; one part c" }, { "id": "eq:sextic-sieve-inductive-bound", "kind": "anchor", "label": "eq:sextic-sieve-inductive-bound", "owner": "lem:sextic-large-sieve", "section": "sec:sextic-row-count", "anchor": { "source": "paper.tex", "line_start": 5069, "line_end": 5070, "char_start": 230695, "char_end": 230734 }, "argument_gist": "", "excerpt": "3.4--3.5]{BGL}. Suppose, simultaneously for $j\\in J$, that for every positive loss D_j(M,N)\\ll_{S,\\varepsilon}(MN)^\\varepsilon \\{M^\\alpha+N+(MN)^{2/3}\\},\\qquad \\alpha>4/3, for $M,N\\ge1$; the unit shells satisfy the same bound directly. The initial estimate gives this with $\\alpha=2$. Write a good evaluation ideal uniquely as \\[ a=a_1a_2^2a_3^3a_4^4a_5^5a" }, { "id": "eq:sextic-sieve-norm-power", "kind": "anchor", "label": "eq:sextic-sieve-norm-power", "owner": "lem:sextic-large-sieve", "section": "sec:sextic-row-count", "anchor": { "source": "paper.tex", "line_start": 5093, "line_end": 5094, "char_start": 231780, "char_end": 231814 }, "argument_gist": "", "excerpt": "Counting the fixed ideals and forgetting their coprimalities in positive sums gives E_j(M,N)\\ll_{S,\\varepsilon}(MN)^\\varepsilon \\max_{\\boldsymbol X} \\min\\left\\{M^{1/3},(M/X_\\ell)^{2/3}\\right\\} D_{j\\ell\\bmod6}(X_\\ell,N). To verify the factor, the number of fixed choices is $O(X/X_\\ell)$. For $\\ell=1$ it is at most a constant times both $(M/X_1)^{1/2}\\" }, { "id": "eq:sextic-sieve-norm-recurrence", "kind": "anchor", "label": "eq:sextic-sieve-norm-recurrence", "owner": "lem:sextic-large-sieve", "section": "sec:sextic-row-count", "anchor": { "source": "paper.tex", "line_start": 5127, "line_end": 5128, "char_start": 233331, "char_end": 233370 }, "argument_gist": "", "excerpt": "(\\Delta_{\\mathrm r}\\Delta_{\\mathrm d})^{-2/3}\\le1. \\] Consequently, for $M,N\\ge1$, D_j(M,N)\\le E_j(M,N)\\ll_{S,\\varepsilon}(MN)^\\varepsilon \\{M+N^{2\\alpha-1}M^{1-\\alpha}+(MN)^{2/3}\\}. Put $h=2-3/(3\\alpha-1)=(2\\alpha-5/3)/(\\alpha-1/3)$ and $\\beta=2-2/(3\\alpha-1)$. If $M\\ge N^h$, the middle term in Equation~\\eqref{eq:sextic-sieve-norm-recurrence} is at most" }, { "id": "eq:sextic-sieve-norm-shell", "kind": "anchor", "label": "eq:sextic-sieve-norm-shell", "owner": "lem:sextic-large-sieve", "section": "sec:sextic-row-count", "anchor": { "source": "paper.tex", "line_start": 5153, "line_end": 5154, "char_start": 234589, "char_end": 234623 }, "argument_gist": "", "excerpt": "4/3}\\le(MN)^{2/3}$ when $M\\le N$ and the analogous inequality when $N\\le M$, proves D_j(M,N)\\ll_{S,\\varepsilon}(MN)^\\varepsilon \\{M+N+(MN)^{2/3}\\}\\qquad(j\\in J). Dyadic subdivision of both norm balls, with Cauchy across the column shells, converts Equation~\\eqref{eq:sextic-sieve-norm-shell} to the same bound for the ball matrix with entries $F_b(a)=" }, { "id": "eq:sextic-row-count", "kind": "anchor", "label": "eq:sextic-row-count", "owner": "prop:sextic-row-count", "section": "sec:sextic-row-count", "anchor": { "source": "paper.tex", "line_start": 5192, "line_end": 5193, "char_start": 236724, "char_end": 236751 }, "argument_gist": "", "excerpt": "losses in Proposition~\\ref{prop:detector-witness}, for all sufficiently large $Z$, \\#\\mathcal B\\ll_{\\mathcal A,\\epsilon} U^{R(\\delta)+\\epsilon}(1+T_1)^{A_{\\mathcal A}}, \\qquad R(\\delta)=\\min\\left\\{1, \\max\\left(1-\\frac\\delta2,\\frac43-\\delta\\right)\\right\\}. The finite order $A_{\\mathcal A}$ is uniform in the physical dyad, the bin, and all moving rows. It" }, { "id": "eq:sextic-witness-spike", "kind": "anchor", "label": "eq:sextic-witness-spike", "owner": "prop:sextic-row-count", "section": "sec:sextic-row-count", "anchor": { "source": "paper.tex", "line_start": 5228, "line_end": 5229, "char_start": 238420, "char_end": 238451 }, "argument_gist": "", "excerpt": "gma}W(q_n/D) \\] for the inverse polynomial of that presentation. The witness gives \\frac12-O(\\epsilon_0)\\le r\\le1+O(1/\\log U), \\qquad |M_u(D;W_u)|^2\\gg_{\\mathcal A,\\epsilon_0} U^{\\delta r-\\epsilon_0}. The notation $W_u$ retains the permitted rowwise norm-profile parameters. The polynomial here is the original one for the displayed presentation, not the pol" }, { "id": "eq:sextic-actual-row-annulus", "kind": "anchor", "label": "eq:sextic-actual-row-annulus", "owner": "prop:sextic-row-count", "section": "sec:sextic-row-count", "anchor": { "source": "paper.tex", "line_start": 5263, "line_end": 5264, "char_start": 240038, "char_end": 240074 }, "argument_gist": "", "excerpt": "such ranges. The original annulus $q_u\\asymp U$ gives the actual residual annulus q_{\\mathfrak k}=\\frac{q_u}{q_{\\mathfrak s}q_{\\mathfrak w}} \\asymp_{\\mathcal A}\\frac UV. We keep this restriction when forming the sum. It will be enlarged to the ball $q_k\\ll_{\\mathcal A}U/V$ only when applying a positive row bound. Let $\\mathcal U_V$ denote the rows of the" }, { "id": "eq:sextic-stratum-cardinality", "kind": "anchor", "label": "eq:sextic-stratum-cardinality", "owner": "prop:sextic-row-count", "section": "sec:sextic-row-count", "anchor": { "source": "paper.tex", "line_start": 5289, "line_end": 5290, "char_start": 241340, "char_end": 241377 }, "argument_gist": "", "excerpt": ")$ and its unit are unique, the number of rows in this powerful-part dyad satisfies \\#\\mathcal U_V\\ll_{\\mathcal A}U V^{-1/2}. Discarding any of the restrictions defining $\\mathcal U_V$ in this upper count can only add rows. For fixed $\\mathfrak w,\\mathfrak s,\\upsilon$, put $v=\\upsilon s_{\\mathfrak s}w$. Multiplicativity in the numerator gives the exact identity \\" }, { "id": "eq:sextic-frozen-mask", "kind": "anchor", "label": "eq:sextic-frozen-mask", "owner": "prop:sextic-row-count", "section": "sec:sextic-row-count", "anchor": { "source": "paper.tex", "line_start": 5298, "line_end": 5299, "char_start": 241692, "char_end": 241721 }, "argument_gist": "", "excerpt": "ilon s_{\\mathfrak s}w$. Multiplicativity in the numerator gives the exact identity \\chi_n(u)^{\\varsigma} =\\chi_n(v)^{\\varsigma}\\chi_n(k)^{\\varsigma}. It includes the zero extensions: if $n$ meets either factor, both sides are zero, and otherwise it is ordinary multiplicativity. In particular, the inverse polynomial for a fixed profile $W$ is the row sum in Lem" }, { "id": "eq:sextic-stratum-fixed-moment", "kind": "anchor", "label": "eq:sextic-stratum-fixed-moment", "owner": "prop:sextic-row-count", "section": "sec:sextic-row-count", "anchor": { "source": "paper.tex", "line_start": 5333, "line_end": 5334, "char_start": 243464, "char_end": 243502 }, "argument_gist": "", "excerpt": "ces of $\\mathfrak w$ and the finitely many choices of $\\mathfrak s,\\upsilon$ yields \\sum_{u\\in\\mathcal U_V}|M_u(D;W)|^2 \\ll_{\\mathcal A,\\epsilon_1}(UD)^{\\epsilon_1}V^{1/2} \\left\\{\\frac UV+D+\\left(\\frac{UD}{V}\\right)^{2/3}\\right\\}. All coefficient sequences in this derivation are fixed within the row sum to which the sieve is applied. They may differ for differen" }, { "id": "eq:sextic-stratum-moment", "kind": "anchor", "label": "eq:sextic-stratum-moment", "owner": "prop:sextic-row-count", "section": "sec:sextic-row-count", "anchor": { "source": "paper.tex", "line_start": 5363, "line_end": 5364, "char_start": 245129, "char_end": 245161 }, "argument_gist": "", "excerpt": "sextic-stratum-fixed-moment} therefore applies to every such integral. This proves \\sum_{u\\in\\mathcal U_V}|M_u(D;W_u)|^2 \\ll_{\\mathcal A,\\epsilon_1}(UD)^{\\epsilon_1}(1+T_1)^{A_{\\mathcal A}} V^{1/2}\\left\\{\\frac UV+D+ \\left(\\frac{UD}{V}\\right)^{2/3}\\right\\}. The derivative order and hence $A_{\\mathcal A}$ are fixed before choosing $\\tau$. The height interv" }, { "id": "eq:sextic-stratum-envelope", "kind": "anchor", "label": "eq:sextic-stratum-envelope", "owner": "prop:sextic-row-count", "section": "sec:sextic-row-count", "anchor": { "source": "paper.tex", "line_start": 5388, "line_end": 5389, "char_start": 246424, "char_end": 246458 }, "argument_gist": "", "excerpt": "so using Equation~\\eqref{eq:sextic-stratum-cardinality}, gives for this subdivision \\#\\mathcal U_V \\ll U^{\\min\\{1-o/2,e_o(r)-\\delta r\\}+O(\\epsilon_0)+\\epsilon_1} (1+T_1)^{A_{\\mathcal A}}, after decreasing the sieve loss denoted by $\\epsilon_1$ if necessary. We compute the largest exponent in this expression first on $0\\le o\\le1$ and $1/2\\le r\\le1$. Put $x" }, { "id": "eq:sextic-envelope-optimization", "kind": "anchor", "label": "eq:sextic-envelope-optimization", "owner": "prop:sextic-row-count", "section": "sec:sextic-row-count", "anchor": { "source": "paper.tex", "line_start": 5421, "line_end": 5422, "char_start": 247648, "char_end": 247687 }, "argument_gist": "", "excerpt": "wo endpoints. It follows, and equality is attained at one of those endpoints, that \\max_{\\substack{0\\le o\\le1\\\\1/2\\le r\\le1}} \\min\\{1-o/2,e_o(r)-\\delta r\\} =\\min\\left\\{1,\\max\\left(1-\\frac\\delta2,\\frac43-\\delta\\right)\\right\\}. This computation is uniform for $0\\le\\delta\\le1$. The functions just used are maxima and minima of affine functions whose slopes are unif" }, { "id": "eq:shared-principal-data", "kind": "anchor", "label": "eq:shared-principal-data", "owner": "sec:shared-analytic-estimates", "section": "sec:shared-analytic-estimates", "anchor": { "source": "paper.tex", "line_start": 5523, "line_end": 5524, "char_start": 252439, "char_end": 252471 }, "argument_gist": "", "excerpt": "),\\qquad c_S=\\widehat W_1(1)M(1/6) \\left(\\operatorname*{Res}_{v=1}\\zeta_F^S(v)\\right)^2/6. Lemma~\\ref{lem:local-euler} makes $H_\\eta$ holomorphic on $\\Re s>7/8$. Its uniform estimate $H_\\eta=1+O(P_0^{-4/5})$ permits a choice of $P_0$ before the target, because the local bound is uniform in the target unit phases. Fix $P_0$ large enough that \\sup_{\\Re s>7" }, { "id": "eq:shared-principal-nonvanishing", "kind": "anchor", "label": "eq:shared-principal-nonvanishing", "owner": "sec:shared-analytic-estimates", "section": "sec:shared-analytic-estimates", "anchor": { "source": "paper.tex", "line_start": 5531, "line_end": 5532, "char_start": 252800, "char_end": 252840 }, "argument_gist": "", "excerpt": "it phases. Fix $P_0$ large enough that \\sup_{\\Re s>7/8}|H_\\eta(s)-1|\\le\\frac12. Enlarging $S$ by these primes leaves the fixed ray group $T$ unchanged, as the calibration in the local identity holds coefficientwise. The excluded set and hence the particular function $H_\\eta$ may differ between applications, but within one application they are exactly those of t" }, { "id": "eq:stage-mellin-exponent", "kind": "anchor", "label": "eq:stage-mellin-exponent", "owner": "def:stage-high-data", "section": "sec:shared-analytic-estimates", "anchor": { "source": "paper.tex", "line_start": 5590, "line_end": 5591, "char_start": 255339, "char_end": 255371 }, "argument_gist": "", "excerpt": "h=1-l_x+\\ell>0, \\qquad C(s)=s+\\frac{l_x}{2}-1+\\frac h6 =s-\\frac56+\\frac{l_x}{3}+\\frac\\ell6. These real numbers are fixed before the target character. For each primitive finite-order target $\\eta$, fix the data $S,\\xi,W_0,W_1$ of Section~\\ref{sec:probe}, independently of $Z$. Data for an exact high representation consist of the following objects and assertions. First, $\\m" }, { "id": "eq:stage-all-height-majorant", "kind": "anchor", "label": "eq:stage-all-height-majorant", "owner": "def:stage-high-data", "section": "sec:shared-analytic-estimates", "anchor": { "source": "paper.tex", "line_start": 5619, "line_end": 5620, "char_start": 256902, "char_end": 256938 }, "argument_gist": "", "excerpt": "eta,\\mathcal K,D} Z^{B_{\\eta,\\mathcal K,D}} (1+|\\Im s|+|\\Im w|+|\\Im z|)^{J_{\\eta,\\mathcal K,D}}. The constants and exponents are independent of $Z,u$ and of any later order of integration by parts. No nonvanishing of $\\mathfrak H$ or of any of its local factors is assumed. Finally, the following identity is asserted for the independently given $\\mathscr I_\\eta(Z)$, with absolu" }, { "id": "eq:stage-high-identity", "kind": "anchor", "label": "eq:stage-high-identity", "owner": "def:stage-high-data", "section": "sec:shared-analytic-estimates", "anchor": { "source": "paper.tex", "line_start": 5639, "line_end": 5640, "char_start": 257625, "char_end": 257655 }, "argument_gist": "", "excerpt": "{L^S(s,\\eta\\overline{\\chi_\\bullet(u)})} \\mathfrak H_{\\eta,u,Z}(s,w,z)\\,dz\\,dw\\,ds, where the sum has the physical restrictions of Equation~\\eqref{eq:probe-poisson-identity}, and \\[ \\mathcal W(X,Y,Z;s,w,z) =X^{1/2-z}Z^{s+z-1}Y^{w-1} e^{(s+z-1)^2}M(z)\\widehat W_1(w). \\] Thus Equation~\\eqref{eq:stage-high-identity} is a hypothesis to be proved for the physica" }, { "id": "eq:external-arithmetic-majorant", "kind": "anchor", "label": "eq:external-arithmetic-majorant", "owner": "lem:external-contour-tails", "section": "sec:shared-analytic-estimates", "anchor": { "source": "paper.tex", "line_start": 5688, "line_end": 5689, "char_start": 259817, "char_end": 259856 }, "argument_gist": "", "excerpt": "Z$ satisfies on $\\mathcal D$ |F_Z(\\boldsymbol t)|\\le C Z^B(1+|\\boldsymbol t|)^J, where $B,J,C$ are fixed independently of $Z$ and of the integer $N$ below. Write $\\widehat{\\boldsymbol t}_j$ for the vector obtained by omitting the $j$th coordinate of $\\boldsymbol t$. Uniformly for $T\\ge1$ and $\\boldsymbol r\\in\\mathcal K$, \\int_{\\mathcal D\\cap\\{\\max" }, { "id": "eq:external-integrated-tail", "kind": "anchor", "label": "eq:external-integrated-tail", "owner": "lem:external-contour-tails", "section": "sec:shared-analytic-estimates", "anchor": { "source": "paper.tex", "line_start": 5698, "line_end": 5698, "char_start": 260262, "char_end": 260297 }, "argument_gist": "", "excerpt": "|>T\\}} K_{\\boldsymbol r,Z}(\\boldsymbol t)|F_Z(\\boldsymbol t)|\\,d\\boldsymbol t &\\ll_N Z^B T^{-N}, \\\\ \\int_{\\mathcal D\\cap\\{t_j=T\\}} K_{\\boldsymbol r,Z}(\\boldsymbol t)|F_Z(\\boldsymbol t)|\\, d\\widehat{\\boldsymbol t}_j &\\ll_N Z^B(1+|T|)^{-N}. The second assertion holds for either sign of $T$ and every $j$, using coordinate Lebesgue measure on $t_j=T$." }, { "id": "eq:external-trace-tail", "kind": "anchor", "label": "eq:external-trace-tail", "owner": "lem:external-contour-tails", "section": "sec:shared-analytic-estimates", "anchor": { "source": "paper.tex", "line_start": 5703, "line_end": 5704, "char_start": 260444, "char_end": 260474 }, "argument_gist": "", "excerpt": "bol r,Z}(\\boldsymbol t)|F_Z(\\boldsymbol t)|\\, d\\widehat{\\boldsymbol t}_j &\\ll_N Z^B(1+|T|)^{-N}. The second assertion holds for either sign of $T$ and every $j$, using coordinate Lebesgue measure on $t_j=T$. Both assertions remain valid after integration over a real contour interval of bounded length when the hypotheses are uniform there and the kernels, domain indicators and arit" }, { "id": "eq:external-two-dimensional-kernel", "kind": "anchor", "label": "eq:external-two-dimensional-kernel", "owner": "sec:shared-analytic-estimates", "section": "sec:shared-analytic-estimates", "anchor": { "source": "paper.tex", "line_start": 5750, "line_end": 5751, "char_start": 262337, "char_end": 262379 }, "argument_gist": "", "excerpt": "2)}_{\\boldsymbol r,Z}(\\boldsymbol t) =e^{-y_1^2}|M(r_z+iy_2)|, \\qquad(y_1,y_2)=(\\Im(s+z),\\Im z). This supplies the trace bound needed for the subsequent $z$-join. The same kernel test handles additional external separating variables. For $m\\ge3$, let $\\boldsymbol y=A\\boldsymbol t$ with a fixed $A\\in\\mathrm{GL}_m(\\mathbb R)$, and multiply the three-factor kernel by a Borel densi" }, { "id": "eq:external-density-contract", "kind": "anchor", "label": "eq:external-density-contract", "owner": "sec:shared-analytic-estimates", "section": "sec:shared-analytic-estimates", "anchor": { "source": "paper.tex", "line_start": 5763, "line_end": 5764, "char_start": 262944, "char_end": 262980 }, "argument_gist": "", "excerpt": "\\boldsymbol v|)^L|v_{\\boldsymbol r,Z}(\\boldsymbol v)|<\\infty \\quad\\text{for every integer }L\\ge0. For $m=3$ the density is one on the zero-dimensional space. The product is again rapidly decreasing in all $m$ coordinates, since $A$ and its inverse are fixed. For the two-factor kernel, the identical argument uses a density on $\\mathbb R^{m-2}$ and $m\\ge2$, with density one when" }, { "id": "eq:stage-central-contours", "kind": "anchor", "label": "eq:stage-central-contours", "owner": "lem:fixed-bin-contour", "section": "sec:shared-analytic-estimates", "anchor": { "source": "paper.tex", "line_start": 5828, "line_end": 5829, "char_start": 266368, "char_end": 266401 }, "argument_gist": "", "excerpt": "on these retained segments of \\Re s=a+16e,\\qquad \\Re w=1-a-6e,\\qquad \\Re z=z_0, up to $O_{\\eta,N}(Z^{B_\\eta}T_1^{-N})$ for every fixed $N$. The finite exponent $B_\\eta$ is fixed before $N$. An empty row set contributes zero. The transformation uses the full holomorphic correction; no decomposition of that correction or subdivision depending on a contour point" }, { "id": "eq:balanced-central-contribution", "kind": "anchor", "label": "eq:balanced-central-contribution", "owner": "sec:shared-analytic-estimates", "section": "sec:shared-analytic-estimates", "anchor": { "source": "paper.tex", "line_start": 5960, "line_end": 5961, "char_start": 272636, "char_end": 272676 }, "argument_gist": "", "excerpt": "+E_{\\mathrm I}(d)+13e +(1+d)\\varepsilon_c+d\\varepsilon_d+\\varepsilon_p} (1+T_1)^{A'_\\eta}. Here we have harmlessly enlarged the bound by $Z^{\\varepsilon_c+\\varepsilon_p}$ so that the losses use the same form as the later general estimate, and \\[ E_{\\mathrm I}(d)=-\\frac34+\\frac{\\delta+R(\\delta)}2 +\\left(d-\\frac12\\right) \\left(R(\\delta)+\\frac\\delta2-\\frac{17}{50}\\ri" }, { "id": "eq:stage-central-bound", "kind": "anchor", "label": "eq:stage-central-bound", "owner": "lem:bin-contour-accounting", "section": "sec:shared-analytic-estimates", "anchor": { "source": "paper.tex", "line_start": 6054, "line_end": 6055, "char_start": 276527, "char_end": 276557 }, "argument_gist": "", "excerpt": "\\right| \\ll_\\eta U^{R+\\delta/2+\\varepsilon_c} Z^{\\ell(z_0-1/2)+g+\\varepsilon_c}(1+T_1)^{A_\\eta}. The bound is uniform in the retained point, row set and moving labels. The number $\\varepsilon_c$ and the bounded range for $R,g$ are fixed before the target; $R,g$ may depend on $d,a$, the pointwise set and the global number $\\beta_*$, but not otherwise on $\\eta$. Let $\\mathcal P_Z" }, { "id": "eq:stage-height-allocation", "kind": "anchor", "label": "eq:stage-height-allocation", "owner": "lem:bin-contour-accounting", "section": "sec:shared-analytic-estimates", "anchor": { "source": "paper.tex", "line_start": 6074, "line_end": 6075, "char_start": 277572, "char_end": 277606 }, "argument_gist": "", "excerpt": "e(3i+1)T_1$. Restrict |v_k|\\le c_kT_1,\\qquad c_k=\\frac1{2m(1+\\max_j|a_{jk}|)}. Coordinates with zero coefficients may be restricted by the same rule. Then every added height is at most $T_1/2$ in total. Fixed bounded enlargements are included by increasing the lower threshold for $Z$. If extra Mellin integrals are used to establish Equation~\\eqref{eq:stage-cen" }, { "id": "eq:stage-high-exponent", "kind": "anchor", "label": "eq:stage-high-exponent", "owner": "lem:bin-contour-accounting", "section": "sec:shared-analytic-estimates", "anchor": { "source": "paper.tex", "line_start": 6092, "line_end": 6093, "char_start": 278467, "char_end": 278497 }, "argument_gist": "", "excerpt": "n} E_{\\sigma_0}(d;R,g)=a-\\sigma_0+h(z_0-1/6)-a l_y-\\ell/2+g +d(R+\\delta/2-z_0). If $\\mathcal B$ is empty its contribution is zero. Otherwise put \\[ E^{\\max}_{\\sigma_0,Z}(d) =\\sup_{(R,g)\\in\\mathcal P_Z}E_{\\sigma_0}(d;R,g). \\] For every $\\varepsilon_d,\\varepsilon_p>0$, the total contribution of $\\mathcal B$ on the central contours is, for sufficiently large $" }, { "id": "eq:stage-central-contribution", "kind": "anchor", "label": "eq:stage-central-contribution", "owner": "lem:bin-contour-accounting", "section": "sec:shared-analytic-estimates", "anchor": { "source": "paper.tex", "line_start": 6107, "line_end": 6108, "char_start": 278999, "char_end": 279036 }, "argument_gist": "", "excerpt": "&\\cdot Z^{(1+d)\\varepsilon_c+d\\varepsilon_d+\\varepsilon_p} (1+T_1)^{A_\\eta}. The supremum is finite because the pairs lie in a fixed bounded range. The power $\\varepsilon_p$ absorbs the pointwise logarithmic multiplicity. No separate integral for a pointwise piece or row set is asserted. All discarded portions and horizontal joins in this contour move are $O_" }, { "id": "eq:principal-correction-bound", "kind": "anchor", "label": "eq:principal-correction-bound", "owner": "lem:principal-residue-interface", "section": "sec:shared-analytic-estimates", "anchor": { "source": "paper.tex", "line_start": 6189, "line_end": 6190, "char_start": 283076, "char_end": 283113 }, "argument_gist": "", "excerpt": "\\ll_{\\eta,e,\\epsilon} Z^{\\ell\\Re z+\\epsilon} (1+|\\Im s|+|\\Im w|+|\\Im z|)^{J_{\\eta,e,\\epsilon}}, with finite $J_{\\eta,e,\\epsilon}$ fixed before any external tail order. Suppose also that $A_\\eta(Z)\\ne0$ for every sufficiently large real $Z$, that $|A_\\eta(Z)|^{-1}\\ll_{\\eta,\\epsilon}Z^\\epsilon$ for every $\\epsilon>0$, and that on $\\Re s=\\beta_*+e$, uniformly in $\\Im s$, \\begin{eq" }, { "id": "eq:principal-correction-residue", "kind": "anchor", "label": "eq:principal-correction-residue", "owner": "lem:principal-residue-interface", "section": "sec:shared-analytic-estimates", "anchor": { "source": "paper.tex", "line_start": 6198, "line_end": 6199, "char_start": 283513, "char_end": 283552 }, "argument_gist": "", "excerpt": "uation} \\mathfrak H_{\\eta,1,Z}(s,1,1/6) =H_\\eta(s)Z^{\\ell/6}A_\\eta(Z)(1+\\mathcal R_{\\eta,Z}(s)). Here either $\\mathcal R_{\\eta,Z}$ is identically zero, or there is a number $\\mu>0$, chosen independently of $\\eta$, such that $|\\mathcal R_{\\eta,Z}(s)|\\ll_\\eta Z^{-\\mu}$ on that entire line. Let $\\mathscr P_\\eta(Z)$ be the $u=1$ term of Equation~\\eqref{eq:stage-high-identity}, and" }, { "id": "eq:principal-remainder-bound", "kind": "anchor", "label": "eq:principal-remainder-bound", "owner": "lem:principal-residue-interface", "section": "sec:shared-analytic-estimates", "anchor": { "source": "paper.tex", "line_start": 6217, "line_end": 6218, "char_start": 284213, "char_end": 284249 }, "argument_gist": "", "excerpt": "1+h)e-l_y/20+\\epsilon} +Z^{C(\\beta_*)+e-h/600+\\epsilon}\\notag\\\\ &+Z^{C(\\beta_*)+e-\\mu+\\epsilon}. The last term is omitted when $\\mathcal R_{\\eta,Z}$ is identically zero. The implied constants and lower thresholds may depend on the target, but the displayed real exponents do not. For $u=1$, the scalar factor in the high identity is \\[ \\frac{\\zeta_F^S(6z)\\" }, { "id": "eq:small-correction-bound", "kind": "anchor", "label": "eq:small-correction-bound", "owner": "lem:outer-row-tails", "section": "sec:shared-analytic-estimates", "anchor": { "source": "paper.tex", "line_start": 6359, "line_end": 6360, "char_start": 290860, "char_end": 290893 }, "argument_gist": "", "excerpt": "e,\\epsilon} Z^{\\ell z_0+\\epsilon}q_u^\\epsilon (1+|\\Im s|+|\\Im w|+|\\Im z|)^{J_{\\eta,e,\\epsilon}}. Then, for every $\\epsilon>0$ and $1\\le U\\le Z^{d_{\\min}}$, |\\mathscr R_\\eta(U;Z)| \\ll_{\\eta,e,\\epsilon} Z^{C(\\beta_*)+h(z_0-1/6)-l_y/2+e+\\epsilon} U^{63/50+\\epsilon}. In particular the sum of these dyads is \\begin{" }, { "id": "eq:small-row-bound", "kind": "anchor", "label": "eq:small-row-bound", "owner": "lem:outer-row-tails", "section": "sec:shared-analytic-estimates", "anchor": { "source": "paper.tex", "line_start": 6367, "line_end": 6368, "char_start": 291104, "char_end": 291130 }, "argument_gist": "", "excerpt": ";Z)| \\ll_{\\eta,e,\\epsilon} Z^{C(\\beta_*)+h(z_0-1/6)-l_y/2+e+\\epsilon} U^{63/50+\\epsilon}. In particular the sum of these dyads is \\ll_{\\eta,e,\\epsilon} Z^{C(\\beta_*)+h(z_0-1/6)-l_y/2+e+(63/50)d_{\\min}+\\epsilon}. For the other end, suppose that for every fixed $v>2$ and $\\epsilon>0$, for every physical row, unifo" }, { "id": "eq:small-row-sum", "kind": "anchor", "label": "eq:small-row-sum", "owner": "lem:outer-row-tails", "section": "sec:shared-analytic-estimates", "anchor": { "source": "paper.tex", "line_start": 6373, "line_end": 6374, "char_start": 291288, "char_end": 291312 }, "argument_gist": "", "excerpt": "egin{equation} \\ll_{\\eta,e,\\epsilon} Z^{C(\\beta_*)+h(z_0-1/6)-l_y/2+e+(63/50)d_{\\min}+\\epsilon}. For the other end, suppose that for every fixed $v>2$ and $\\epsilon>0$, for every physical row, uniformly in all imaginary parts on $(\\Re s,\\Re w,\\Re z)=(2,2,v)$, |\\mathfrak H_{\\eta,u,Z}(s,w,z)| \\ll_{\\eta,v,\\epsilon} Z^{\\ell v+\\epsilon}q_u^\\epsilon (1+|\\Im s|+|\\" }, { "id": "eq:large-correction-bound", "kind": "anchor", "label": "eq:large-correction-bound", "owner": "lem:outer-row-tails", "section": "sec:shared-analytic-estimates", "anchor": { "source": "paper.tex", "line_start": 6384, "line_end": 6385, "char_start": 291651, "char_end": 291684 }, "argument_gist": "", "excerpt": "a,v,\\epsilon} Z^{\\ell v+\\epsilon}q_u^\\epsilon (1+|\\Im s|+|\\Im w|+|\\Im z|)^{J_{\\eta,v,\\epsilon}}. Write $B_0=l_x/2+1+l_y$. Then, for every dyad $U$, |\\mathscr R_\\eta(U;Z)| \\ll_{\\eta,v,\\epsilon}Z^{B_0+hv+\\epsilon}U^{1+\\epsilon-v}. For every fixed $\\zeta>0$, if $\\epsilonZ^{h+\\zeta}$ is \\begin{equa" }, { "id": "eq:large-row-bound", "kind": "anchor", "label": "eq:large-row-bound", "owner": "lem:outer-row-tails", "section": "sec:shared-analytic-estimates", "anchor": { "source": "paper.tex", "line_start": 6390, "line_end": 6391, "char_start": 291853, "char_end": 291879 }, "argument_gist": "", "excerpt": "begin{equation} |\\mathscr R_\\eta(U;Z)| \\ll_{\\eta,v,\\epsilon}Z^{B_0+hv+\\epsilon}U^{1+\\epsilon-v}. For every fixed $\\zeta>0$, if $\\epsilonZ^{h+\\zeta}$ is \\ll_{\\eta,v,\\epsilon} Z^{B_0+(h+\\zeta)(1+\\epsilon)-\\zeta v+\\epsilon}. Thus a fixed sufficiently large $v$ makes this last contribution smaller tha" }, { "id": "eq:large-row-sum", "kind": "anchor", "label": "eq:large-row-sum", "owner": "lem:outer-row-tails", "section": "sec:shared-analytic-estimates", "anchor": { "source": "paper.tex", "line_start": 6397, "line_end": 6398, "char_start": 292063, "char_end": 292087 }, "argument_gist": "", "excerpt": "zeta}$ is \\ll_{\\eta,v,\\epsilon} Z^{B_0+(h+\\zeta)(1+\\epsilon)-\\zeta v+\\epsilon}. Thus a fixed sufficiently large $v$ makes this last contribution smaller than any prescribed power of $Z$. Every row under consideration has nonprincipal numerator by the classification preceding Lemma~\\ref{lem:principal-residue-interface}. The denominator" }, { "id": "eq:late-height-hypothesis", "kind": "anchor", "label": "eq:late-height-hypothesis", "owner": "lem:late-height-closure", "section": "sec:eleven-twelfths-conclusion", "anchor": { "source": "paper.tex", "line_start": 6536, "line_end": 6537, "char_start": 298654, "char_end": 298687 }, "argument_gist": "", "excerpt": "|J_\\eta(Z)-f_\\eta(Z)| \\ll_{\\eta,N} Z^{C(\\beta_*)-m}(1+T_1)^{A_\\eta} +Z^{B_\\eta}T_1^{-N}, where $A_\\eta\\ge0$ and $B_\\eta$ are finite and independent of $N$. Suppose there is a number $\\tau_{0,\\eta}>0$, fixed after the real choices and the finite profile orders for the target but before $N$ and $Z$, such that the estimate is valid for sufficiently large $Z$ whenever $T_1=Z" }, { "id": "eq:hecke-dirichlet-factorization", "kind": "anchor", "label": "eq:hecke-dirichlet-factorization", "owner": "prop:hecke-dirichlet-transfer", "section": "sec:eleven-twelfths-conclusion", "anchor": { "source": "paper.tex", "line_start": 6750, "line_end": 6751, "char_start": 308385, "char_end": 308425 }, "argument_gist": "", "excerpt": "thcal S_{\\mathbb Q}}(s,\\eta) =L^{S_{\\mathbb Q}}(s,\\chi) L^{S_{\\mathbb Q}}(s,\\chi\\chi_{-3}). The product character on the right may be imprimitive; deletion of $S_{\\mathbb Q}$ makes the identity independent of that choice. Every deleted factor is nonzero for $\\Re s>0$. Both Dirichlet factors are holomorphic in $0<\\Re s<1$. A zero of $L(s,\\chi)$ there would therefore give" }, { "id": "part:seven-eighths", "kind": "anchor", "label": "part:seven-eighths", "owner": "sec:eleven-twelfths-conclusion", "section": "sec:eleven-twelfths-conclusion", "anchor": { "source": "paper.tex", "line_start": 6808, "line_end": 6809, "char_start": 311072, "char_end": 311098 }, "argument_gist": "", "excerpt": "rincipal pole. This proves the theorem. \\part{The seven-eighths zero-free half-plane} \\section{The compensated probe} Theorem~\\ref{thm:eleven-twelfths} applies to every primitive character entering the supremum in Equation~\\eqref{old-eq:1.1b}, and therefore gives $\\beta_*\\le11/12$. Suppose for contradiction throughout Part~II that $\\beta_*>7/8$, and put \\b" }, { "id": "eq:part-II-bootstrap", "kind": "anchor", "label": "eq:part-II-bootstrap", "owner": "sec:compensation", "section": "sec:compensation", "anchor": { "source": "paper.tex", "line_start": 6816, "line_end": 6817, "char_start": 311412, "char_end": 311440 }, "argument_gist": "", "excerpt": "le11/12$. Suppose for contradiction throughout Part~II that $\\beta_*>7/8$, and put \\Delta:=\\beta_*-\\frac78,\\qquad 0<\\Delta\\le\\frac1{24}, \\qquad \\kappa:=2\\beta_*-1=\\frac34+2\\Delta\\le\\frac56. The exact bin ceiling in Section~\\ref{sec:detector} applies, since $\\beta_*>7/8>51/100$. For every retained nonprincipal row and its bin $(i,a)$ it gives \\la" }, { "id": "eq:part-II-bin-ceiling", "kind": "anchor", "label": "eq:part-II-bin-ceiling", "owner": "sec:compensation", "section": "sec:compensation", "anchor": { "source": "paper.tex", "line_start": 6824, "line_end": 6825, "char_start": 311737, "char_end": 311767 }, "argument_gist": "", "excerpt": "ta_*>7/8>51/100$. For every retained nonprincipal row and its bin $(i,a)$ it gives a\\le\\beta_*,\\qquad \\delta=2a-1\\le\\kappa\\le\\frac56. Indeed, the finite set defining $M_i(u)$ consists of the floor $51/100$ and real parts of actual zeros, all at most $\\beta_*$. This uses only the definition of the supremum, not its attainment. The boundary case $\\delta=5/6$ remai" }, { "id": "eq:part-II-mellin-target", "kind": "anchor", "label": "eq:part-II-mellin-target", "owner": "sec:compensation", "section": "sec:compensation", "anchor": { "source": "paper.tex", "line_start": 6834, "line_end": 6835, "char_start": 312210, "char_end": 312242 }, "argument_gist": "", "excerpt": "plication of Proposition~\\ref{lem:continuation-criterion}, write throughout Part~II C(s)=C_{\\mathrm{II}}(s):=s-\\frac{11}{16},\\qquad C(7/8)=\\frac3{16}. Our task is to construct a normalized probe $J_{\\mathrm{II},\\eta}$ from the finite expression below. It is distinct from the balanced probe $J_{\\mathrm I,\\eta}$. Its low-side target is $|J_{\\mathrm{II},\\eta}(Z)|\\l" }, { "id": "old-eq:5.1", "kind": "anchor", "label": "old-eq:5.1", "owner": "sec:compensation", "section": "sec:compensation", "anchor": { "source": "paper.tex", "line_start": 6866, "line_end": 6867, "char_start": 313624, "char_end": 313642 }, "argument_gist": "", "excerpt": "{23}{48},\\\\ X=Z^{l_x},\\qquad Y=Z^{l_y},\\qquad M=l_x+l_y=\\frac56,\\qquad M+\\ell=1. Fix positive slot lengths $\\ell_1,\\ldots,\\ell_K$ of total length $\\ell$, and put $P_i=Z^{\\ell_i}$. Each physical slot has a nonnegative, nonzero smooth annular weight $W_i(q_p/P_i)$. Its allowed prime set is \\[ \\mathcal P_i(Z)=\\{p\\text{ prime}:p\\notin S,\\ p\\in1_T," }, { "id": "eq:compensated-probe-definition", "kind": "anchor", "label": "eq:compensated-probe-definition", "owner": "sec:compensation", "section": "sec:compensation", "anchor": { "source": "paper.tex", "line_start": 6894, "line_end": 6895, "char_start": 315058, "char_end": 315097 }, "argument_gist": "", "excerpt": "zed probe $J_{\\mathrm{II},\\eta}$. Define the modified probe by the finite identity I_{\\eta,\\mathrm{modified}}(Z) =\\sum_{(p_i)\\in\\prod_i\\mathcal P_i(Z)}\\prod_{i=1}^K W_i(q_{p_i}/P_i) \\sum_{J\\subseteq\\{1,\\ldots,K\\}}&(-1)^{|J|}q_{p_J}^{-3/2} \\overline{\\eta(p_{J^c})}\\\\ &\\cdot I_{\\eta;p_{J^c}} (X/q_{p_J},Y/q_{p_J},Zq_{p_{J^c}}). \\end{equa" }, { "id": "old-eq:1.2", "kind": "anchor", "label": "old-eq:1.2", "owner": "sec:additional-arithmetic", "section": "sec:additional-arithmetic", "anchor": { "source": "paper.tex", "line_start": 6937, "line_end": 6938, "char_start": 317092, "char_end": 317110 }, "argument_gist": "", "excerpt": "egin{equation} Q_{\\psi,i}=Z^{-z_i/2}\\sum_{p\\ {\\rm prime}} \\psi(p)\\nu_i(p)W_i(q_p/Z^{z_i}). Here $\\nu_i$ is a fixed finite-ray character or a fixed finite linear combination of such characters. It is independent of the row and of the other selected primes. Distinct slots have disjoint underlying prime supports before any common mask or row zero extension is imposed. For" }, { "id": "old-eq:1.3", "kind": "anchor", "label": "old-eq:1.3", "owner": "sec:additional-arithmetic", "section": "sec:additional-arithmetic", "anchor": { "source": "paper.tex", "line_start": 6955, "line_end": 6956, "char_start": 318120, "char_end": 318138 }, "argument_gist": "", "excerpt": "e rows are elements $00\\), the number of tupl" }, { "id": "eq:marked-completion", "kind": "anchor", "label": "eq:marked-completion", "owner": "sec:marked", "section": "sec:marked", "anchor": { "source": "paper.tex", "line_start": 7318, "line_end": 7319, "char_start": 335218, "char_end": 335246 }, "argument_gist": "", "excerpt": "mooth annular \\(W\\), put \\(V_*(y)=y^{1/2}W(y)\\) and define the completed marked sum \\mathcal T_{\\mathfrak d}(X;k)= \\sum_{\\substack{n\\ {\\rm sf}\\\\b}} \\frac{\\bar\\alpha(n)\\gamma_2(n)\\Psi_k(n)\\, \\bar\\alpha(b)^3\\Psi_k(b)^3}{\\sqrt{q_n}\\,q_b} \\mathfrak d(nb^3)V_*(q_nq_b^3/X). Both primal ideals avoid \\(S\\); they may share primes. The mark belongs to the whole in" }, { "id": "par:reflection-marks", "kind": "anchor", "label": "par:reflection-marks", "owner": "sec:marked", "section": "sec:marked", "anchor": { "source": "paper.tex", "line_start": 7354, "line_end": 7355, "char_start": 336976, "char_end": 337004 }, "argument_gist": "", "excerpt": "t shared good primes; no moving good prime enters \\(L\\). \\subsection{Whole-index marked reflection} The next corollary adds the product-form marks to the exact reflection in Part I. Its branch data are important: an inactive marked prime is absent from the conductor, so the marked transform is not obtained by multiplying one fixed-conductor dual sum by independent local factors. \\begin{corolla" }, { "id": "eq:marked-reflection", "kind": "anchor", "label": "eq:marked-reflection", "owner": "cor:marked-reflection", "section": "sec:marked", "anchor": { "source": "paper.tex", "line_start": 7374, "line_end": 7375, "char_start": 338087, "char_end": 338115 }, "argument_gist": "", "excerpt": "al squarefree and cube ideals may share primes. There is an exact finite expansion \\mathcal T_{\\mathcal Q}(X;\\Psi,V) =\\sum_{\\mathcal B}\\zeta_{\\mathcal Q} \\sum_{0\\ne\\mu\\in\\lambda^{-4}\\mathcal O} &\\frac{d(\\mu)\\alpha(\\mu)\\vartheta(\\lambda^4\\mu)}{\\sqrt{q_\\mu}} \\prod_{p\\in\\mathcal A_0}B_p(\\lambda^4\\mu)\\\\ &\\cdot\\prod_{q\\in\\mathcal A_{\\mathcal Q}} q_q^{-1/2}\\" }, { "id": "eq:marked-reflection-phase", "kind": "anchor", "label": "eq:marked-reflection-phase", "owner": "cor:marked-reflection", "section": "sec:marked", "anchor": { "source": "paper.tex", "line_start": 7398, "line_end": 7399, "char_start": 339125, "char_end": 339159 }, "argument_gist": "", "excerpt": "p,\\epsilon_p,\\omega_{p,j}\\) computed using this branch's whole \\(c\\), the scalar is \\zeta_{\\mathcal Q}={}&-\\frac{i}{81}\\bar\\alpha(c)^2 \\widehat\\phi(h_0)\\bar\\kappa_F \\prod_{p\\in\\mathcal I_0}(1-q_p^{-1}) \\prod_{q\\in\\mathcal I_{\\mathcal Q}}q_q^{-1}\\\\ &\\cdot\\prod_{p\\in\\mathcal A_0}\\chi_p(\\sigma_p)^{-2}\\omega_{p,j_p} \\prod_{q\\in\\mathcal A_{\\" }, { "id": "eq:reflection-pair-phase", "kind": "anchor", "label": "eq:reflection-pair-phase", "owner": "sec:marked", "section": "sec:marked", "anchor": { "source": "paper.tex", "line_start": 7446, "line_end": 7447, "char_start": 341737, "char_end": 341769 }, "argument_gist": "", "excerpt": "(p,q\\), Equation~\\eqref{eq:reflection-cross-prime-phase} and cubic reciprocity give \\chi_p(q)^{2j_p+2}\\chi_q(p)^{2j_q+2} =\\left(\\frac qp\\right)_3^{j_p+j_q+2}. For an active mark the exponent is \\(j_q=0\\); its changed sign occurs only in its one-prime scalar. A residual row prime with \\(j_p=1\\) and a marked prime therefore have pair factor \\((q/p)_3^3=1\\)." }, { "id": "eq:cubic-large-sieve", "kind": "anchor", "label": "eq:cubic-large-sieve", "owner": "sec:marked", "section": "sec:marked", "anchor": { "source": "paper.tex", "line_start": 7491, "line_end": 7492, "char_start": 344169, "char_end": 344197 }, "argument_gist": "", "excerpt": "~\\ref{lem:completed-quadratic-reduction}. We add Heath--Brown's cubic large sieve: \\sum_{\\substack{a\\equiv1\\ (3)\\\\q_a\\le U}}^{*} \\left|\\sum_{\\substack{b\\equiv1\\ (3)\\\\q_b\\le V}}^{*} c_b\\left(\\frac ba\\right)_3\\right|^2 \\ll_\\epsilon (UV)^\\epsilon \\{U+V+(UV)^{2/3}\\}\\sum_b^{*}|c_b|^2, where both stars mean squarefree in \\(\\mathcal O\\), and the coeff" }, { "id": "old-eq:4.6b", "kind": "anchor", "label": "old-eq:4.6b", "owner": "lem:hybrid-energy", "section": "sec:marked", "anchor": { "source": "paper.tex", "line_start": 7523, "line_end": 7524, "char_start": 345783, "char_end": 345802 }, "argument_gist": "", "excerpt": "e \\(a\\) is independent of \\(k,n,b\\), and \\(\\beta\\) is independent of \\(k,P\\). Then &\\sum_k\\left| \\sum_P\\frac{a(P)}{\\sqrt{q_P}}1_{(P,k)=1} \\sum_{n,b}\\beta(n,b)\\chi_k(nb)^3 \\chi_P(n)^{-2}1_{(P,b)=1} \\right|^2\\\\ &\\qquad\\ll_\\epsilon (KNBL)^\\epsilon(K+NB)B\\{N+L+(NL)^{2/3}\\}. Fixed restrictions on the \\(k\\)-set and fixed" }, { "id": "eq:hybrid-mark-mask-reduction", "kind": "anchor", "label": "eq:hybrid-mark-mask-reduction", "owner": "lem:hybrid-energy", "section": "sec:marked", "anchor": { "source": "paper.tex", "line_start": 7561, "line_end": 7562, "char_start": 347226, "char_end": 347263 }, "argument_gist": "", "excerpt": "eros and \\(|\\beta_D|\\le1\\). Factoring \\(q_P^{-1/2}=q_D^{-1/2}q_{P'}^{-1/2}\\) gives \\mathcal H\\ll (KNBL)^\\epsilon\\sum_D\\frac1{q_D} \\sum_{k'}\\left| \\sum_{n,b}\\beta_D(n,b)\\chi_{k'}(nb)^3 \\sum_{P'}\\frac{a_D(P')}{\\sqrt{q_{P'}}} \\chi_{P'}(n)^{-2}1_{(P',b)=1} \\right|^2. Here \\(q_{k'}\\ll K/q_D\\), \\(q_{P'}\\asymp L/q_D\\), and all inher" }, { "id": "eq:hybrid-after-quadratic-reduction", "kind": "anchor", "label": "eq:hybrid-after-quadratic-reduction", "owner": "lem:hybrid-energy", "section": "sec:marked", "anchor": { "source": "paper.tex", "line_start": 7594, "line_end": 7595, "char_start": 348747, "char_end": 348790 }, "argument_gist": "", "excerpt": "tive output, inserted in Equation~\\eqref{eq:hybrid-mark-mask-reduction}, is at most (KNBL)^\\epsilon\\sum_D\\frac1{q_D}T\\sum_{q_t\\asymp T} \\sum_{\\substack{r\\ {\\rm sf},\\ (r,S)=1\\\\q_r\\ll\\min(K/q_D,C)}} &\\left\\{\\frac K{q_Dq_r}+\\frac{NG}{C^2}\\right\\}\\frac C{q_r} \\sum_{c',m,h}|\\beta_D(rc'm,rc'ht^2)|^2\\\\ &\\quad\\cdot\\left| \\sum_{P'}\\frac{a_{D,r}(P')}{\\sqrt{q_{P'}}}" }, { "id": "old-eq:4.6c", "kind": "anchor", "label": "old-eq:4.6c", "owner": "lem:hybrid-energy", "section": "sec:marked", "anchor": { "source": "paper.tex", "line_start": 7635, "line_end": 7636, "char_start": 350617, "char_end": 350636 }, "argument_gist": "", "excerpt": "wed by the \\(O(G/C)\\) choices of \\(h\\), give, on \\(q_D\\asymp D_0,\\ q_r\\asymp r_0\\), (KNBL)^\\epsilon\\frac GC \\left\\{\\frac N{r_0}+\\frac L{D_0} +\\left(\\frac{NL}{r_0D_0}\\right)^{2/3}\\right\\}. The fixed mask at \\(ht\\) is a coefficient restriction in this cubic sieve, not a row-dependent mask. The reduction supplies \\(T\\sum_t\\), with \\(O(T)\\) possible \\(t\\), and" }, { "id": "par:reflected-energy-data", "kind": "anchor", "label": "par:reflected-energy-data", "owner": "sec:marked", "section": "sec:marked", "anchor": { "source": "paper.tex", "line_start": 7663, "line_end": 7664, "char_start": 351800, "char_end": 351833 }, "argument_gist": "", "excerpt": "ion also specifies the dyadic contribution used in the statement. \\paragraph{Reflected block data.} Fix \\(f,\\rho\\) and put \\(Q=\\log_Zq_{\\mathfrak r_\\rho}\\). For an element row \\(0T_d+\\tau_{\\rm ref} \\] have arbitrarily small total size, with a fixed polynomial height cost, after a su" }, { "id": "old-eq:4.5", "kind": "anchor", "label": "old-eq:4.5", "owner": "lem:reflected-energy", "section": "sec:marked", "anchor": { "source": "paper.tex", "line_start": 7775, "line_end": 7776, "char_start": 357079, "char_end": 357097 }, "argument_gist": "", "excerpt": "\\(\\tau_{\\rm ref}\\). Every retained dyad has coefficient exponent, per active tuple, -v/2-\\ell_b-e_\\lambda/3-(S_0+B_0+z_a)/2. Define u=\\min\\{v,z_a,(v+z_a)/3\\}, and E_{\\rm ref}={}&O/2+\\max(H,v+\\ell_b)-S_0-B_0+z_a-u-\\ell_b -2e_\\lambda/3\\\\ &-\\tfra" }, { "id": "old-eq:4.7", "kind": "anchor", "label": "old-eq:4.7", "owner": "lem:reflected-energy", "section": "sec:marked", "anchor": { "source": "paper.tex", "line_start": 7779, "line_end": 7780, "char_start": 357178, "char_end": 357196 }, "argument_gist": "", "excerpt": "} -v/2-\\ell_b-e_\\lambda/3-(S_0+B_0+z_a)/2. Define u=\\min\\{v,z_a,(v+z_a)/3\\}, and E_{\\rm ref}={}&O/2+\\max(H,v+\\ell_b)-S_0-B_0+z_a-u-\\ell_b -2e_\\lambda/3\\\\ &-\\tfrac12(T_d-v-3\\ell_b-e_\\lambda)_+. For every \\(\\epsilon>0\\), the contribut" }, { "id": "old-eq:4.6", "kind": "anchor", "label": "old-eq:4.6", "owner": "lem:reflected-energy", "section": "sec:marked", "anchor": { "source": "paper.tex", "line_start": 7783, "line_end": 7784, "char_start": 357260, "char_end": 357278 }, "argument_gist": "", "excerpt": "e u=\\min\\{v,z_a,(v+z_a)/3\\}, and E_{\\rm ref}={}&O/2+\\max(H,v+\\ell_b)-S_0-B_0+z_a-u-\\ell_b -2e_\\lambda/3\\\\ &-\\tfrac12(T_d-v-3\\ell_b-e_\\lambda)_+. For every \\(\\epsilon>0\\), the contribution with fixed \\(k_{\\rm pow},k_{\\rm sup}\\) satisfies \\[ \\sum_{k_{\\rm res}\\ {\\rm s" }, { "id": "eq:reflection-actual-annuli", "kind": "anchor", "label": "eq:reflection-actual-annuli", "owner": "lem:reflected-energy", "section": "sec:marked", "anchor": { "source": "paper.tex", "line_start": 7865, "line_end": 7866, "char_start": 361080, "char_end": 361115 }, "argument_gist": "", "excerpt": "\\ell_b+e_\\lambda-T_d}. \\] Multiplicativity gives the exact identity \\frac{q_\\mu X}{q_c^2} =Yy_ny_b^3y_R^{-2}\\prod_{i\\ {\\rm active}}y_i^{-2}. It remains true if \\(N_F\\) shares primes with \\(b\\). All frozen factors in \\(Y\\) are their actual products, and every moving norm remains a coordinate. The actual row cutoff and the individual dual and" }, { "id": "old-eq:4.6a", "kind": "anchor", "label": "old-eq:4.6a", "owner": "lem:reflected-energy", "section": "sec:marked", "anchor": { "source": "paper.tex", "line_start": 7934, "line_end": 7935, "char_start": 364532, "char_end": 364551 }, "argument_gist": "", "excerpt": "is bounded by \\[ Z^{-v-2\\ell_b-2e_\\lambda/3-S_0-B_0+\\epsilon}\\mathcal N, \\] where \\mathcal N={}& \\sum_{\\substack{k_{\\rm res}\\ {\\rm sf}\\\\q_{k_{\\rm res}}\\ll Z^H}} \\left| \\sum_{\\boldsymbol p} \\frac{a(\\boldsymbol p)}{\\sqrt{q_P}}1_{(P,k_{\\rm res})=1} \\sum_{\\substack{n\\ {\\rm sf}\\\\b}} \\beta(n,b)\\chi_{k_{\\rm res}}(nb)^3 \\chi_P(n)^{-2}1_{(P,b)=1} \\rig" }, { "id": "eq:common-width-charge", "kind": "anchor", "label": "eq:common-width-charge", "owner": "sec:marked", "section": "sec:marked", "anchor": { "source": "paper.tex", "line_start": 8028, "line_end": 8029, "char_start": 368660, "char_end": 368690 }, "argument_gist": "", "excerpt": "_H^2C_SC_k^2C_OC_f\\). The preceding inequality and the definition of \\(T_d\\) imply T_d-S_0-B_0 \\le 2M-O+Q+V-N_*+2z_a+ \\frac{\\log C_{\\rm width}}{\\log Z}. For empty slot lists, \\(z_a=u=0\\). Choose \\(\\eta>0\\) and then take \\(Z\\) so large that the two displayed constant errors are at most \\(\\eta\\) and \\(e_\\lambda\\ge-\\eta\\); the latter follows from \\(\\log Z\\g" }, { "id": "eq:completed-moment", "kind": "anchor", "label": "eq:completed-moment", "owner": "sec:marked", "section": "sec:marked", "anchor": { "source": "paper.tex", "line_start": 8053, "line_end": 8054, "char_start": 369741, "char_end": 369768 }, "argument_gist": "", "excerpt": "and the local small-power losses sufficiently small in terms of \\(\\epsilon\\) proves \\sum_{04$ bounds their sum by $(\\mathcal Q/N)^A$. For $N<\\mathcal Q$, use the point count in the two annuli. A nonempty subunit annulus has $N$" }, { "id": "old-eq:5.10", "kind": "anchor", "label": "old-eq:5.10", "owner": "prop:probe-low", "section": "sec:compensated-low", "anchor": { "source": "paper.tex", "line_start": 8571, "line_end": 8572, "char_start": 392964, "char_end": 392983 }, "argument_gist": "", "excerpt": "n{equation} |I_{\\eta,\\mathrm{modified}}(Z)| \\ll Z^{l_x/2+b/12+\\epsilon}=Z^{3/16+\\epsilon}. The exponent is independent of the target and the slot mesh; the constant and lower threshold may depend on the fixed arithmetic data, slot system, and smooth tests. For a fixed rescaled subset and tuple, put $Q=q_{b_*}X'Y'$. Its separation is Equatio" }, { "id": "old-eq:5.6", "kind": "anchor", "label": "old-eq:5.6", "owner": "prop:probe-low", "section": "sec:compensated-low", "anchor": { "source": "paper.tex", "line_start": 8610, "line_end": 8611, "char_start": 394266, "char_end": 394284 }, "argument_gist": "", "excerpt": "n} \\sum_{q_m\\ll Q}|A_{m,\\sigma,v}(Y')|^2 \\ll (1+|v|)^{J_{\\mathrm{Gr}}}(Q/Y')P_a^{1/6}Z^\\epsilon. Cauchy--Schwarz in the row sum, Lemma~\\ref{lem:probe-row-norm}, and the integrability of $(1+|v|)^{J_{\\mathrm{Gr}}/2}|\\widehat W_0(iv)|$ now bound this tuple's unscaled separation by $Z^\\epsilon$ times \\[ Q^{-1/2}\\bigl[(Q/Y')P_a^{1/6}\\bigr]^{1/2} \\bigl[Z^{M'}" }, { "id": "old-eq:5.5", "kind": "anchor", "label": "old-eq:5.5", "owner": "prop:probe-low", "section": "sec:compensated-low", "anchor": { "source": "paper.tex", "line_start": 8632, "line_end": 8633, "char_start": 395001, "char_end": 395019 }, "argument_gist": "", "excerpt": "^{-d/2}$. The total additional exponent is consequently f(d)=d-3d/2-d/2=-d\\le0. Choose the small powers in the component estimates so that their sum lies within the prescribed $\\epsilon$. Summing the finitely many subsets proves Equation~\\eqref{old-eq:5.10}. For the already defined $C=C_{\\mathrm{II}}$ of Equation~\\eqref{eq:part-II-mellin-target}, the normaliza" }, { "id": "old-eq:5.11", "kind": "anchor", "label": "old-eq:5.11", "owner": "prop:probe-low", "section": "sec:compensated-low", "anchor": { "source": "paper.tex", "line_start": 8644, "line_end": 8645, "char_start": 395500, "char_end": 395519 }, "argument_gist": "", "excerpt": "C_{\\mathrm{II}}(s)=l_x/2+s-1+h/6=s-11/16,\\\\ C_{\\mathrm{II}}(7/8)=3/16=l_x/2+b/12. Thus, under the contradiction assumed in Part~II, this low bound is smaller than $Z^{C_{\\mathrm{II}}(\\beta_*)}$ by the exact power $\\Delta=\\beta_*-7/8$, apart from the arbitrarily prescribed $\\epsilon$. \\section{The local compensation and its errors} \\label{sec:local-com" }, { "id": "old-eq:5.13", "kind": "anchor", "label": "old-eq:5.13", "owner": "sec:local-compensation", "section": "sec:local-compensation", "anchor": { "source": "paper.tex", "line_start": 8680, "line_end": 8681, "char_start": 397258, "char_end": 397277 }, "argument_gist": "", "excerpt": "\\] Thus, on the same locus, the exact local multiplier of the two-term operation is Q^{z-1}\\mathcal B_p,\\qquad \\mathcal B_p=\\overline{\\eta(p)}Q^xP_p^*/P_p-Q^{-w}. Define the combined local replacement by G_p= \\frac{(\\overline{\\eta(p)}Q^x-Q^{-w})(1-V)(1-W)P_p^* -Q^{-w}(1-VW)}{1-D}. At points in the Euler" }, { "id": "old-eq:5.13c", "kind": "anchor", "label": "old-eq:5.13c", "owner": "sec:local-compensation", "section": "sec:local-compensation", "anchor": { "source": "paper.tex", "line_start": 8685, "line_end": 8686, "char_start": 397431, "char_end": 397451 }, "argument_gist": "", "excerpt": "eta(p)}Q^xP_p^*/P_p-Q^{-w}. Define the combined local replacement by G_p= \\frac{(\\overline{\\eta(p)}Q^x-Q^{-w})(1-V)(1-W)P_p^* -Q^{-w}(1-VW)}{1-D}. At points in the Euler regions where the raw quotient in Equation~\\eqref{old-eq:5.13} is defined, one has $G_p=H_p\\mathcal B_p$. The displayed formula defines $G_p$ also where that quotient is not" }, { "id": "eq:holomorphic-selected-tuple", "kind": "anchor", "label": "eq:holomorphic-selected-tuple", "owner": "sec:local-compensation", "section": "sec:local-compensation", "anchor": { "source": "paper.tex", "line_start": 8707, "line_end": 8708, "char_start": 398319, "char_end": 398356 }, "argument_gist": "", "excerpt": "l B_p. \\] The correction used for contour moves is defined without these quotients: \\mathfrak H_{\\eta,u,Z}(x,w,z)= \\sum_{(p_i)\\in\\prod_i\\mathcal P_i(Z)} \\prod_{i=1}^K\\bigl[W_i(q_{p_i}/P_i)q_{p_i}^{z-1}G_{p_i}\\bigr] \\prod_{\\substack{p\\notin S\\\\p\\notin\\{p_1,\\ldots,p_K\\}}}H_p. This is the full correction after the scalar quotient in Equation~\\eqref{eq:probe-high} h" }, { "id": "eq:selected-local-operation", "kind": "anchor", "label": "eq:selected-local-operation", "owner": "sec:local-compensation", "section": "sec:local-compensation", "anchor": { "source": "paper.tex", "line_start": 8717, "line_end": 8718, "char_start": 398853, "char_end": 398888 }, "argument_gist": "", "excerpt": "ts,p_K\\}$. On the absolute starting lines the selected operation replaces $P_p$ by \\overline{\\eta(p)}Q^{x+z-1}P_p^*-Q^{z-w-1}P_p =\\frac{1-D}{(1-V)(1-W)}Q^{z-1}G_p. The equality follows by substituting $P_p-P_p^*=(1-V)^{-1}+W/(1-W)$ into Equation~\\eqref{old-eq:5.13c}. It extracts the same scalar local factor $(1-V)^{-1}(1-W)^{-1}(1-D)$ as at an unselected prime." }, { "id": "eq:unselected-local-product", "kind": "anchor", "label": "eq:unselected-local-product", "owner": "sec:local-compensation", "section": "sec:local-compensation", "anchor": { "source": "paper.tex", "line_start": 8738, "line_end": 8739, "char_start": 399979, "char_end": 400014 }, "argument_gist": "", "excerpt": "defect bounds in Lemma~\\ref{lem:local-euler} give, uniformly in the selected tuple, \\prod_{\\substack{p\\notin S\\\\p\\notin\\mathcal P}}|H_p| \\le \\prod_{\\substack{p\\notin S\\\\p\\nmid u}} (1+Cq_p^{-1-\\epsilon_H}) \\prod_{p\\mid u}(1+Cq_p^{-\\epsilon_H}) \\ll_\\epsilon q_u^\\epsilon. The first positive product converges by the ideal count, and the second satisfies the" }, { "id": "eq:compensated-all-height-majorant", "kind": "anchor", "label": "eq:compensated-all-height-majorant", "owner": "sec:local-compensation", "section": "sec:local-compensation", "anchor": { "source": "paper.tex", "line_start": 8764, "line_end": 8765, "char_start": 401417, "char_end": 401459 }, "argument_gist": "", "excerpt": "finite tuple sum and Equation~\\eqref{eq:unselected-local-product} consequently give |\\mathfrak H_{\\eta,u,Z}(x,w,z)| \\ll_\\epsilon q_u^\\epsilon\\prod_iP_i^{z_r+B} \\ll_\\epsilon q_u^\\epsilon Z^{B'}. for a fixed $B'$ on that box. For $q_u\\le Z^D$ this is Equation~\\eqref{eq:stage-all-height-majorant}, even with height exponent zero. All constants here are fixed before" }, { "id": "eq:compensated-poisson-identity", "kind": "anchor", "label": "eq:compensated-poisson-identity", "owner": "sec:local-compensation", "section": "sec:local-compensation", "anchor": { "source": "paper.tex", "line_start": 8777, "line_end": 8778, "char_start": 401940, "char_end": 401979 }, "argument_gist": "", "excerpt": "ith Equations~\\eqref{eq:probe-poisson-identity} and \\eqref{eq:probe-high} now gives I_{\\eta,\\mathrm{modified}}(Z) ={}&\\frac1{(2\\pi i)^3}\\int_{(3)}\\int_{(3)}\\int_{(2)} \\mathcal W(X,Y,Z;x,w,z)\\\\ &\\quad\\cdot\\sum_u^{(6)}q_u^{-z}\\overline{\\xi(u)} \\frac{\\zeta_F^S(6z)L^S(w,\\chi_\\bullet(u))} {L^S(x,\\eta\\overline{\\chi_\\bullet(u)})} \\mathfrak H_{" }, { "id": "old-eq:5.13b", "kind": "anchor", "label": "old-eq:5.13b", "owner": "sec:local-compensation", "section": "sec:local-compensation", "anchor": { "source": "paper.tex", "line_start": 8808, "line_end": 8809, "char_start": 403450, "char_end": 403470 }, "argument_gist": "", "excerpt": "$v^{-1}W=Q^{-w}$ and $\\mathcal E_p=P_p^*+D$ gives the exact normalized cancellation G_p+v^{-1}H_p={}& \\frac{(1-V)(1-W)}{1-D} \\Bigl[(v^{-1}-Q^{-w})\\{V/(1-V)-D\\}\\\\ &\\hspace{17mm}+(\\overline{\\eta(p)}Q^x+v^{-1}-Q^{-w}) \\mathcal E_p\\Bigr]. This is an identity of holomorphic local expressions in the Euler regions. Within t" }, { "id": "old-eq:5.13e", "kind": "anchor", "label": "old-eq:5.13e", "owner": "prop:probe-errors", "section": "sec:local-compensation", "anchor": { "source": "paper.tex", "line_start": 8883, "line_end": 8884, "char_start": 406880, "char_end": 406900 }, "argument_gist": "", "excerpt": "\\] Then, for every subset $I$ of the slots and the same points $w\\in\\mathcal W_u$, \\left|L^S(w,\\chi_\\bullet(u))\\prod_{i\\in I}\\mathcal D_i(u)\\right| \\ll U^{a-1/2+O(e)+\\epsilon}(3+T_1)^C \\prod_{i\\in I}P_i^{z_r-1/2+O(e)+\\epsilon}. The main parts retain the physical row, all zero masks, and the fixed identity-ray restriction. In particular, their central norm" }, { "id": "eq:dynamic-compensated-decomposition", "kind": "anchor", "label": "eq:dynamic-compensated-decomposition", "owner": "prop:probe-errors", "section": "sec:local-compensation", "anchor": { "source": "paper.tex", "line_start": 8898, "line_end": 8899, "char_start": 407504, "char_end": 407548 }, "argument_gist": "", "excerpt": ", and on this dynamic region only, the full correction has the finite decomposition \\mathfrak H_{\\eta,u,Z} =\\mathcal H_{\\eta,u}\\prod_{i=1}^K(\\mathcal Q_i+\\mathcal D_i) =\\mathcal H_{\\eta,u} \\sum_{I\\subseteq\\{1,\\ldots,K\\}} \\prod_{i\\in I}\\mathcal D_i\\prod_{i\\notin I}\\mathcal Q_i. In this region $\\vartheta=(-w_r)_+\\le6e$. T" }, { "id": "old-eq:5.13d", "kind": "anchor", "label": "old-eq:5.13d", "owner": "prop:probe-errors", "section": "sec:local-compensation", "anchor": { "source": "paper.tex", "line_start": 9007, "line_end": 9008, "char_start": 412436, "char_end": 412456 }, "argument_gist": "", "excerpt": "at $2,3$. It follows that, for any set $J_0$ of distinct selected ramified labels, q_{\\mathfrak f_u}\\ll_S q_{\\operatorname{rad}(u)} \\le q_u\\prod_{p\\in J_0}q_p^{-(j_p-1)}. The functional equation for the primitive numerator \\cite[Equation (1.1)]{GZ}, the assumed reflected bound, and Stirling give a cost $q_{\\mathfrak f_u}^{A_*}U^\\epsilon(3+T_1)^C$, where \\[" }, { "id": "eq:principal-slot-factorization", "kind": "anchor", "label": "eq:principal-slot-factorization", "owner": "sec:local-compensation", "section": "sec:local-compensation", "anchor": { "source": "paper.tex", "line_start": 9059, "line_end": 9060, "char_start": 414921, "char_end": 414960 }, "argument_gist": "", "excerpt": "The lower bound and the disjoint supports give the separate principal factorization \\mathfrak H_{\\eta,1,Z}(x,w,z) =\\mathcal H_{\\eta,1}(x,w,z)\\prod_{i=1}^K\\mathcal B_i(1;x,w,z) \\qquad\\text{in the second Euler region}. This does not extend the central main/error decomposition beyond its stated dynamic region. The four error exponents are now \\[ -x_r,\\qquad -6z_r," }, { "id": "eq:principal-local-approximation", "kind": "anchor", "label": "eq:principal-local-approximation", "owner": "sec:local-compensation", "section": "sec:local-compensation", "anchor": { "source": "paper.tex", "line_start": 9073, "line_end": 9074, "char_start": 415406, "char_end": 415446 }, "argument_gist": "", "excerpt": "\\] Each is at most $-7/8$ in that region. Hence the principal multiplier satisfies \\mathcal B_p=-1+O(Q^{-7/8}), uniformly in the imaginary parts and the target unit phase. This is the local estimate used to normalize the principal residue. It also gives the absolute principal correction required by the shared residue estimate. Indeed $G_p=H_p\\mathcal B_p=O(1)$" }, { "id": "eq:principal-local-tuple-bound", "kind": "anchor", "label": "eq:principal-local-tuple-bound", "owner": "sec:local-compensation", "section": "sec:local-compensation", "anchor": { "source": "paper.tex", "line_start": 9085, "line_end": 9086, "char_start": 416048, "char_end": 416086 }, "argument_gist": "", "excerpt": "ery fixed real box in the second Euler region and uniformly in all imaginary parts, |\\mathfrak H_{\\eta,1,Z}(x,w,z)| \\le\\sum_{(p_i)\\in\\prod_i\\mathcal P_i(Z)} \\prod_i\\bigl[|W_i(q_{p_i}/P_i)|q_{p_i}^{z_r-1}|G_{p_i}|\\bigr] \\prod_{\\substack{p\\notin S\\\\p\\notin\\{p_1,\\ldots,p_K\\}}}|H_p| \\ll \\prod_iP_i^{z_r}=Z^{\\ell z_r}. This uses the separate principal-row" }, { "id": "eq:absolute-local-tuple-bound", "kind": "anchor", "label": "eq:absolute-local-tuple-bound", "owner": "lem:compensated-absolute-local-bounds", "section": "sec:local-compensation", "anchor": { "source": "paper.tex", "line_start": 9122, "line_end": 9123, "char_start": 417551, "char_end": 417588 }, "argument_gist": "", "excerpt": "{enumerate} In either case the positive sum over the full selected tuples satisfies \\sum_{(p_i)\\in\\prod_i\\mathcal P_i(Z)} \\prod_i\\bigl[|W_i(q_{p_i}/P_i)|q_{p_i}^{z_r-1}|G_{p_i}|\\bigr] \\prod_{\\substack{p\\notin S\\\\p\\notin\\{p_1,\\ldots,p_K\\}}}|H_p| \\ll_\\epsilon U^\\epsilon\\prod_iP_i^{z_r}. The constants may depend on the fixed data, $e$, and, in the secon" }, { "id": "eq:small-line-positive-slot", "kind": "anchor", "label": "eq:small-line-positive-slot", "owner": "lem:compensated-absolute-local-bounds", "section": "sec:local-compensation", "anchor": { "source": "paper.tex", "line_start": 9175, "line_end": 9176, "char_start": 419642, "char_end": 419677 }, "argument_gist": "", "excerpt": "divisor-many, while a slot contains $O(P_i)$ ideals. Its positive sum is therefore \\sum_{p\\in\\mathcal P_i(Z)} |W_i(q_p/P_i)|q_p^{z_r-1}|G_p| \\ll P_i^{z_r}+U^\\epsilon P_i^{z_r-1/2} \\ll U^\\epsilon P_i^{z_r}. Using the positive majorant for the unselected product and distributing the requested $\\epsilon$ among the fixed number of slots proves Equation~\\eqref" }, { "id": "old-eq:4.1", "kind": "anchor", "label": "old-eq:4.1", "owner": "lem:marked", "section": "sec:inverse", "anchor": { "source": "paper.tex", "line_start": 9257, "line_end": 9258, "char_start": 423423, "char_end": 423441 }, "argument_gist": "", "excerpt": "Suppose \\[ r+2z\\le m-c_1,\\qquad 2r+8z\\le3m-c_2. \\] Then, for every \\(\\epsilon>0\\), \\sum_{\\substack{u\\in\\mathcal O\\\\00\\), define \\mathcal E={}&Z^{-V}\\sum_f w(f) \\sum_{\\substack{k\\in\\mathc" }, { "id": "eq:canonical-energy", "kind": "anchor", "label": "eq:canonical-energy", "owner": "lem:canonical-moment", "section": "sec:inverse", "anchor": { "source": "paper.tex", "line_start": 9324, "line_end": 9325, "char_start": 426989, "char_end": 427016 }, "argument_gist": "", "excerpt": "red} For every smooth annular \\(W\\) and every \\(\\epsilon>0\\), define \\mathcal E={}&Z^{-V}\\sum_f w(f) \\sum_{\\substack{k\\in\\mathcal O\\\\00$, p_j\\bigl(\\chi(\\boldsymbol y)K(Rm(\\boldsymbol y))\\bigr) \\le C'_{A,j}\\mathfrak S_{m(A,j)}\\min(1,R^{-A}). Indeed $y_i\\partial_{y_i}$ acting on the kernel is the fixed exponent of $y_i$ in $m$ times $r\\partial_r$, and $m$ is bounded above and below on the cutoff support. There is no e" }, { "id": "eq:masked-poisson", "kind": "anchor", "label": "eq:masked-poisson", "owner": "lem:masked-poisson", "section": "sec:inverse", "anchor": { "source": "paper.tex", "line_start": 9728, "line_end": 9729, "char_start": 447359, "char_end": 447384 }, "argument_gist": "", "excerpt": "ourier transform for the self-dual measure and the kernel \\(e(-zy)\\). For \\(K>0\\), &\\sum_{k\\in\\mathcal O}\\psi(k)1_{(k,R)=1}\\Phi(q_k/K)\\\\ &\\quad=\\frac{K\\gamma(\\psi;m)}{\\sqrt{q_m}} \\sum_{d\\mid\\operatorname{rad}R}\\frac{\\mu(d)\\psi(d)}{q_d} \\sum_{h\\in\\mathcal O}\\bar\\psi(h) \\mathcal F\\Phi\\!\\left(\\frac{Kq_h}{q_dq_m}\\right), where \\(\\" }, { "id": "eq:inverse-fixed-invariant", "kind": "anchor", "label": "eq:inverse-fixed-invariant", "owner": "lem:canonical-moment", "section": "sec:inverse", "anchor": { "source": "paper.tex", "line_start": 9807, "line_end": 9808, "char_start": 450911, "char_end": 450945 }, "argument_gist": "", "excerpt": "arting node. The two hypotheses at a node with margin \\(c_{\\rm node}\\) are exactly F-M-Q-z_0\\ge c_{\\rm node},\\qquad 4F-3M-6z_0\\ge c_{\\rm node}. In particular \\(M+z_0+c_{\\rm node}\\le F\\). The number \\(Q\\) is the actual logarithmic norm of the already fixed puncture radical, not a dyadic center. The parameter \\(M\\) is the fixed logarithmic row length; the first P" }, { "id": "eq:marked-completion-inversion", "kind": "anchor", "label": "eq:marked-completion-inversion", "owner": "lem:canonical-moment", "section": "sec:inverse", "anchor": { "source": "paper.tex", "line_start": 9841, "line_end": 9842, "char_start": 452443, "char_end": 452481 }, "argument_gist": "", "excerpt": "f the cube factor in Equation~\\eqref{eq:marked-completion} gives the exact identity &Z^{-N/2}\\sum_{n\\ {\\rm sf}}a(n)\\chi_n(k)\\chi_n(f)^4 \\mathfrak d(n)W(q_n/Z^N)\\\\ &\\quad= \\sum_h\\frac{\\mu(h)\\bar\\alpha(h)^3\\Psi_k(h)^3}{q_h} \\mathcal T_{\\mathfrak d_h}(Z^N/q_h^3;k), \\qquad \\mathfrak d_h(A)=\\mathfrak d(h^3A). To check both th" }, { "id": "eq:inverse-terminal-width", "kind": "anchor", "label": "eq:inverse-terminal-width", "owner": "lem:canonical-moment", "section": "sec:inverse", "anchor": { "source": "paper.tex", "line_start": 9884, "line_end": 9885, "char_start": 454654, "char_end": 454687 }, "argument_gist": "", "excerpt": "ed-energy}, keep the actual residual-row dyad and choose the same threshold so that H\\le M-O+\\eta,\\qquad T_d-S_0-B_0\\le2M-O+Q+V-N_*+2z_a+\\eta. These are Equations~\\eqref{eq:actual-residual-row-dyad} and \\eqref{eq:common-width-charge}, not estimates for an enlarged row range. Substituting \\(N_*=N-3\\widehat h\\) and the first inequality in Equation~\\eqref{eq:inverse" }, { "id": "old-eq:4.9", "kind": "anchor", "label": "old-eq:4.9", "owner": "lem:canonical-moment", "section": "sec:inverse", "anchor": { "source": "paper.tex", "line_start": 9892, "line_end": 9893, "char_start": 455027, "char_end": 455045 }, "argument_gist": "", "excerpt": "t h\\) and the first inequality in Equation~\\eqref{eq:inverse-fixed-invariant} gives T_d-S_0-B_0 \\le M-O+2z_a-z_0-c_{\\rm node}+5d+4\\eta. Indeed, the additional terms are \\(2V+3\\widehat h+\\eta<5d+4\\eta\\). Take the threshold also to ensure \\(e_\\lambda\\ge-\\eta\\), and discard the whole reflected tail dyads as in Lemma~\\ref{lem:reflected-energy}. Thus every retained dy" }, { "id": "old-eq:4.10", "kind": "anchor", "label": "old-eq:4.10", "owner": "lem:canonical-moment", "section": "sec:inverse", "anchor": { "source": "paper.tex", "line_start": 9910, "line_end": 9911, "char_start": 455791, "char_end": 455810 }, "argument_gist": "", "excerpt": "a/3+\\tau_{\\rm ref}\\). Equation~\\eqref{old-eq:4.9} and the first invariant then give E_{\\rm ref}\\le F-2c_{\\rm node}+5d+\\tfrac{17}{3}\\eta+\\tau_{\\rm ref}. If \\(u\\ne z_a\\), its definition implies \\(u\\ge v/2\\). The retained dual bound now first gives \\[ E_{\\rm ref}\\le O/2+(T_d-S_0-B_0)/2+z_a +\\tfrac76\\eta+\\tfrac12\\tau_{\\rm ref}. \\] Using Equation~\\" }, { "id": "old-eq:4.11", "kind": "anchor", "label": "old-eq:4.11", "owner": "lem:canonical-moment", "section": "sec:inverse", "anchor": { "source": "paper.tex", "line_start": 9921, "line_end": 9922, "char_start": 456198, "char_end": 456217 }, "argument_gist": "", "excerpt": "g Equation~\\eqref{old-eq:4.9}, \\(z_a\\le z_0\\), and then the second invariant yields E_{\\rm ref}\\le F-M/4-3c_{\\rm node}/4+(5/2)d +\\tfrac{19}{6}\\eta+\\tfrac12\\tau_{\\rm ref}. Use the reflected estimate with aggregate local loss \\(\\pi_{\\rm ref}\\), including its freely chosen exponent, the label divisor bound, and the finitely many logarithmic sums." }, { "id": "old-eq:4.12", "kind": "anchor", "label": "old-eq:4.12", "owner": "lem:canonical-moment", "section": "sec:inverse", "anchor": { "source": "paper.tex", "line_start": 9945, "line_end": 9946, "char_start": 457406, "char_end": 457425 }, "argument_gist": "", "excerpt": "herwise retain it on the residual column. The basic identity for the completion is 1_{p\\mid nb^3}=1_{p\\mid b}+1_{p\\nmid b}1_{p\\mid n}. More generally, for an ordered list of extracted factors \\(D_1,\\ldots,D_h\\), the exact priority identity is 1_{p\\mid D_1\\cdots D_hn} =\\sum_{j=1}^h1_{p\\mid D_j}\\prod_{aJ+d\\), gives \\int_{\\mathbb R^d}|\\widehat{\\mathfrak H}(\\boldsymbol t)| (1+|\\boldsymbol t|)^J\\,d\\boldsymbol t \\ll_{J,d,\\mathrm{box}}\\max_{|\\boldsymbol a|\\le2m_0} \\|D^{\\boldsymbol a}\\mathfrak H\\|_\\infty. For a fixed radial Schwartz Fourier kernel \\(K\\), every \\(\\sup_{x\\ge0}|(x" }, { "id": "eq:inverse-raw-tail", "kind": "anchor", "label": "eq:inverse-raw-tail", "owner": "lem:canonical-moment", "section": "sec:inverse", "anchor": { "source": "paper.tex", "line_start": 10127, "line_end": 10128, "char_start": 467030, "char_end": 467057 }, "argument_gist": "", "excerpt": "kernel-profile}. The raw tail estimate we shall use is, for \\(Y\\ge1\\) and \\(A>1\\), \\sum_{h\\ne0:\\,a q_h>Y}|K(aq_h)| \\ll_A(1+a^{-1})Y^{1-A}\\qquad(a>0). Indeed, the shell \\(2^jY0\\) and \\(\\epsilon>0\\), put \\[ H=\\max(2U,D^{1+c}),\\qquad P=(H/U)^{1/6}. \\] Then \\sum_u|M_u(D;W)|^2 \\ll \\frac HP(UD)^\\epsilon \\ll \\max\\{U,U^{1/6}D^{5(1+c)/6}\\}(UD)^\\epsilon. The implied constant depends only on \\(c,\\epsilon\\), the fixed arithmetic data, and finitely many seminorms of \\(W\\). In particular, for \\(D=U^r\\) with \\(r\\) in a fixed bounded nonnegativ" }, { "id": "eq:inverse-scale-supremum", "kind": "anchor", "label": "eq:inverse-scale-supremum", "owner": "lem:inverse-amplification", "section": "sec:inverse", "anchor": { "source": "paper.tex", "line_start": 12394, "line_end": 12395, "char_start": 574787, "char_end": 574820 }, "argument_gist": "", "excerpt": "ef{eq:raw-moment} and Lemma~\\ref{lem:smooth-calculus} imply the rowwise scale bound \\sum_{\\substack{v\\in\\mathcal O\\\\00\\) and n_1+n_2+6\\kappa z=A+(6\\kappa-1)z\\le M. \\end{e" }, { "id": "old-eq:3.9", "kind": "anchor", "label": "old-eq:3.9", "owner": "lem:plain", "section": "sec:plain", "anchor": { "source": "paper.tex", "line_start": 12562, "line_end": 12563, "char_start": 582759, "char_end": 582777 }, "argument_gist": "", "excerpt": "bound on the conductor. \\item \\(z>0\\) and n_1+n_2+6\\kappa z=A+(6\\kappa-1)z\\le M. If \\(\\kappa<1\\), assume in addition that \\(\\beta_*\\le(1+\\kappa)/2\\). If \\(\\kappa=1\\), no zero-free hypothesis is required. For an empty slot list, \\(Q=1\\). If \\(z=0\\) and the list contains zero-length slots, their scales are bounded and absolute co" }, { "id": "old-eq:2.1a", "kind": "anchor", "label": "old-eq:2.1a", "owner": "sec:plain", "section": "sec:plain", "anchor": { "source": "paper.tex", "line_start": 12627, "line_end": 12627, "char_start": 586192, "char_end": 586211 }, "argument_gist": "", "excerpt": "mathfrak R}(X) &=\\sum_{d\\mid\\mathfrak R}\\mu(d)\\psi_k^0(d)q_d^{-1/2} T_k(X/q_d), \\\\ Q_{k,i,\\mathfrak R} &=Q_{k,i}^{\\,0} -P_i^{-1/2}\\sum_{\\substack{p\\mid\\mathfrak R\\\\p\\ {\\rm prime}}} \\psi_k^0(p)\\nu_i(p)W_i^{\\rm slot}(q_p/P_i), \\qquad P_i=Z^{z_i}. For example, the first identity follows by inserting \\(\\sum_{d\\mid(n,\\mathfrak" }, { "id": "eq:prime-mask-erasure", "kind": "anchor", "label": "eq:prime-mask-erasure", "owner": "sec:plain", "section": "sec:plain", "anchor": { "source": "paper.tex", "line_start": 12633, "line_end": 12634, "char_start": 586390, "char_end": 586419 }, "argument_gist": "", "excerpt": "thfrak R\\\\p\\ {\\rm prime}}} \\psi_k^0(p)\\nu_i(p)W_i^{\\rm slot}(q_p/P_i), \\qquad P_i=Z^{z_i}. For example, the first identity follows by inserting \\(\\sum_{d\\mid(n,\\mathfrak R)}\\mu(d)\\) and writing \\(n=dl\\). The quotient \\(l\\) is unrestricted at primes of \\(d\\); the original natural zero extension remains on it. Apply these identities simultaneously to both plain factors and all" }, { "id": "old-eq:2.1b", "kind": "anchor", "label": "old-eq:2.1b", "owner": "sec:plain", "section": "sec:plain", "anchor": { "source": "paper.tex", "line_start": 12663, "line_end": 12664, "char_start": 587520, "char_end": 587539 }, "argument_gist": "", "excerpt": "R}q_d^{-1/2} \\le \\prod_{p\\mid\\mathfrak R}(1-q_p^{-1/2})^{-1} \\ll_{\\epsilon_1,B} Z^{\\epsilon_1}. This follows from Lemma~\\ref{lem:deleted-euler-factors}. The corresponding mass for a frozen slot is also \\(Z^{\\epsilon_1}\\): on its annular support, \\(P_i^{-1/2}\\ll q_p^{-1/2}\\), and the same product bounds the sum over \\(p\\mid\\mathfrak R\\). The number of slots is fixed. Minkowsk" }, { "id": "old-eq:2.1c", "kind": "anchor", "label": "old-eq:2.1c", "owner": "sec:plain", "section": "sec:plain", "anchor": { "source": "paper.tex", "line_start": 12687, "line_end": 12688, "char_start": 588787, "char_end": 588806 }, "argument_gist": "", "excerpt": "thcal S\\) gives C_kq_{\\mathfrak R_{0,k}} \\ll_{\\mathcal S,\\tau_{\\rm fixed}} Z^M. Indeed, each good prime contributes at most once, either to the primitive conductor or to the redundant radical. Their union is contained in the union of the row radical and the declared moving radical, whose norm is at most \\(q_kZ^q\\). By Lemma~\\ref{lem:fixed-numerator-ray}, the u" }, { "id": "old-eq:2.1d", "kind": "anchor", "label": "old-eq:2.1d", "owner": "sec:plain", "section": "sec:plain", "anchor": { "source": "paper.tex", "line_start": 12754, "line_end": 12755, "char_start": 592015, "char_end": 592034 }, "argument_gist": "", "excerpt": "}}{Xq_{h_0}},\\qquad d_0\\mid\\mathfrak R_{0,k},\\quad \\operatorname{rad}(h_0)\\mid\\mathfrak R_{0,k}. By Equation~\\eqref{old-eq:2.1c}, there is a fixed \\(C_{\\mathcal A}\\ge1\\) such that \\[ C_kq_{\\mathfrak R_{0,k}}\\le C_{\\mathcal A}Z^M,\\qquad Y\\le C_{\\mathcal A}Z^{M-\\log_ZX}. \\] The profile \\(W^\\sharp\\) can be partitioned into smooth annuli. Below its main scale, central normalizati" }, { "id": "eq:centered-reflection-length", "kind": "anchor", "label": "eq:centered-reflection-length", "owner": "sec:plain", "section": "sec:plain", "anchor": { "source": "paper.tex", "line_start": 12803, "line_end": 12804, "char_start": 594354, "char_end": 594391 }, "argument_gist": "", "excerpt": "C_{\\rm ref}}{\\log Z} \\le M-n+\\xi \\qquad\\left(\\frac{\\log C_{\\rm ref}}{\\log Z}\\le\\frac\\xi2\\right). This is a linear bound also when \\(M-n\\) is slightly negative; if its right side is negative, no such retained scale exists. The smaller upper-annulus range changes only the fixed decay, seminorm, and height orders used for the discarded tail. For \\(z=0\\), reflect each original fac" }, { "id": "old-eq:2.1h", "kind": "anchor", "label": "old-eq:2.1h", "owner": "sec:plain", "section": "sec:plain", "anchor": { "source": "paper.tex", "line_start": 12819, "line_end": 12820, "char_start": 595119, "char_end": 595138 }, "argument_gist": "", "excerpt": "lengths satisfy n_i^*\\le M/2+\\xi,\\qquad A^*=n_1^*+n_2^*\\le M+2\\xi\\le M+\\delta, provided \\(2\\xi\\le\\delta\\). We call this the padded zero-slot core. Here \\(\\delta\\) is the positive padding parameter chosen below after the width floor and step. There is no repeated reflection at the boundary \\(M/2\\). Reflection and scale suprema are taken before centering; on a" }, { "id": "sec:weighted", "kind": "anchor", "label": "sec:weighted", "owner": "sec:plain", "section": "sec:plain", "anchor": { "source": "paper.tex", "line_start": 12831, "line_end": 12832, "char_start": 595748, "char_end": 595768 }, "argument_gist": "", "excerpt": "tive inducing character belongs to \\(\\Theta\\). \\subsection{Prime estimates and the induction order} For \\(\\kappa<1\\), put \\(s_\\kappa=(1+\\kappa)/2\\). Under the hypothesis \\(s_\\kappa\\ge\\beta_*\\), the global part of Lemma~\\ref{lem:logarithmic-control} gives, for every fixed \\(e>0\\), \\frac{L'}{L}(s_\\kappa+e+it,\\psi) \\ll_e \\log\\!\\bigl(2Q_\\psi(3+|t|)^2\\bigr) \\en" }, { "id": "old-eq:3.4", "kind": "anchor", "label": "old-eq:3.4", "owner": "sec:plain", "section": "sec:plain", "anchor": { "source": "paper.tex", "line_start": 12839, "line_end": 12840, "char_start": 596046, "char_end": 596064 }, "argument_gist": "", "excerpt": ">0\\), \\frac{L'}{L}(s_\\kappa+e+it,\\psi) \\ll_e \\log\\!\\bigl(2Q_\\psi(3+|t|)^2\\bigr) for a primitive nonprincipal inducing character \\(\\psi\\). To estimate a prime annulus of scale \\(P\\), apply Mellin inversion to a smooth von Mangoldt sum and move its contour to \\(\\Re s=s_\\kappa+e\\). There are no poles in the region of the shift, and the Mellin transform has arbitra" }, { "id": "old-eq:3.5", "kind": "anchor", "label": "old-eq:3.5", "owner": "sec:plain", "section": "sec:plain", "anchor": { "source": "paper.tex", "line_start": 12876, "line_end": 12877, "char_start": 597920, "char_end": 597938 }, "argument_gist": "", "excerpt": "1}p_j(\\boldsymbol W)^b (1+|\\boldsymbol\\omega|)^h \\qquad(k\\in\\mathcal R_z,\\ z>0). Here \\(j\\) may be rounded up to an integer and \\(2s_\\kappa-1=\\kappa\\). The orders and \\(C\\) may depend on the fixed data, slot count, loss, and any fixed requested internal logarithmic or normalized-twist derivatives, but not on \\(Z\\), moving labels, rows, or the numerical heights." }, { "id": "old-eq:3.6", "kind": "anchor", "label": "old-eq:3.6", "owner": "sec:plain", "section": "sec:plain", "anchor": { "source": "paper.tex", "line_start": 12934, "line_end": 12935, "char_start": 601156, "char_end": 601174 }, "argument_gist": "", "excerpt": "e z\\), whence z\\le\\frac{M}{6\\kappa}\\le\\frac{2M}{9}, \\qquad \\kappa z\\le\\frac M6. Thus the extra terminal exponent for positive slots is at most \\(\\rho/6\\). These terminal exponents can be made smaller than the reserved final loss by choosing \\(\\rho,\\delta\\) sufficiently small. For a width above the floor, the two-transform estimate below will first prove the un" }, { "id": "old-eq:2.2", "kind": "anchor", "label": "old-eq:2.2", "owner": "sec:plain", "section": "sec:plain", "anchor": { "source": "paper.tex", "line_start": 12945, "line_end": 12946, "char_start": 601608, "char_end": 601626 }, "argument_gist": "", "excerpt": "xplain now why the remaining inputs can be compared with that range. Set L=M/4. For \\(A>5M/6\\), the comparison lengths \\(L,A-z-L\\) are both at least \\(L\\). In the positive-slot case, \\((6\\kappa-1)z\\le M-A2L\\). If one original plain length is \\(b0\\), use \\((6\\kappa-1)z\\le M-A\\), \\(A>5M/6\\), and \\(z0\\), for the stated masks, and for the finit" }, { "id": "old-eq:2.18f", "kind": "anchor", "label": "old-eq:2.18f", "owner": "lem:centered-lattice-cancellation", "section": "sec:plain", "anchor": { "source": "paper.tex", "line_start": 14592, "line_end": 14593, "char_start": 675658, "char_end": 675678 }, "argument_gist": "", "excerpt": "_2}^{it}D_{\\mathbf b}(l_1,l_2) \\right| \\ll Z^{\\epsilon_1}(1+|t|)^{2J}T^{1/2}Z^{-r}. Without a nonnegative lower length, the same expression is \\(O(T^{1/2})\\). First omit the mask. The primary generators representing ideals outside \\(\\mathcal S\\), with the fixed character weight \\(\\vartheta\\), are a finite weighted collection of residue cl" }, { "id": "old-eq:2.18", "kind": "anchor", "label": "old-eq:2.18", "owner": "sec:plain", "section": "sec:plain", "anchor": { "source": "paper.tex", "line_start": 14717, "line_end": 14718, "char_start": 680831, "char_end": 680850 }, "argument_gist": "", "excerpt": "\\mathfrak t,\\mathbf J}(h')|\\\\ &\\hspace{15mm}\\ll Z^{a_0-b_2-r+2\\theta_N+\\epsilon_1}. Here and below a fixed polynomial in the separated heights is understood. To check the bound, the raw coefficients and volumes give the reference exponent \\(a_0-b_2+e_1+e_2\\) and reduction \\(-\\widetilde r_1-\\widetilde r_2\\), while the divisor count gives \\(t_-\\). Equation~\\eqref{eq" }, { "id": "old-eq:2.18h", "kind": "anchor", "label": "old-eq:2.18h", "owner": "sec:plain", "section": "sec:plain", "anchor": { "source": "paper.tex", "line_start": 14731, "line_end": 14732, "char_start": 681378, "char_end": 681398 }, "argument_gist": "", "excerpt": "-\\widetilde r_1)_+ &=(t_- -\\widetilde r_2)-\\max(r,\\widetilde r_1)\\\\ &\\le\\omega_2-r. Adding \\(e_1+e_2\\) leaves at most \\(e_1+(e_2+\\omega_2)-r\\le2\\theta_N-r\\). The divisor-bounded allocations contribute only \\(\\epsilon_1\\), proving Equation~\\eqref{old-eq:2.18}. This applies to the artificial terms even when quotients retain \\(\\mathfrak t\\)-primes, because the coeffici" }, { "id": "old-eq:2.19", "kind": "anchor", "label": "old-eq:2.19", "owner": "sec:plain", "section": "sec:plain", "anchor": { "source": "paper.tex", "line_start": 14762, "line_end": 14763, "char_start": 682816, "char_end": 682835 }, "argument_gist": "", "excerpt": "aggregate support perturbations, by A-\\tfrac56M-\\tfrac23v-(L-v)_+ \\le (A-M)_+. Indeed, for \\(0\\le v\\le L\\) the left side is \\(A-5M/6-L+v/3\\), which increases with \\(v\\); for \\(v\\ge L\\) it is \\(A-5M/6-2v/3\\), which decreases. Its maximum is attained at \\(v=L=M/4\\) and equals \\(A-M\\). For positive slots, Equation~\\eqref{old-eq:3.9} implies \\(A\\le M\\). For the ze" }, { "id": "old-eq:2.1j", "kind": "anchor", "label": "old-eq:2.1j", "owner": "sec:plain", "section": "sec:plain", "anchor": { "source": "paper.tex", "line_start": 14825, "line_end": 14826, "char_start": 686259, "char_end": 686278 }, "argument_gist": "", "excerpt": "he positive-slot affine expression is at most |\\Delta M|+|\\Delta A|+5|\\Delta z|, because \\(0\\le6\\kappa-1\\le5\\). The total change of \\(z\\) is aggregated before this inequality is used. These observations give one numerical \\(C_*\\), enlarged for the fixed number of operations per stage, that covers all frequency, normalization, and boundary errors by \\(C_*\\xi\\)." }, { "id": "old-eq:2.1i", "kind": "anchor", "label": "old-eq:2.1i", "owner": "sec:plain", "section": "sec:plain", "anchor": { "source": "paper.tex", "line_start": 14885, "line_end": 14886, "char_start": 689458, "char_end": 689477 }, "argument_gist": "", "excerpt": "\\frac{\\epsilon}{16C_*D}\\right\\},\\qquad \\epsilon_0<\\frac{\\epsilon}{16C_*D}. These choices make the strict drop at least \\(\\sigma/2\\), preserve the comparison margins and padded core, and are uniform for \\(\\kappa\\in[3/4,1]\\). \\paragraph{Choices for a fixed slot system.} Now fix any \\(Z\\)-independent \\(N\\), arithmetic data, and profile windows. Form \\(H_N\\)" }, { "id": "sec:amplitudes", "kind": "anchor", "label": "sec:amplitudes", "owner": "sec:refined-row-counts", "section": "sec:refined-row-counts", "anchor": { "source": "paper.tex", "line_start": 14995, "line_end": 14996, "char_start": 695478, "char_end": 695500 }, "argument_gist": "", "excerpt": "gle height allocation in Equation~\\eqref{eq:stage-height-allocation}. \\subsection{Prime amplitudes} For a main physical slot of scale \\(P_i=Z^{\\ell_i}\\), write its central factor as \\[ Q_i(u;z)=P_i^{-1/2}\\sum_{p\\in1_T} \\overline{\\chi_p(u)}W_i(q_p/P_i)(q_p/P_i)^{z-1}. \\] The main part \\(\\mathcal Q_i\\) in Proposition~\\ref{prop:probe-errors} is exactly \\[ \\mathcal Q_i(u;z)=-\\sum_{p\\in1_T}\\over" }, { "id": "old-eq:6.6", "kind": "anchor", "label": "old-eq:6.6", "owner": "sec:refined-row-counts", "section": "sec:refined-row-counts", "anchor": { "source": "paper.tex", "line_start": 15090, "line_end": 15091, "char_start": 699685, "char_end": 699703 }, "argument_gist": "", "excerpt": "gin{equation} q=\\frac{\\sum_i\\ell_i g_i}{\\ell},\\qquad 0\\le q\\le\\delta/2,\\qquad \\ell=\\sum_i\\ell_i. This \\(q\\) records prime amplitude; it is not the conductor exponent denoted \\(q\\) in the auxiliary fourth moment. \\subsection{Selected prime slots and the row counts} Use \\(\\Delta\\) and \\(\\kappa\\) from Equation~\\eqref{eq:part-II-bootstrap}: \\[ 0<\\Delta\\le\\fra" }, { "id": "sec:selection", "kind": "anchor", "label": "sec:selection", "owner": "sec:refined-row-counts", "section": "sec:refined-row-counts", "anchor": { "source": "paper.tex", "line_start": 15095, "line_end": 15096, "char_start": 699887, "char_end": 699908 }, "argument_gist": "", "excerpt": "denoted \\(q\\) in the auxiliary fourth moment. \\subsection{Selected prime slots and the row counts} Use \\(\\Delta\\) and \\(\\kappa\\) from Equation~\\eqref{eq:part-II-bootstrap}: \\[ 0<\\Delta\\le\\frac1{24},\\qquad \\kappa=2\\beta_*-1=\\frac34+2\\Delta\\in(3/4,5/6],\\qquad \\alpha:=\\frac56. \\] The parameter \\(\\kappa\\) remains dynamic; \\(\\alpha\\) is the fixed slope from sixth-power amplification. Since \\(\\ka" }, { "id": "eq:plain-zero-loss-capacity", "kind": "anchor", "label": "eq:plain-zero-loss-capacity", "owner": "sec:refined-row-counts", "section": "sec:refined-row-counts", "anchor": { "source": "paper.tex", "line_start": 15177, "line_end": 15178, "char_start": 703971, "char_end": 704006 }, "argument_gist": "", "excerpt": "row width one, is z_P(m)=\\frac{1-2m}{6\\kappa} =\\frac{1-2m}{9/2+12\\Delta}. Indeed the moment is applied to two copies of the plain witness, so its condition is \\(2m+6\\kappa z\\le1\\). For an inverse witness of length \\(r<1\\), put \\(z_M(r)=(1-r)/2\\). We use this capacity only where the second strict inequality in Lemma~\\ref{lem:marked} also has a fixed margin" }, { "id": "eq:no-slot-inverse-count", "kind": "anchor", "label": "eq:no-slot-inverse-count", "owner": "prop:detector-counts", "section": "sec:refined-row-counts", "anchor": { "source": "paper.tex", "line_start": 15282, "line_end": 15283, "char_start": 708490, "char_end": 708522 }, "argument_gist": "", "excerpt": "erms are at most \\(\\epsilon_m/2\\). Thus division by the inverse witness spike gives \\#\\mathcal B&\\ll U^{e(r)-\\delta r+\\epsilon}(1+T_1)^{A_{\\mathcal A}},\\\\ e(r)-\\delta r&= 1-\\delta r,&0\\le r\\le1,\\\\ 1-\\alpha+(\\alpha-\\delta)r,&r\\ge1. In particular this is not an application of the strict marked moment w" }, { "id": "eq:long-count", "kind": "anchor", "label": "eq:long-count", "owner": "prop:detector-counts", "section": "sec:refined-row-counts", "anchor": { "source": "paper.tex", "line_start": 15308, "line_end": 15309, "char_start": 709358, "char_end": 709379 }, "argument_gist": "", "excerpt": "is is at most L(t)=1-\\alpha+(\\alpha-\\delta)t =1-\\delta+(\\alpha-\\delta)(t-1) up to \\(O(\\epsilon)\\). This includes \\(r=1\\), where the two branches in Equation~\\eqref{eq:no-slot-inverse-count} agree. If \\(m\\ge1/2\\), the zero-slot case of Lemma~\\ref{lem:plain} gives count exponent \\(1-2\\delta m\\le1-\\delta\\). Since \\(R_{\\rm short}(t)\\ge1-\\delta\\), this also sati" }, { "id": "eq:selected-counts", "kind": "anchor", "label": "eq:selected-counts", "owner": "prop:detector-counts", "section": "sec:refined-row-counts", "anchor": { "source": "paper.tex", "line_start": 15335, "line_end": 15336, "char_start": 710455, "char_end": 710481 }, "argument_gist": "", "excerpt": ")&=1-\\delta\\left\\{\\frac{4x}{9} +\\left(2-\\frac{8x}{9}\\right)(t-r)\\right\\}. Both the actual plain exponent \\(1-2\\delta m-2q(1-2m)/(9/2+12\\Delta)\\) and its baseline version decrease with \\(m\\): their derivatives are respectively \\[ -2\\delta+\\frac{4q}{9/2+12\\Delta}<0,\\qquad -2\\delta+\\frac{8q}{9}<0. \\] The \\(O(\\epsilon)\\) replacement of \\(m\\) is therefore leg" }, { "id": "eq:capacity-comparison", "kind": "anchor", "label": "eq:capacity-comparison", "owner": "prop:detector-counts", "section": "sec:refined-row-counts", "anchor": { "source": "paper.tex", "line_start": 15402, "line_end": 15403, "char_start": 712486, "char_end": 712516 }, "argument_gist": "", "excerpt": ") &=\\frac{24q(1-2m)\\Delta}{(9/2)(9/2+12\\Delta)}\\\\ &\\le\\frac{\\Delta}{4}+O(\\epsilon). For the inequality use \\(m\\ge1/3-O(\\epsilon)\\), \\(2q\\le1\\), and the lower bound four for each denominator. The constants in the \\(O(\\epsilon)\\) term are uniform. The weighted average R_{\\rm short}(t) &=\\frac{(2-8x/9)A_I(r)+(1-x)S_t(r)}{D_x}\\\\ &=" }, { "id": "eq:short-crossing-count", "kind": "anchor", "label": "eq:short-crossing-count", "owner": "prop:detector-counts", "section": "sec:refined-row-counts", "anchor": { "source": "paper.tex", "line_start": 15415, "line_end": 15416, "char_start": 712871, "char_end": 712902 }, "argument_gist": "", "excerpt": "=\\frac{(2-8x/9)A_I(r)+(1-x)S_t(r)}{D_x}\\\\ &=1-\\delta+\\frac{\\delta P_x}{D_x}(3/2-t) is independent of \\(r\\): the \\(r\\)-coefficients cancel, and the weights sum to \\(D_x\\). It is the common value at \\(r_*(t)\\). Since \\(S_t\\) is increasing in \\(r\\) and \\(A_I\\) is decreasing, the chosen short count is at most \\(R_{\\rm short}(t)+\\Delta/4+O(\\epsilon)\\). \\paragraph{Smal" }, { "id": "sec:principal-signal", "kind": "anchor", "label": "sec:principal-signal", "owner": "sec:conclusion", "section": "sec:conclusion", "anchor": { "source": "paper.tex", "line_start": 15548, "line_end": 15549, "char_start": 718979, "char_end": 719007 }, "argument_gist": "", "excerpt": "ning real losses, is made in the concluding order of choices. \\subsection{The principal normalizer} We verify the two correction hypotheses of Lemma~\\ref{lem:principal-residue-interface}. First, Equation~\\eqref{eq:principal-local-tuple-bound} gives \\[ |\\mathfrak H_{\\eta,1,Z}(s,w,z)|\\ll Z^{\\ell\\Re z} \\] uniformly in all imaginary parts on every fixed real box in the second Euler region. In par" }, { "id": "eq:principal-scalar", "kind": "anchor", "label": "eq:principal-scalar", "owner": "sec:conclusion", "section": "sec:conclusion", "anchor": { "source": "paper.tex", "line_start": 15575, "line_end": 15576, "char_start": 720005, "char_end": 720032 }, "argument_gist": "", "excerpt": "arge \\(Z\\), with a common lower threshold allowed to depend on the fixed data. Set A_T(Z)=(-1)^K Z^{-\\ell/6}\\prod_{i=1}^K S_i(Z). Using \\(\\log P_i=\\ell_i\\log Z\\) and \\(\\sum_i\\ell_i=\\ell\\), we obtain \\[ A_T(Z)\\sim \\frac{(-1)^K}{|T|^K(\\log Z)^K} \\prod_{i=1}^K\\left\\{\\frac1{\\ell_i} \\int_0^\\infty W_i(y)y^{-5/6}\\,dy\\right\\}. \\] Consequently \\(A_T(Z)\\ne0\\) on t" }, { "id": "eq:part-II-normalized-low", "kind": "anchor", "label": "eq:part-II-normalized-low", "owner": "sec:conclusion", "section": "sec:conclusion", "anchor": { "source": "paper.tex", "line_start": 15644, "line_end": 15645, "char_start": 722736, "char_end": 722769 }, "argument_gist": "", "excerpt": ":probe-low} and the subpower bound for \\(A_T^{-1}\\) give, for every \\(\\epsilon>0\\), |J_{\\mathrm{II},\\eta}(Z)| \\ll_{\\eta,\\epsilon} Z^{C(7/8)+\\epsilon}. The same scalar is used for the high comparison. Choose the fixed cutoff as required for Equation~\\eqref{eq:shared-principal-nonvanishing}. The associated \\(H_\\eta\\) is holomorphic on \\(\\Re s>7/8\\), satisfies the" }, { "id": "old-eq:5.14", "kind": "anchor", "label": "old-eq:5.14", "owner": "sec:conclusion", "section": "sec:conclusion", "anchor": { "source": "paper.tex", "line_start": 15673, "line_end": 15674, "char_start": 723789, "char_end": 723808 }, "argument_gist": "", "excerpt": "(\\beta_*)+e-\\kappa_P+\\epsilon}, \\] where the two geometric margins are m_w:=\\frac{l_y}{20}=\\frac{23}{960},\\qquad m_z:=\\frac h{600}=\\frac{13}{9600}. Lemma~\\ref{lem:principal-residue-interface} includes the residue paths, the scalar Jacobian, and the rightward shift of the main integral. In particular the last term is the error estimated on its origina" }, { "id": "eq:common-high-exponent", "kind": "anchor", "label": "eq:common-high-exponent", "owner": "lem:high-bin", "section": "sec:conclusion", "anchor": { "source": "paper.tex", "line_start": 15726, "line_end": 15727, "char_start": 726266, "char_end": 726297 }, "argument_gist": "", "excerpt": ", and \\(\\beta_*\\), but not otherwise on the target. For each occurring pair define E(d) &=a-\\frac78+h\\left(\\frac{17}{50}-\\frac16\\right) -a l_y-(1-a)\\ell-(\\delta/2-q)\\ell +d\\left(R+\\frac\\delta2-\\frac{17}{50}\\right)\\\\ &=C_0+\\frac23\\delta+\\frac q6-h(1-R) +(d-h)\\left(R+\\frac\\delta2-\\frac{17}{50}\\right), \\qquad C_0=-\\frac1{48}. \\end{split" }, { "id": "eq:floor-bound", "kind": "anchor", "label": "eq:floor-bound", "owner": "sec:conclusion", "section": "sec:conclusion", "anchor": { "source": "paper.tex", "line_start": 15846, "line_end": 15847, "char_start": 731710, "char_end": 731732 }, "argument_gist": "", "excerpt": "no zero witness. The ideal count \\(R=1\\), together with \\(q\\le\\delta_0/2\\), gives E(h)\\le C_0+\\frac34\\delta_0=-\\frac7{1200}<0. Its frequency slope \\(R+\\delta_0/2-17/50\\) is positive, so this bounds every \\(d\\le h\\). A small extension above \\(h\\) will be controlled below. \\subsection{Small and large row norms} Set \\(d_{\\min}=1/100\\). Lemma~\\re" }, { "id": "sec:tails", "kind": "anchor", "label": "sec:tails", "owner": "sec:conclusion", "section": "sec:conclusion", "anchor": { "source": "paper.tex", "line_start": 15853, "line_end": 15854, "char_start": 731979, "char_end": 731996 }, "argument_gist": "", "excerpt": "\\). A small extension above \\(h\\) will be controlled below. \\subsection{Small and large row norms} Set \\(d_{\\min}=1/100\\). Lemma~\\ref{lem:compensated-absolute-local-bounds} bounds the positive sum of the full selected tuples, with every \\(G_p\\) retained before taking absolute values. On the small-row lines \\[ \\Re s=\\beta_*+e,\\qquad \\Re w=1/2,\\qquad \\Re z=17/50 \\] Equation~\\eqref{eq:absolute-" }, { "id": "eq:small-frequency-margin", "kind": "anchor", "label": "eq:small-frequency-margin", "owner": "sec:conclusion", "section": "sec:conclusion", "anchor": { "source": "paper.tex", "line_start": 15876, "line_end": 15877, "char_start": 732953, "char_end": 732986 }, "argument_gist": "", "excerpt": "lected moment is used here. The \\(d\\)-independent part of the relative exponent is h\\left(\\frac{17}{50}-\\frac16\\right)-\\frac{l_y}{2} =-\\frac{79}{800}. The row exponent in Equation~\\eqref{eq:small-row-bound} is \\(63/50<2\\). Thus the small dyadic sum in Equation~\\eqref{eq:small-row-sum}, measured relative to \\(C(\\beta_*)\\), has exponent at most \\[ -\\frac{79}{800}" }, { "id": "sec:adaptive-endpoint", "kind": "anchor", "label": "sec:adaptive-endpoint", "owner": "sec:conclusion", "section": "sec:conclusion", "anchor": { "source": "paper.tex", "line_start": 15920, "line_end": 15921, "char_start": 734619, "char_end": 734648 }, "argument_gist": "", "excerpt": "local-compensation}, not the central main/error factorization. \\subsection{The endpoint inequality} We next bound \\(E(h)\\), first without adjustable losses or height factors. They will be restored with a quantified order of choices. Recall \\[ \\kappa=\\frac34+2\\Delta,\\qquad \\alpha=\\frac56. \\] At the live upper endpoint \\(\\delta=\\alpha\\), take \\(t=3/2\\) in Proposition~\\ref{prop:detector-witness}." }, { "id": "eq:large-delta-endpoint", "kind": "anchor", "label": "eq:large-delta-endpoint", "owner": "sec:conclusion", "section": "sec:conclusion", "anchor": { "source": "paper.tex", "line_start": 15946, "line_end": 15947, "char_start": 735660, "char_end": 735691 }, "argument_gist": "", "excerpt": "C_0+\\left(\\frac34-h\\right)\\delta+O(\\epsilon) =-\\frac1{48}-\\frac{\\delta}{16}+O(\\epsilon)<0. Now assume \\(1/50<\\delta<\\alpha\\). Put \\(x=q/\\delta\\in[0,1/2]\\), and use \\(D_x,P_x,R_{\\rm short},L\\) from Proposition~\\ref{prop:detector-counts}. The parameter \\(\\kappa\\) remains the dynamic value \\(3/4+2\\Delta\\), not the upper bound \\(5/6\\). Thus the actual plain capacity used i" }, { "id": "eq:target-cutoff", "kind": "anchor", "label": "eq:target-cutoff", "owner": "sec:conclusion", "section": "sec:conclusion", "anchor": { "source": "paper.tex", "line_start": 15966, "line_end": 15967, "char_start": 736438, "char_end": 736462 }, "argument_gist": "", "excerpt": "thcal J=(\\alpha-\\delta)D_x+\\delta P_x,\\qquad t=1+\\frac{\\delta P_x}{2\\mathcal J},\\qquad R_*=L(t). The denominator is positive. Indeed \\[ \\frac{37}{18}\\le D_x\\le3,\\qquad \\frac79\\le P_x\\le2,\\qquad D_x-P_x=1+x-\\frac{8x^2}{9}>0, \\] and consequently \\[ \\frac{35}{54}\\le\\mathcal J\\le\\frac52. \\] Because \\(0<\\delta<\\alpha\\), one has \\(\\mathcal J>\\delta P_x>0\\), so \\(10, for every occurring \\((R,q)\\) and every moderate \\(d\\), with a fixed fraction of \\(m_{\\rm high}\\) still reserved. 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Milne, Class Field Theory, course notes, version 4.03, 6 August 2020. \\url{https://www.jmilne.org/math/CourseNotes/CFT.pdf}.", "urls": [ "https://www.jmilne.org/math/CourseNotes/CFT.pdf" ], "argument_gist": "Finite-order Hecke/ray-class identification and Kummer character construction.", "anchor": { "source": "paper.tex", "line_start": 16623, "line_end": 16628, "char_start": 764086, "char_end": 764244 }, "input_role": "used", "role_evidence": "paper.tex lines 55–59; bibliography[22] at lines 16623–16628", "gist_origin": "source_audit_annotation" }, { "id": "external:OpenAICubicFirstMoment", "kind": "external", "label": "[23] OpenAICubicFirstMoment", "bibliography_key": "OpenAICubicFirstMoment", "bibliography_number": "23", "title": "OpenAI, An unconditional first moment for cubic Gauss sums, OpenAI Math Release preprint \\href{https://github.com/openai/math/blob/main/preprints/An-unconditional-first-moment-for-cubic-Gauss-sums-September-25-2026/paper.pdf}{OAI:An-unconditional-first-moment-for-cubic-Gauss-sums-September-25-2026}, 2026.", "urls": [ "https://github.com/openai/math/blob/main/preprints/An-unconditional-first-moment-for-cubic-Gauss-sums-September-25-2026/paper.pdf" ], "argument_gist": "Theorem 1.1 of the independent cubic first moment manuscript is explicitly not used.", "anchor": { "source": "paper.tex", "line_start": 16628, "line_end": 16635, "char_start": 764244, "char_end": 764592 }, "input_role": "excluded", "role_evidence": "paper.tex lines 125–129; bibliography[23] at lines 16628–16635", "gist_origin": "source_audit_annotation" }, { "id": "external:Patterson", "kind": "external", "label": "[24] Patterson", "bibliography_key": "Patterson", "bibliography_number": "24", "title": "S. J. Patterson, A cubic analogue of the theta series, Journal f\\\"ur die reine und angewandte Mathematik 296 (1977), 125--161. \\href{https://doi.org/10.1515/crll.1977.296.125}{doi:10.1515/crll.1977.296.125}.", "urls": [ "https://doi.org/10.1515/crll.1977.296.125" ], "argument_gist": "Original theta/coefficient calculations underlying DR; the paper says its formal input remains DR.", "anchor": { "source": "paper.tex", "line_start": 16635, "line_end": 16642, "char_start": 764592, "char_end": 764837 }, "input_role": "background", "role_evidence": "paper.tex lines 120–124; bibliography[24] at lines 16635–16642", "gist_origin": "source_audit_annotation" }, { "id": "external:Riemann", "kind": "external", "label": "[25] Riemann", "bibliography_key": "Riemann", "bibliography_number": "25", "title": "B. Riemann, Ueber die Anzahl der Primzahlen unter einer gegebenen Gr\\\"osse, Monatsberichte der K\\\"oniglichen Preussischen Akademie der Wissenschaften zu Berlin (1859), 671--680. Manuscript, transcription, and translation at \\url{https://www.claymath.org/library/historical/riemann/r.html}.", "urls": [ "https://www.claymath.org/library/historical/riemann/r.html" ], "argument_gist": "Historical connection of zeta zeros and primes.", "anchor": { "source": "paper.tex", "line_start": 16642, "line_end": 16650, "char_start": 764837, "char_end": 765153 }, "input_role": "background", "role_evidence": "paper.tex lines 87–91; bibliography[25] at lines 16642–16650", "gist_origin": "source_audit_annotation" }, { "id": "external:TaoNonresidues", "kind": "external", "label": "[26] TaoNonresidues", "bibliography_key": "TaoNonresidues", "bibliography_number": "26", "title": "T. Tao, The Elliott--Halberstam conjecture implies the Vinogradov least quadratic nonresidue conjecture, Algebra \\& Number Theory 9 (2015), no. 4, 1005--1034. \\href{https://doi.org/10.2140/ant.2015.9.1005}{doi:10.2140/ant.2015.9.1005}; \\href{https://msp.org/ant/2015/9-4/ant-v9-n4-p09-p.pdf}{publisher PDF}.", "urls": [ "https://doi.org/10.2140/ant.2015.9.1005", "https://msp.org/ant/2015/9-4/ant-v9-n4-p09-p.pdf" ], "argument_gist": "Vinogradov least nonresidue conjecture contextual citation.", "anchor": { "source": "paper.tex", "line_start": 16650, "line_end": 16658, "char_start": 765153, "char_end": 765503 }, "input_role": "background", "role_evidence": "paper.tex lines 331–335; bibliography[26] at lines 16650–16658", "gist_origin": "source_audit_annotation" }, { "id": "external:TZ", "kind": "external", "label": "[27] TZ", "bibliography_key": "TZ", "bibliography_number": "27", "title": "J. Thorner and A. Zaman, A unified and improved Chebotarev density theorem, Algebra \\& Number Theory 13 (2019), no. 5, 1039--1068. \\href{https://doi.org/10.2140/ant.2019.13.1039}{doi:10.2140/ant.2019.13.1039}; \\href{https://arxiv.org/abs/1803.02823}{arXiv:1803.02823}.", "urls": [ "https://doi.org/10.2140/ant.2019.13.1039", "https://arxiv.org/abs/1803.02823" ], "argument_gist": "Fixed-extension/Frobenius prime counting and nontrivial sextic detection; its classical zero-free region statement is background.", "anchor": { "source": "paper.tex", "line_start": 16658, "line_end": 16665, "char_start": 765503, "char_end": 765802 }, "input_role": "used", "role_evidence": "paper.tex lines 97–101; bibliography[27] at lines 16658–16665", "gist_origin": "source_audit_annotation" }, { "id": "external:ValleePoussin", "kind": "external", "label": "[28] ValleePoussin", "bibliography_key": "ValleePoussin", "bibliography_number": "28", "title": "C.-J. de la Vall\\'ee Poussin, Recherches analytiques sur la th\\'eorie des nombres premiers. Premi\\`ere partie: La fonction $\\zeta(s)$ de Riemann et les nombres premiers en g\\'en\\'eral, Annales de la Soci\\'et\\'e scientifique de Bruxelles 20, deuxi\\`eme partie (1896), 183--256. Reprinted in Recherches analytiques sur la th\\'eorie des nombres premiers, Hayez, Brussels, 1897, \\url{https://books.google.com/books?id=7e0GAAAAYAAJ}.", "urls": [ "https://books.google.com/books?id=7e0GAAAAYAAJ" ], "argument_gist": "Historical prime-number theorem/nonvanishing on Re(s)=1.", "anchor": { "source": "paper.tex", "line_start": 16665, "line_end": 16676, "char_start": 765802, "char_end": 766279 }, "input_role": "background", "role_evidence": "paper.tex lines 90–94; bibliography[28] at lines 16665–16676", "gist_origin": "source_audit_annotation" } ], "edges": [ { "from": "thm:main", "to": "sec:introduction", "type": "internal_reference", "evidence": "ta(s)$. A pole at $s=1$ for a principal character is allowed. In particular, Theorem~\\ref{thm:main} resolves the quasi-Riemann hypothesis affirmatively: the supremum of the real parts of the nontrivial zeros of $\\zeta$, namely it", "anchor": { "source": "paper.tex", "line_start": 113, "line_end": 113, "char_start": 4858, "char_end": 4872 }, "origin": "source_reference", "context": "narrative" }, { "from": "prop:completed-reflection", "to": "sec:introduction", "type": "internal_reference", "evidence": "nuscript \\cite[Theorem~1.1]{OpenAICubicFirstMoment}; that result is also not used here. Proposition~\\ref{prop:completed-reflection} derives the reflection used here while retaining the characters' zero-on-nonunit restrictions. The character large-sieve argumen", "anchor": { "source": "paper.tex", "line_start": 128, "line_end": 128, "char_start": 5808, "char_end": 5839 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:sextic-large-sieve", "to": "sec:introduction", "type": "internal_reference", "evidence": "ubic estimate \\cite[Theorem 2]{HB}. We prove the precise sextic specialization needed here in Lemma~\\ref{lem:sextic-large-sieve}, using paired residue characters to carry out that recursion in the primary-generator convention. An Eisenstein-integer form is a", "anchor": { "source": "paper.tex", "line_start": 137, "line_end": 137, "char_start": 6307, "char_end": 6335 }, "origin": "source_reference", "context": "narrative" }, { "from": "part:eleven-twelfths", "to": "sec:introduction", "type": "internal_reference", "evidence": "a completed cubic-theta sum but using a different normalized sum in the continuation argument. Part~\\ref{part:eleven-twelfths} proves the corresponding $11/12$ assertion in Theorem~\\ref{thm:eleven-twelfths}, already obtaining a fixed zero-free half-plane f", "anchor": { "source": "paper.tex", "line_start": 161, "line_end": 161, "char_start": 7716, "char_end": 7742 }, "origin": "source_reference", "context": "narrative" }, { "from": "thm:eleven-twelfths", "to": "sec:introduction", "type": "internal_reference", "evidence": "ion argument. Part~\\ref{part:eleven-twelfths} proves the corresponding $11/12$ assertion in Theorem~\\ref{thm:eleven-twelfths}, already obtaining a fixed zero-free half-plane for both families. Part~\\ref{part:seven-eighths} starts from that conclusion and", "anchor": { "source": "paper.tex", "line_start": 162, "line_end": 162, "char_start": 7797, "char_end": 7822 }, "origin": "source_reference", "context": "narrative" }, { "from": "part:seven-eighths", "to": "sec:introduction", "type": "internal_reference", "evidence": "m~\\ref{thm:eleven-twelfths}, already obtaining a fixed zero-free half-plane for both families. Part~\\ref{part:seven-eighths} starts from that conclusion and introduces prime compensation, asymmetric scales, and two additional moment estimates. The contri", "anchor": { "source": "paper.tex", "line_start": 164, "line_end": 164, "char_start": 7895, "char_end": 7919 }, "origin": "source_reference", "context": "narrative" }, { "from": "sec:analytic-reduction", "to": "sec:introduction", "type": "internal_reference", "evidence": "ntinue every target reciprocal across a common positive distance to the left of $\\beta_*$. Section~\\ref{sec:analytic-reduction} gives the precise criterion. For each target $\\eta$ and large real scale $Z$, it compares a normalized character sum with a Melli", "anchor": { "source": "paper.tex", "line_start": 181, "line_end": 181, "char_start": 8848, "char_end": 8876 }, "origin": "source_reference", "context": "narrative" }, { "from": "sec:arithmetic", "to": "sec:introduction", "type": "internal_reference", "evidence": "-character constants and lower thresholds may depend on it. \\paragraph{The completed sum.} Section~\\ref{sec:arithmetic} sets up the sextic residue characters over $\\mathcal O$, always retaining their zero values on nonunits. The base sum in Section~", "anchor": { "source": "paper.tex", "line_start": 198, "line_end": 198, "char_start": 9703, "char_end": 9723 }, "origin": "source_reference", "context": "narrative" }, { "from": "sec:probe", "to": "sec:introduction", "type": "internal_reference", "evidence": "aracters over $\\mathcal O$, always retaining their zero values on nonunits. The base sum in Section~\\ref{sec:probe} averages smoothed cubic-theta Fourier coefficients against these characters and a fixed target $\\eta$. After the fixed rescaling", "anchor": { "source": "paper.tex", "line_start": 200, "line_end": 200, "char_start": 9853, "char_end": 9868 }, "origin": "source_reference", "context": "narrative" }, { "from": "prop:completed-reflection", "to": "sec:introduction", "type": "internal_reference", "evidence": "ors. The resulting completed base sum has two exact representations. The reflection in Proposition~\\ref{prop:completed-reflection} transforms its theta coefficients while retaining the character zeros in the resulting formula. Poisson summation transforms the", "anchor": { "source": "paper.tex", "line_start": 211, "line_end": 212, "char_start": 10527, "char_end": 10558 }, "origin": "source_reference", "context": "narrative" }, { "from": "sec:local-euler", "to": "sec:introduction", "type": "internal_reference", "evidence": ",a\\in\\mathcal O$ and $u$ is sixth-power-free: every prime valuation of $u$ is at most five. Section~\\ref{sec:local-euler} then expresses the contribution of each such row through a quotient of Hecke $L$-functions and a controlled Euler product. For $u", "anchor": { "source": "paper.tex", "line_start": 217, "line_end": 218, "char_start": 10862, "char_end": 10883 }, "origin": "source_reference", "context": "narrative" }, { "from": "part:eleven-twelfths", "to": "sec:introduction", "type": "internal_reference", "evidence": "ient contains the reciprocal of the target function. \\paragraph{The balanced first stage.} In Part~\\ref{part:eleven-twelfths}, the two averaging scales are both $Z^{1/2}$. On the reflected side, the base sum separates into a completed theta row and an add", "anchor": { "source": "paper.tex", "line_start": 223, "line_end": 223, "char_start": 11127, "char_end": 11153 }, "origin": "source_reference", "context": "narrative" }, { "from": "prop:balanced-low", "to": "sec:introduction", "type": "internal_reference", "evidence": "ired planar large-sieve bound. The Cauchy--Schwarz inequality combines the two norms in Proposition~\\ref{prop:balanced-low}. This direct estimate does not use the later inverse-moment recursion. On the Poisson side, the intermediate rows are grouped by", "anchor": { "source": "paper.tex", "line_start": 230, "line_end": 230, "char_start": 11612, "char_end": 11635 }, "origin": "source_reference", "context": "narrative" }, { "from": "sec:detector", "to": "sec:introduction", "type": "internal_reference", "evidence": "n bounded-height rectangles for the finite family of Hecke twists that each row determines. Section~\\ref{sec:detector} assigns zero-free rectangles to these families. When a row has a selected zero above the detector's fixed floor for the real part", "anchor": { "source": "paper.tex", "line_start": 235, "line_end": 235, "char_start": 11907, "char_end": 11925 }, "origin": "source_reference", "context": "narrative" }, { "from": "part:eleven-twelfths", "to": "sec:introduction", "type": "internal_reference", "evidence": "ne without those coefficients. They have one common row character and one common twist height. Part~\\ref{part:eleven-twelfths} uses only the inverse polynomial in its row count. After the part of the row with prime valuations at least two is fixed, the sex", "anchor": { "source": "paper.tex", "line_start": 241, "line_end": 241, "char_start": 12282, "char_end": 12308 }, "origin": "source_reference", "context": "narrative" }, { "from": "prop:sextic-row-count", "to": "sec:introduction", "type": "internal_reference", "evidence": "uations at least two is fixed, the sextic large sieve applies to its squarefree factor; Proposition~\\ref{prop:sextic-row-count} gives the resulting count. Rows at the floor and the remaining small and large norm ranges are bounded directly. The principal r", "anchor": { "source": "paper.tex", "line_start": 244, "line_end": 244, "char_start": 12500, "char_end": 12527 }, "origin": "source_reference", "context": "narrative" }, { "from": "thm:eleven-twelfths", "to": "sec:introduction", "type": "internal_reference", "evidence": "stimate verify the common continuation criterion with $C_{\\mathrm I}(s)=s-2/3$. This proves Theorem~\\ref{thm:eleven-twelfths}. The entire family is needed even for the consequence about $\\zeta$, because the Poisson rows introduce finite-order Hecke twists", "anchor": { "source": "paper.tex", "line_start": 253, "line_end": 253, "char_start": 12963, "char_end": 12988 }, "origin": "source_reference", "context": "narrative" }, { "from": "part:seven-eighths", "to": "sec:introduction", "type": "internal_reference", "evidence": "rows introduce finite-order Hecke twists of the target. \\paragraph{The refined second stage.} Part~\\ref{part:seven-eighths} retains the completed support and the shared arithmetic identities, but it modifies the base sum. Selected prime factors provide", "anchor": { "source": "paper.tex", "line_start": 258, "line_end": 258, "char_start": 13178, "char_end": 13202 }, "origin": "source_reference", "context": "narrative" }, { "from": "sec:compensated-low", "to": "sec:introduction", "type": "internal_reference", "evidence": "ized sum again follows directly from reflected energy and an additive mean-square estimate; Section~\\ref{sec:compensated-low} proves it for the modified sum. The high estimate uses both polynomials supplied by the zero detector. For a smooth compactly su", "anchor": { "source": "paper.tex", "line_start": 265, "line_end": 265, "char_start": 13706, "char_end": 13731 }, "origin": "source_reference", "context": "narrative" }, { "from": "sec:refined-row-counts", "to": "sec:introduction", "type": "internal_reference", "evidence": "mon row character and twist height. Bounds for their moments limit the number of such rows. Section~\\ref{sec:refined-row-counts} combines the two bounds, using integer powers and selected prime factors in the larger intermediate norm ranges. Small and large", "anchor": { "source": "paper.tex", "line_start": 278, "line_end": 279, "char_start": 14442, "char_end": 14470 }, "origin": "source_reference", "context": "narrative" }, { "from": "sec:inverse", "to": "sec:introduction", "type": "internal_reference", "evidence": "e floor class, remain direct estimates. The two moment bounds require separate inductions. Section~\\ref{sec:inverse} proves the second-moment estimate for an inverse polynomial multiplied by sums over disjoint prime sets (Lemma~\\ref{lem:marked}).", "anchor": { "source": "paper.tex", "line_start": 284, "line_end": 284, "char_start": 14719, "char_end": 14736 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:marked", "to": "sec:introduction", "type": "internal_reference", "evidence": "second-moment estimate for an inverse polynomial multiplied by sums over disjoint prime sets (Lemma~\\ref{lem:marked}). Its recursive step uses two finite Poisson transformations to shorten the row range, with reflected energy providing the termin", "anchor": { "source": "paper.tex", "line_start": 286, "line_end": 286, "char_start": 14848, "char_end": 14864 }, "origin": "source_reference", "context": "narrative" }, { "from": "sec:plain", "to": "sec:introduction", "type": "internal_reference", "evidence": "ansformations to shorten the row range, with reflected energy providing the terminal bound. Section~\\ref{sec:plain} proves a mean-square estimate for products of two plain polynomials, again with permitted prime factors (Lemma~\\ref{lem:plain}).", "anchor": { "source": "paper.tex", "line_start": 288, "line_end": 288, "char_start": 15012, "char_end": 15027 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:plain", "to": "sec:introduction", "type": "internal_reference", "evidence": "an-square estimate for products of two plain polynomials, again with permitted prime factors (Lemma~\\ref{lem:plain}). Ordinary Hecke reflection reduces the length ranges at each stage; the recursive step uses two finite Poisson transformations.", "anchor": { "source": "paper.tex", "line_start": 290, "line_end": 290, "char_start": 15139, "char_end": 15154 }, "origin": "source_reference", "context": "narrative" }, { "from": "sec:conclusion", "to": "sec:introduction", "type": "internal_reference", "evidence": "y for that centered expression, not for every row in the finite inducing-character family. Section~\\ref{sec:conclusion} combines the resulting row counts with the compensated expansion, its local errors, the contour tails, and the principal residue.", "anchor": { "source": "paper.tex", "line_start": 305, "line_end": 305, "char_start": 15982, "char_end": 16002 }, "origin": "source_reference", "context": "narrative" }, { "from": "thm:main", "to": "cor:nonresidues-square-roots", "type": "candidate_proof_dependency", "evidence": "returns a square root of $a$ or reports that none exists. By Theorem~\\ref{thm:main} and the functional equation, the nontrivial zeros of every primitive Dirichlet $L$-function lie in $1/8\\le\\Re s\\le7/8$. They ther", "anchor": { "source": "paper.tex", "line_start": 340, "line_end": 340, "char_start": 17656, "char_end": 17670 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:signal-euler-contract", "to": "lem:continuation-criterion", "type": "candidate_proof_dependency", "evidence": "a<\\Delta_0$, so $\\beta_*-\\epsilon_*>\\sigma_0$. We also need control as $Z\\downarrow0$. By Equation~\\eqref{eq:signal-euler-contract}, $H_\\eta$ is bounded on $\\Re s>\\sigma_0$. The reciprocal Euler product is absolutely and uniformly bounded on $\\Re s\\ge2$. On eve", "anchor": { "source": "paper.tex", "line_start": 449, "line_end": 449, "char_start": 22185, "char_end": 22217 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:common-residue", "to": "lem:continuation-criterion", "type": "candidate_proof_dependency", "evidence": "d uniformly bounded on $\\Re s\\ge2$. On every fixed strip $2\\le\\Re s\\le B$, the Gaussian in Equation~\\eqref{eq:common-residue} is $O_B(e^{-(\\Im s)^2})$. Cauchy's theorem on rectangles therefore moves the contour to any fixed $B>2$, with horizontal integral", "anchor": { "source": "paper.tex", "line_start": 452, "line_end": 452, "char_start": 22405, "char_end": 22430 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:uniform-saving", "to": "lem:continuation-criterion", "type": "candidate_proof_dependency", "evidence": "ce $B$ is arbitrary, the signal has arbitrarily rapid power decay at zero. This bound and Equation~\\eqref{eq:uniform-saving} show that \\[ F_\\eta(s):=\\int_0^\\infty f_\\eta(Z)Z^{-C(s)}\\,\\frac{dZ}{Z} \\] converges locally uniformly on $\\Re s>\\beta_*-\\epsilon", "anchor": { "source": "paper.tex", "line_start": 460, "line_end": 460, "char_start": 22742, "char_end": 22767 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:uniform-saving", "to": "lem:continuation-criterion", "type": "candidate_proof_dependency", "evidence": "ubset, choose $B$ larger than all occurring real parts for the integral near zero, and use Equation~\\eqref{eq:uniform-saving} near infinity. Thus $F_\\eta$ is holomorphic on that half-plane. We identify this Mellin transform without moving a contour acros", "anchor": { "source": "paper.tex", "line_start": 466, "line_end": 466, "char_start": 23019, "char_end": 23044 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:common-residue", "to": "lem:continuation-criterion", "type": "candidate_proof_dependency", "evidence": "s>1). \\] The function $A_\\eta(2+it)$ is continuous and integrable in $t$. Writing $Z=e^u$, Equation~\\eqref{eq:common-residue} becomes \\[ e^{-(2+c)u}f_\\eta(e^u) =\\frac1{2\\pi}\\int_{\\mathbb R}e^{itu}A_\\eta(2+it)\\,dt. \\] The left side is integrable in $u$", "anchor": { "source": "paper.tex", "line_start": 477, "line_end": 477, "char_start": 23384, "char_end": 23409 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:signal-euler-contract", "to": "lem:continuation-criterion", "type": "candidate_proof_dependency", "evidence": "^{(s-5/6)^2} \\frac{H_\\eta(s)}{L_F^{\\mathcal S}(s,\\eta)} \\qquad(\\Re s>1). \\] Equation~\\eqref{eq:signal-euler-contract} implies $|H_\\eta(s)|\\ge1/2$ on $\\Re s>\\sigma_0$. Consequently \\[ e^{-(s-5/6)^2}\\frac{F_\\eta(s)}{H_\\eta(s)} \\qquad(\\Re s>\\beta_*", "anchor": { "source": "paper.tex", "line_start": 491, "line_end": 491, "char_start": 23848, "char_end": 23880 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:continuation-criterion", "to": "sec:analytic-reduction", "type": "internal_reference", "evidence": "are nonzero there. This contradicts the continuation. The high estimate in Proposition~\\ref{lem:continuation-criterion} is measured relative to $C(\\beta_*)$, not to $C(\\sigma_0)$. This distinction matters when a row contribution reaches the low scal", "anchor": { "source": "paper.tex", "line_start": 504, "line_end": 504, "char_start": 24407, "char_end": 24439 }, "origin": "source_reference", "context": "narrative" }, { "from": "part:eleven-twelfths", "to": "sec:analytic-reduction", "type": "internal_reference", "evidence": "eaches the low scale but still has a power saving relative to the hypothetical rightmost zero. Part~\\ref{part:eleven-twelfths} uses \\[ \\sigma_0=\\frac{11}{12},\\qquad C(s)=s-\\frac23, \\qquad C(\\sigma_0)=\\frac14, \\] whereas Part~\\ref{part:seven-eighths} uses", "anchor": { "source": "paper.tex", "line_start": 507, "line_end": 507, "char_start": 24650, "char_end": 24676 }, "origin": "source_reference", "context": "narrative" }, { "from": "part:seven-eighths", "to": "sec:analytic-reduction", "type": "internal_reference", "evidence": "uses \\[ \\sigma_0=\\frac{11}{12},\\qquad C(s)=s-\\frac23, \\qquad C(\\sigma_0)=\\frac14, \\] whereas Part~\\ref{part:seven-eighths} uses \\[ \\sigma_0=\\frac78,\\qquad C(s)=s-\\frac{11}{16}, \\qquad C(\\sigma_0)=\\frac3{16}. \\] Only the positive power margins must be", "anchor": { "source": "paper.tex", "line_start": 512, "line_end": 512, "char_start": 24777, "char_end": 24801 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:continuation-criterion", "to": "sec:first-stage", "type": "internal_reference", "evidence": ". A zero detector and the sextic large sieve control those rows. These estimates verify Proposition~\\ref{lem:continuation-criterion} with boundary $11/12$. The fixed family in Equation~\\eqref{old-eq:1.1b} is essential even when the desired consequence concerns", "anchor": { "source": "paper.tex", "line_start": 549, "line_end": 549, "char_start": 26266, "char_end": 26298 }, "origin": "source_reference", "context": "narrative" }, { "from": "old-eq:1.1b", "to": "sec:first-stage", "type": "internal_reference", "evidence": "y Proposition~\\ref{lem:continuation-criterion} with boundary $11/12$. The fixed family in Equation~\\eqref{old-eq:1.1b} is essential even when the desired consequence concerns $\\zeta$: the Poisson representation introduces finite-order Hecke twists", "anchor": { "source": "paper.tex", "line_start": 551, "line_end": 551, "char_start": 26352, "char_end": 26371 }, "origin": "source_reference", "context": "narrative" }, { "from": "part:seven-eighths", "to": "sec:first-stage", "type": "internal_reference", "evidence": "et. Proving the family-wide assertion also supplies the precise input used at the beginning of Part~\\ref{part:seven-eighths}. \\section{Arithmetic and analytic preliminaries} The two parts use the same arithmetic conventions and an", "anchor": { "source": "paper.tex", "line_start": 555, "line_end": 555, "char_start": 26612, "char_end": 26636 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:smooth-calculus", "to": "sec:arithmetic", "type": "internal_reference", "evidence": "ds for every logarithmic derivative that is invoked. The precise separation norm is given in Lemma~\\ref{lem:smooth-calculus}. In a product of two plain factors, both factors have the same row character, including the same fixed finite-ray twist. Conjug", "anchor": { "source": "paper.tex", "line_start": 723, "line_end": 723, "char_start": 35172, "char_end": 35197 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:quadratic-four-term", "to": "lem:quadratic-four-term", "type": "candidate_proof_dependency", "evidence": "rmalized right side therefore tends to $(2|c|)^{-1}\\sum_{y\\bmod2}e(-cy^2/4)$, which proves Equation~\\eqref{eq:quadratic-four-term}. The representatives $0,1,\\omega,\\omega^2$ modulo $2$ give the asserted four-term expression. Multiplication by an odd $v$ perm", "anchor": { "source": "paper.tex", "line_start": 920, "line_end": 920, "char_start": 43715, "char_end": 43745 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:reciprocity-four-class", "to": "lem:fixed-gauss-phase", "type": "internal_reference", "evidence": "e $S$, including noncoprime pairs, define $\\mathcal R$ by the bicharacter $\\mathfrak r$ of Equation~\\eqref{eq:reciprocity-four-class}. Define, on every primary index outside $S$, \\[ G(c)=\\overline{\\chi_c(4)}\\Gamma(c). \\] It has modulus one and factors through a", "anchor": { "source": "paper.tex", "line_start": 943, "line_end": 943, "char_start": 44778, "char_end": 44811 }, "origin": "source_reference", "context": "statement" }, { "from": "eq:reciprocity-four-class", "to": "lem:fixed-gauss-phase", "type": "internal_reference", "evidence": "to\\mathcal R(t,t)$ is a character of the fixed ray group. In the square-class notation of Equation~\\eqref{eq:reciprocity-four-class}, it is $\\mathfrak r(-1,n)$, where $-1$ denotes its residue square class modulo $4$, not the ideal ray class of the unit ideal $(-", "anchor": { "source": "paper.tex", "line_start": 961, "line_end": 961, "char_start": 45481, "char_end": 45514 }, "origin": "source_reference", "context": "statement" }, { "from": "lem:prime-gauss-identities", "to": "lem:fixed-gauss-phase", "type": "candidate_proof_dependency", "evidence": "s for $j=1$ and $j=2$ is $(\\chi_a(b)\\chi_b(a))^3=\\mathcal R(a,b)$. The prime identities from Lemma~\\ref{lem:prime-gauss-identities}, followed by Equation~\\eqref{eq:G-quadratic-refinement}, therefore prove Equation~\\eqref{eq:signal} by induction on the number of", "anchor": { "source": "paper.tex", "line_start": 1049, "line_end": 1049, "char_start": 49387, "char_end": 49419 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:G-quadratic-refinement", "to": "lem:fixed-gauss-phase", "type": "candidate_proof_dependency", "evidence": "al R(a,b)$. The prime identities from Lemma~\\ref{lem:prime-gauss-identities}, followed by Equation~\\eqref{eq:G-quadratic-refinement}, therefore prove Equation~\\eqref{eq:signal} by induction on the number of prime factors. For later element rows, fix", "anchor": { "source": "paper.tex", "line_start": 1050, "line_end": 1050, "char_start": 49442, "char_end": 49475 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:signal", "to": "lem:fixed-gauss-phase", "type": "candidate_proof_dependency", "evidence": "gauss-identities}, followed by Equation~\\eqref{eq:G-quadratic-refinement}, therefore prove Equation~\\eqref{eq:signal} by induction on the number of prime factors. For later element rows, fix the generators $\\pi_{\\mathfrak p}$ from Lem", "anchor": { "source": "paper.tex", "line_start": 1051, "line_end": 1051, "char_start": 49502, "char_end": 49519 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:fixed-numerator-ray", "to": "sec:arithmetic", "type": "internal_reference", "evidence": "ime factors. For later element rows, fix the generators $\\pi_{\\mathfrak p}$ from Lemma~\\ref{lem:fixed-numerator-ray}. Every $0\\ne m\\in\\mathcal O$ has a unique expression \\[ m=u m_S m_{\\mathrm{good}},\\qquad m_S=\\prod_{\\mathfrak p\\in S}\\pi_{\\mat", "anchor": { "source": "paper.tex", "line_start": 1055, "line_end": 1055, "char_start": 49652, "char_end": 49681 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:late-fourier-tail", "to": "lem:smooth-calculus", "type": "internal_reference", "evidence": "inate is fixed, is estimated by this pointwise bound, not solely by the integrated tail in Equation~\\eqref{eq:late-fourier-tail}. The following joint version will also be useful. For measurable functions $a(v),b(w)$ for which the right side is finite, and", "anchor": { "source": "paper.tex", "line_start": 1177, "line_end": 1177, "char_start": 55225, "char_end": 55253 }, "origin": "source_reference", "context": "statement" }, { "from": "eq:pointwise-mellin-tail", "to": "lem:smooth-calculus", "type": "internal_reference", "evidence": "iously fixed height order. On a join of bounded real length at height comparable to $T_1$, Equation~\\eqref{eq:pointwise-mellin-tail} with order $N+\\lceil J\\rceil$ gives $Z^BT_1^{-N}$ times the corresponding finite weighted derivative norm of the external test, p", "anchor": { "source": "paper.tex", "line_start": 1206, "line_end": 1206, "char_start": 56674, "char_end": 56706 }, "origin": "source_reference", "context": "statement" }, { "from": "eq:parameter-sobolev", "to": "lem:smooth-calculus", "type": "candidate_proof_dependency", "evidence": "^2\\ll\\int_{I+[-1,1]}(|F|^2+|F'|^2)$. Applying this successively in each coordinate proves Equation~\\eqref{eq:parameter-sobolev}. It can be summed over rows before the derivative integrals, since all terms are nonnegative. A logarithmic scale range of leng", "anchor": { "source": "paper.tex", "line_start": 1234, "line_end": 1234, "char_start": 58015, "char_end": 58043 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:late-fourier-tail", "to": "lem:smooth-calculus", "type": "candidate_proof_dependency", "evidence": "H+d_t}$. On $|\\boldsymbol t|>T_1$ one has $(1+|\\boldsymbol t|)^{-N}\\le T_1^{-N}$, proving Equation~\\eqref{eq:late-fourier-tail}. The Mellin shift is immediate from its definition. To prove Equation~\\eqref{eq:pointwise-mellin-tail}, put $F_\\sigma(u)=e^{\\si", "anchor": { "source": "paper.tex", "line_start": 1248, "line_end": 1248, "char_start": 58824, "char_end": 58852 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:pointwise-mellin-tail", "to": "lem:smooth-calculus", "type": "candidate_proof_dependency", "evidence": "eqref{eq:late-fourier-tail}. The Mellin shift is immediate from its definition. To prove Equation~\\eqref{eq:pointwise-mellin-tail}, put $F_\\sigma(u)=e^{\\sigma u}W(e^u)$. For $|T|\\ge1$, integration by parts gives \\[ \\mathcal MW(\\sigma+iT)=(-iT)^{-N}", "anchor": { "source": "paper.tex", "line_start": 1249, "line_end": 1249, "char_start": 58925, "char_end": 58957 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:joint-gaussian-slice", "to": "lem:smooth-calculus", "type": "candidate_proof_dependency", "evidence": "le $e^{\\sigma u}$ at $-\\infty$ and Schwartz decay at $+\\infty$ suffice in the second. For Equation~\\eqref{eq:joint-gaussian-slice}, use \\[ 1+|T|\\le(1+|T+v|)(1+|v|),\\qquad 1+|T|+|v|+|w|\\le2(1+|T+v|)(1+|v|)(1+|w|). \\] Move $(1+|T|)^N$ to the left, bound the we", "anchor": { "source": "paper.tex", "line_start": 1264, "line_end": 1264, "char_start": 59585, "char_end": 59616 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:smooth-calculus", "to": "lem:gaussian-annular", "type": "internal_reference", "evidence": "ussian-annular-sum} \\sum_{k\\in\\mathbb Z}e^{A|k|}p_j(w_k)<\\infty. Consequently Lemma~\\ref{lem:smooth-calculus} may be applied on each annulus after $y=e^k x$, and the resulting weighted separation norms are summable. For fixed $\\kappa>0$, $", "anchor": { "source": "paper.tex", "line_start": 1305, "line_end": 1305, "char_start": 61461, "char_end": 61486 }, "origin": "source_reference", "context": "statement" }, { "from": "eq:pointwise-mellin-tail", "to": "lem:gaussian-annular", "type": "internal_reference", "evidence": "irectly, $\\mathcal MW_{\\mathrm G}(s)=e^{s^2}$, and the weighted logarithmic derivatives in Equation~\\eqref{eq:pointwise-mellin-tail} are finite uniformly on every compact real strip. Such a partition is obtained by normalizing the inte", "anchor": { "source": "paper.tex", "line_start": 1313, "line_end": 1313, "char_start": 61845, "char_end": 61877 }, "origin": "source_reference", "context": "statement" }, { "from": "lem:smooth-calculus", "to": "lem:gaussian-annular", "type": "candidate_proof_dependency", "evidence": "A|k|}$ for every fixed $A$. The first assertion follows, and the separation norms follow from Lemma~\\ref{lem:smooth-calculus}. In the profile's logarithmic Fourier transform, rescaling an annulus inserts only the unit phase $e^{-ikt}$, so it does not cha", "anchor": { "source": "paper.tex", "line_start": 1329, "line_end": 1329, "char_start": 62518, "char_end": 62543 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:hecke-growth", "to": "lem:hecke-strip-growth", "type": "candidate_proof_dependency", "evidence": "the strip, with the same constant for every $Q$. Multiplication by $Q^{3/5}|s+2|^2$ proves Equation~\\eqref{eq:hecke-growth}. For the principal function, the factor $(s-1)/(s+1)$ removes its only pole in the strip and is bounded on the boundary. Its fu", "anchor": { "source": "paper.tex", "line_start": 1519, "line_end": 1519, "char_start": 71520, "char_end": 71543 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:hecke-strip-growth", "to": "lem:logarithmic-control", "type": "candidate_proof_dependency", "evidence": "of the regularizing factor in the principal case. The value $g(2+it)$ is uniformly bounded. Lemma~\\ref{lem:hecke-strip-growth} and Euler convergence to the right of $11/10$ give \\[ \\Re g(s)=\\log|\\mathcal L(s)|\\ll\\log\\mathcal C \\qquad(|s-(2+it)|\\le", "anchor": { "source": "paper.tex", "line_start": 1560, "line_end": 1560, "char_start": 73574, "char_end": 73602 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:disk-control", "to": "lem:logarithmic-control", "type": "candidate_proof_dependency", "evidence": "heta)\\ll_{e,\\epsilon}\\mathcal C^\\epsilon$. Applying this to both $e^g$ and $e^{-g}$ proves Equation~\\eqref{eq:disk-control}. Cauchy's estimate between radii $R_6$ and $R_8$ gives the stated bound for $g'=\\mathcal L'/\\mathcal L$. For the global asserti", "anchor": { "source": "paper.tex", "line_start": 1585, "line_end": 1585, "char_start": 74686, "char_end": 74709 }, "origin": "source_reference", "context": "proof" }, { "from": "old-eq:4.3", "to": "prop:completed-reflection", "type": "candidate_proof_dependency", "evidence": "ine{d_j(-\\mu)}$, as in their Equation (5.16). Those formulas give the support inclusion in Equation~\\eqref{old-eq:4.3}. For completeness, the two possible magnitudes from $\\tau$ at $\\mu=u\\lambda^k n b^3$ are \\[ 3^{k/6+8/3}|b|\\quad\\hbox{and}\\quad3", "anchor": { "source": "paper.tex", "line_start": 1947, "line_end": 1947, "char_start": 90247, "char_end": 90265 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:reflection-infinity-coefficient", "to": "prop:completed-reflection", "type": "candidate_proof_dependency", "evidence": "defined to be zero directly; no value of $\\chi_n(\\lambda)$ is evaluated there. Combining Equations~\\eqref{eq:reflection-infinity-coefficient} and \\eqref{eq:reflection-cubic-sextic-conversion} on the retained domain now identifies the direct Mellin series with the product", "anchor": { "source": "paper.tex", "line_start": 2062, "line_end": 2062, "char_start": 96088, "char_end": 96130 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:reflection-cubic-sextic-conversion", "to": "prop:completed-reflection", "type": "candidate_proof_dependency", "evidence": "n(\\lambda)$ is evaluated there. Combining Equations~\\eqref{eq:reflection-infinity-coefficient} and \\eqref{eq:reflection-cubic-sextic-conversion} on the retained domain now identifies the direct Mellin series with the product $T$. The dual coefficient functions $d_H$ keep t", "anchor": { "source": "paper.tex", "line_start": 2063, "line_end": 2063, "char_start": 96135, "char_end": 96180 }, "origin": "source_reference", "context": "proof" }, { "from": "old-eq:4.3", "to": "prop:completed-reflection", "type": "candidate_proof_dependency", "evidence": "ith the product $T$. The dual coefficient functions $d_H$ keep the full source support in Equation~\\eqref{old-eq:4.3}; no $S$-mask is placed on them. Fix $h_0$ and declare $p$ active precisely when $h_p\\ne0$. Equation~\\eqref{eq:reflection-local-f", "anchor": { "source": "paper.tex", "line_start": 2065, "line_end": 2065, "char_start": 96345, "char_end": 96363 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:reflection-local-fourier", "to": "prop:completed-reflection", "type": "candidate_proof_dependency", "evidence": "no $S$-mask is placed on them. Fix $h_0$ and declare $p$ active precisely when $h_p\\ne0$. Equation~\\eqref{eq:reflection-local-fourier} forces all primes with $j_p\\ne0$ to be active. Choose once for each $h_0\\bmod L$ a representative and a reduced expression $\\lam", "anchor": { "source": "paper.tex", "line_start": 2069, "line_end": 2069, "char_start": 96465, "char_end": 96500 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:reflection-translated-theta", "to": "prop:completed-reflection", "type": "candidate_proof_dependency", "evidence": "ctive prime is primary, this unit depends only on $h_0$. For $r=\\prod_{p\\ {\\rm active}}p$, Equation~\\eqref{eq:reflection-translated-theta} then gives z_h=\\frac ac,\\qquad c=c_Fr,\\qquad a=a_Fr+\\lambda^2c_F\\sum_{p\\mid", "anchor": { "source": "paper.tex", "line_start": 2076, "line_end": 2077, "char_start": 96925, "char_end": 96963 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:reflection-cusp-fraction", "to": "prop:completed-reflection", "type": "candidate_proof_dependency", "evidence": "bmod p$ to be lifted with $h_p\\equiv0\\pmod{M^2}$. Restrict $r$ to one class modulo $M^2$. Equation~\\eqref{eq:reflection-cusp-fraction} shows that $a$ is then fixed modulo $Mc_F$; in fact its moving terms are divisible by $M^2c_F$, and $a_Fr$ is fixed modulo $M^2$,", "anchor": { "source": "paper.tex", "line_start": 2095, "line_end": 2095, "char_start": 97911, "char_end": 97946 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:reflection-cusp-fraction", "to": "prop:completed-reflection", "type": "candidate_proof_dependency", "evidence": "lected additive phase.} Write $x=\\lambda^4\\mu$ and $D_F=\\lambda^3c_F$. At each $p\\mid r$, Equation~\\eqref{eq:reflection-cusp-fraction} gives $a\\equiv\\sigma_ph_p\\pmod p$. Hence $\\delta'\\equiv\\sigma_p^{-1}h_p^{-1}\\pmod p$. Additive Chinese remaindering for the cop", "anchor": { "source": "paper.tex", "line_start": 2203, "line_end": 2203, "char_start": 102598, "char_end": 102633 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:reflection-multiplier", "to": "prop:completed-reflection", "type": "candidate_proof_dependency", "evidence": "y many $D_F$ is therefore a fixed modulus for all the characters $\\vartheta$. Conjugating Equation~\\eqref{eq:reflection-multiplier} contributes $\\bar\\kappa_F\\prod_{p\\mid r}\\chi_p(\\sigma_p)^{-2}\\chi_p(h_p)^{-2}$. For $j\\ne0$, multiplication by Equation~\\eqref{eq", "anchor": { "source": "paper.tex", "line_start": 2225, "line_end": 2225, "char_start": 103592, "char_end": 103624 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:reflection-local-fourier", "to": "prop:completed-reflection", "type": "candidate_proof_dependency", "evidence": "ppa_F\\prod_{p\\mid r}\\chi_p(\\sigma_p)^{-2}\\chi_p(h_p)^{-2}$. For $j\\ne0$, multiplication by Equation~\\eqref{eq:reflection-local-fourier} leaves the local sum \\[ q_p^{-1/2}\\tau_{p,j}^{-} \\sum_{h\\ne0}\\chi_p(h)^{-j-2}e(\\epsilon_ph^{-1}x/p). \\] Upon putting $y=h^{-1}$", "anchor": { "source": "paper.tex", "line_start": 2228, "line_end": 2228, "char_start": 103745, "char_end": 103780 }, "origin": "source_reference", "context": "proof" }, { "from": "par:reflection-marks", "to": "prop:completed-reflection", "type": "candidate_proof_dependency", "evidence": "is proves the simultaneous branch compatibility in the statement. The marked reflection in Section~\\ref{par:reflection-marks} will use the following dependence on pairs of active primes. Since \\[ \\epsilon_p=-\\lambda^{-5}(c/p)^{-2}\\pmod p, \\] there is a", "anchor": { "source": "paper.tex", "line_start": 2264, "line_end": 2264, "char_start": 105442, "char_end": 105468 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:reflection-infinity-coefficient", "to": "prop:completed-reflection", "type": "candidate_proof_dependency", "evidence": "First use the whole-index mask to restrict to $(nb,\\mathfrak E)=1$ as above, then insert Equations~\\eqref{eq:reflection-infinity-coefficient} and \\eqref{eq:reflection-cubic-sextic-conversion}. Using $q_{\\lambda^{-3}}=27^{-1}$ and $\\bar\\alpha(\\lambda^{-3})=-i$ gives \\beg", "anchor": { "source": "paper.tex", "line_start": 2305, "line_end": 2305, "char_start": 107035, "char_end": 107077 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:reflection-cubic-sextic-conversion", "to": "prop:completed-reflection", "type": "candidate_proof_dependency", "evidence": "$(nb,\\mathfrak E)=1$ as above, then insert Equations~\\eqref{eq:reflection-infinity-coefficient} and \\eqref{eq:reflection-cubic-sextic-conversion}. Using $q_{\\lambda^{-3}}=27^{-1}$ and $\\bar\\alpha(\\lambda^{-3})=-i$ gives J", "anchor": { "source": "paper.tex", "line_start": 2306, "line_end": 2306, "char_start": 107082, "char_end": 107127 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:reflection-cusp-expansion", "to": "prop:completed-reflection", "type": "candidate_proof_dependency", "evidence": "-\\delta'/c}. \\] In particular the derivative removes $C_H(v)$ from every cusp expansion in Equation~\\eqref{eq:reflection-cusp-expansion}. Substituting that expansion and then $v\\mapsto1/(q_cv)$ in the integral gives, for $\\Re s<0$, the contribution \\", "anchor": { "source": "paper.tex", "line_start": 2332, "line_end": 2332, "char_start": 108195, "char_end": 108231 }, "origin": "source_reference", "context": "proof" }, { "from": "old-eq:4.3", "to": "prop:completed-reflection", "type": "candidate_proof_dependency", "evidence": "We justify both the continuation and the contour operations here. The coefficient bound in Equation~\\eqref{old-eq:4.3} implies absolute convergence of the reflected series when $\\Re(1-s)>1$: the separate majorants are \\[ \\sum_n^{\\rm sf}q_n^{-(1-\\R", "anchor": { "source": "paper.tex", "line_start": 2351, "line_end": 2351, "char_start": 109023, "char_end": 109041 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:reflection-direct-mellin", "to": "prop:completed-reflection", "type": "candidate_proof_dependency", "evidence": "hat $J$ is entire and decreases faster than every power of $|\\Im s|$ on each fixed strip. Equation~\\eqref{eq:reflection-direct-mellin}, with reciprocal gamma functions on its right, continues $T$ to an entire function. Its direct series is bounded on any line to", "anchor": { "source": "paper.tex", "line_start": 2369, "line_end": 2369, "char_start": 109858, "char_end": 109893 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:reflection-reflected-mellin", "to": "prop:completed-reflection", "type": "candidate_proof_dependency", "evidence": "$T$ to an entire function. Its direct series is bounded on any line to the right of $1$. Equation~\\eqref{eq:reflection-reflected-mellin} and Stirling's formula give a polynomial bound on any line to the left of $0$; the exponential parts of the two gamma products ca", "anchor": { "source": "paper.tex", "line_start": 2372, "line_end": 2372, "char_start": 110050, "char_end": 110088 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:reflection", "to": "prop:completed-reflection", "type": "candidate_proof_dependency", "evidence": "strip. This proves the stated polynomial strip growth. Shift the integral on the left of Equation~\\eqref{eq:reflection} from $\\Re s=a$ to $\\Re s=1/2-\\sigma<0$ with $\\sigma>1/2$. The entire continuation, polynomial strip bound, and rapid Mellin decay", "anchor": { "source": "paper.tex", "line_start": 2384, "line_end": 2384, "char_start": 110828, "char_end": 110849 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:pointwise-mellin-tail", "to": "prop:completed-reflection", "type": "candidate_proof_dependency", "evidence": "The entire continuation, polynomial strip bound, and rapid Mellin decay justify the shift: Equation~\\eqref{eq:pointwise-mellin-tail} applied to the weighted logarithmic derivatives of $V$ makes the horizontal joins tend to zero. Inserting Equation~\\eqref{eq:ref", "anchor": { "source": "paper.tex", "line_start": 2387, "line_end": 2387, "char_start": 111008, "char_end": 111040 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:reflection-reflected-mellin", "to": "prop:completed-reflection", "type": "candidate_proof_dependency", "evidence": "eighted logarithmic derivatives of $V$ makes the horizontal joins tend to zero. Inserting Equation~\\eqref{eq:reflection-reflected-mellin} divided by Equation~\\eqref{eq:reflection-direct-mellin}, and putting $t=1/2-s$, gives the gamma quotient $R(t)$ and the factor \\[", "anchor": { "source": "paper.tex", "line_start": 2390, "line_end": 2390, "char_start": 111157, "char_end": 111195 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:reflection-direct-mellin", "to": "prop:completed-reflection", "type": "candidate_proof_dependency", "evidence": "joins tend to zero. Inserting Equation~\\eqref{eq:reflection-reflected-mellin} divided by Equation~\\eqref{eq:reflection-direct-mellin}, and putting $t=1/2-s$, gives the gamma quotient $R(t)$ and the factor \\[ -\\frac i{81}\\bar\\alpha(c)^2q_\\mu^{-1/2} \\left(\\", "anchor": { "source": "paper.tex", "line_start": 2391, "line_end": 2391, "char_start": 111216, "char_end": 111251 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:reflection-additive-phase", "to": "prop:completed-reflection", "type": "candidate_proof_dependency", "evidence": "line, so it may be interchanged with the integral. Summing its finite local factors using Equation~\\eqref{eq:reflection-additive-phase} and the computed Gauss sums proves Equation~\\eqref{eq:reflection} and Equation~\\eqref{eq:reflection-row-phase}. Since $|\\phi|\\le", "anchor": { "source": "paper.tex", "line_start": 2402, "line_end": 2403, "char_start": 111698, "char_end": 111734 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:reflection", "to": "prop:completed-reflection", "type": "candidate_proof_dependency", "evidence": "ors using Equation~\\eqref{eq:reflection-additive-phase} and the computed Gauss sums proves Equation~\\eqref{eq:reflection} and Equation~\\eqref{eq:reflection-row-phase}. Since $|\\phi|\\le1$, we have $|\\widehat\\phi(h_0)|\\le1$; every other factor in the s", "anchor": { "source": "paper.tex", "line_start": 2403, "line_end": 2403, "char_start": 111779, "char_end": 111800 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:reflection-row-phase", "to": "prop:completed-reflection", "type": "candidate_proof_dependency", "evidence": "tion-additive-phase} and the computed Gauss sums proves Equation~\\eqref{eq:reflection} and Equation~\\eqref{eq:reflection-row-phase}. Since $|\\phi|\\le1$, we have $|\\widehat\\phi(h_0)|\\le1$; every other factor in the scalar has absolute value at most one. This p", "anchor": { "source": "paper.tex", "line_start": 2404, "line_end": 2404, "char_start": 111814, "char_end": 111845 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:pointwise-mellin-tail", "to": "prop:completed-reflection", "type": "candidate_proof_dependency", "evidence": "estimate is uniform also for bounded $u$, since this closed strip has no numerator pole. Equation~\\eqref{eq:pointwise-mellin-tail} makes $\\widehat V$ rapidly decreasing pointwise on every vertical strip. Rectangular contour shifts are therefore valid within $", "anchor": { "source": "paper.tex", "line_start": 2417, "line_end": 2417, "char_start": 112440, "char_end": 112472 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:reflection-kernel", "to": "prop:completed-reflection", "type": "candidate_proof_dependency", "evidence": "rceil+j+2}(V),\\quad x^{-A}\\mathfrak M_{A,\\lceil4A\\rceil+j+2}(V). \\] Combining them proves Equation~\\eqref{eq:reflection-kernel}. The rapid large-$x$ bound, with $A$ arbitrarily large, also makes every dual sum in Equation~\\eqref{eq:reflection} absolutely co", "anchor": { "source": "paper.tex", "line_start": 2429, "line_end": 2429, "char_start": 112992, "char_end": 113020 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:reflection", "to": "prop:completed-reflection", "type": "candidate_proof_dependency", "evidence": "rnel}. The rapid large-$x$ bound, with $A$ arbitrarily large, also makes every dual sum in Equation~\\eqref{eq:reflection} absolutely convergent when $V^\\sharp$ is represented on the zero line. Thus the shift of the kernel contour does not require a t", "anchor": { "source": "paper.tex", "line_start": 2431, "line_end": 2431, "char_start": 113115, "char_end": 113136 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:reflection-annular-fourier", "to": "prop:completed-reflection", "type": "candidate_proof_dependency", "evidence": "our does not require a termwise shift of a conditionally convergent Dirichlet series. For Equation~\\eqref{eq:reflection-annular-fourier}, set $f_Y(v)=\\chi(e^v)V^\\sharp(Ye^v)$. This is supported on a fixed compact interval. Integrating its Fourier transform by part", "anchor": { "source": "paper.tex", "line_start": 2436, "line_end": 2436, "char_start": 113342, "char_end": 113379 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:reflection-kernel", "to": "prop:completed-reflection", "type": "candidate_proof_dependency", "evidence": "at f_Y(u)|\\,du \\ll_J\\sum_{j=0}^{J+2}\\|f_Y^{(j)}\\|_{L^1(\\mathbb R)}. \\] Leibniz's rule and Equation~\\eqref{eq:reflection-kernel} bound each norm by the right side of Equation~\\eqref{eq:reflection-annular-fourier}, because $e^v$ stays in a fixed compact subin", "anchor": { "source": "paper.tex", "line_start": 2444, "line_end": 2444, "char_start": 113707, "char_end": 113735 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:reflection-annular-fourier", "to": "prop:completed-reflection", "type": "candidate_proof_dependency", "evidence": "bniz's rule and Equation~\\eqref{eq:reflection-kernel} bound each norm by the right side of Equation~\\eqref{eq:reflection-annular-fourier}, because $e^v$ stays in a fixed compact subinterval of $(0,\\infty)$. Finally, for $|u|\\ge1$, $B+2$ integrations by parts give \\[", "anchor": { "source": "paper.tex", "line_start": 2445, "line_end": 2445, "char_start": 113782, "char_end": 113819 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:reflection-test-seminorm", "to": "prop:completed-reflection", "type": "candidate_proof_dependency", "evidence": "ding undifferentiated integral. On $-A\\le\\eta\\le1/4$, $x^\\eta\\le1+x^{-A}+x^{1/4}$, proving Equation~\\eqref{eq:reflection-test-seminorm}. The identity $\\widehat{x^{iu_0}V}(\\eta+iu)=\\widehat V(\\eta+i(u+u_0))$ and $1+|u|\\le(1+|u+u_0|)(1+|u_0|)$ give the asserted norm", "anchor": { "source": "paper.tex", "line_start": 2456, "line_end": 2456, "char_start": 114193, "char_end": 114228 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:fixed-numerator-ray", "to": "lem:reflection-row-sectors", "type": "internal_reference", "evidence": "ments prime to $S$. Use the fixed generators $\\pi_{\\mathfrak l}$ for $\\mathfrak l\\in S$ from Lemma~\\ref{lem:fixed-numerator-ray}. For $0\\ne m\\in\\mathcal O$, write uniquely \\[ m=u m_Sm_{\\rm good},\\qquad m_S=\\prod_{\\mathfrak l\\in S}\\pi_{\\mathfrak l}^{v_{\\mat", "anchor": { "source": "paper.tex", "line_start": 2475, "line_end": 2475, "char_start": 115140, "char_end": 115169 }, "origin": "source_reference", "context": "statement" }, { "from": "prop:completed-reflection", "to": "lem:reflection-row-sectors", "type": "internal_reference", "evidence": "ray characters with conductors supported on $S$. Therefore a single $\\mathfrak E,L$ in Proposition~\\ref{prop:completed-reflection} works for all these sectors. The local prime set contains every $p\\mid m_{\\rm good}$, with exponent $v_p(m)$ modulo six, even wh", "anchor": { "source": "paper.tex", "line_start": 2502, "line_end": 2502, "char_start": 116312, "char_end": 116343 }, "origin": "source_reference", "context": "statement" }, { "from": "lem:fixed-gauss-phase", "to": "lem:reflection-row-sectors", "type": "candidate_proof_dependency", "evidence": "egin{proof} If $(A,m_{\\rm good})=1$, multiplicativity and the symmetric reciprocity factor in Lemma~\\ref{lem:fixed-gauss-phase} give Equation~\\eqref{eq:reflection-row-reduction}. If a good prime is shared, both sides are zero: the left side is the zero ext", "anchor": { "source": "paper.tex", "line_start": 2510, "line_end": 2510, "char_start": 116657, "char_end": 116684 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:reflection-row-reduction", "to": "lem:reflection-row-sectors", "type": "candidate_proof_dependency", "evidence": "plicativity and the symmetric reciprocity factor in Lemma~\\ref{lem:fixed-gauss-phase} give Equation~\\eqref{eq:reflection-row-reduction}. If a good prime is shared, both sides are zero: the left side is the zero extension of $\\chi_A(m)$, and its local factor on the", "anchor": { "source": "paper.tex", "line_start": 2511, "line_end": 2511, "char_start": 116699, "char_end": 116734 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:fixed-numerator-ray", "to": "lem:reflection-row-sectors", "type": "candidate_proof_dependency", "evidence": "s $d=u\\prod_{\\mathfrak l\\in S}\\pi_{\\mathfrak l}^{e_{\\mathfrak l}}$, $0\\le e_{\\mathfrak l}<6$, Lemma~\\ref{lem:fixed-numerator-ray} identifies $A\\mapsto\\chi_A(d)$ with a finite-ray character whose conductor is supported at primes over $6d$, all in $S$. Once th", "anchor": { "source": "paper.tex", "line_start": 2520, "line_end": 2520, "char_start": 117253, "char_end": 117282 }, "origin": "source_reference", "context": "proof" }, { "from": "prop:completed-reflection", "to": "lem:reflection-common-profile", "type": "internal_reference", "evidence": "kernel profile] Let $V$ satisfy the hypotheses of Proposition~\\ref{prop:completed-reflection}. Fix an integer $d\\ge1$, a compact set $\\Omega\\subset\\mathbb R^d$, and real exponents $a_1,\\ldots,a_d$. Let $W$ be one smooth pr", "anchor": { "source": "paper.tex", "line_start": 2542, "line_end": 2542, "char_start": 118433, "char_end": 118464 }, "origin": "source_reference", "context": "statement" }, { "from": "eq:reflection-kernel", "to": "lem:reflection-common-profile", "type": "candidate_proof_dependency", "evidence": "sharp$ at $Yh(\\boldsymbol y)$. It produces no additional power of $Y$. Leibniz's rule and Equation~\\eqref{eq:reflection-kernel} prove Equation~\\eqref{eq:reflection-common-profile}. Applying Lemma~\\ref{lem:smooth-calculus} with $j=J+d+2$ gives Equation~\\eqr", "anchor": { "source": "paper.tex", "line_start": 2574, "line_end": 2574, "char_start": 119841, "char_end": 119869 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:reflection-common-profile", "to": "lem:reflection-common-profile", "type": "candidate_proof_dependency", "evidence": "o additional power of $Y$. Leibniz's rule and Equation~\\eqref{eq:reflection-kernel} prove Equation~\\eqref{eq:reflection-common-profile}. Applying Lemma~\\ref{lem:smooth-calculus} with $j=J+d+2$ gives Equation~\\eqref{eq:reflection-common-measure}. Here", "anchor": { "source": "paper.tex", "line_start": 2575, "line_end": 2575, "char_start": 119885, "char_end": 119921 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:smooth-calculus", "to": "lem:reflection-common-profile", "type": "candidate_proof_dependency", "evidence": "n~\\eqref{eq:reflection-kernel} prove Equation~\\eqref{eq:reflection-common-profile}. Applying Lemma~\\ref{lem:smooth-calculus} with $j=J+d+2$ gives Equation~\\eqref{eq:reflection-common-measure}. Here is the norm consequence, including its orde", "anchor": { "source": "paper.tex", "line_start": 2576, "line_end": 2576, "char_start": 119939, "char_end": 119964 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:reflection-common-measure", "to": "lem:reflection-common-profile", "type": "candidate_proof_dependency", "evidence": "reflection-common-profile}. Applying Lemma~\\ref{lem:smooth-calculus} with $j=J+d+2$ gives Equation~\\eqref{eq:reflection-common-measure}. Here is the norm consequence, including its order of operations. Let $\\mathcal I$ be the original row set, and sup", "anchor": { "source": "paper.tex", "line_start": 2577, "line_end": 2577, "char_start": 119995, "char_end": 120031 }, "origin": "source_reference", "context": "proof" }, { "from": "prop:completed-reflection", "to": "sec:completed-reflection", "type": "internal_reference", "evidence": "nstead contributes its scalar once, not automatically its square. The Gaussian test in Proposition~\\ref{prop:completed-reflection} is directly admissible there and in Lemma~\\ref{lem:reflection-common-profile}: only the joint factor $W$ in the latter lemma need", "anchor": { "source": "paper.tex", "line_start": 2611, "line_end": 2611, "char_start": 122042, "char_end": 122073 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:reflection-common-profile", "to": "sec:completed-reflection", "type": "internal_reference", "evidence": "ssian test in Proposition~\\ref{prop:completed-reflection} is directly admissible there and in Lemma~\\ref{lem:reflection-common-profile}: only the joint factor $W$ in the latter lemma needs compact logarithmic support. If the compact-support calculus is instead app", "anchor": { "source": "paper.tex", "line_start": 2612, "line_end": 2612, "char_start": 122116, "char_end": 122151 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:gaussian-annular", "to": "sec:completed-reflection", "type": "internal_reference", "evidence": "arithmic support. If the compact-support calculus is instead applied to the Gaussian itself, Lemma~\\ref{lem:gaussian-annular} supplies the required absolutely summable annular decomposition. [Lattice kernel tails]\\label{lem:lattice-kernel-ta", "anchor": { "source": "paper.tex", "line_start": 2615, "line_end": 2615, "char_start": 122316, "char_end": 122342 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:reflection-kernel", "to": "lem:lattice-kernel-tail", "type": "candidate_proof_dependency", "evidence": "{-1}U^{1-A}))$, which implies the display because $U\\ge1$. The last assertion follows from Equation~\\eqref{eq:reflection-kernel}. \\subsection{A quadratic norm for completed indices} The next estimate bounds the quadratic character left by refle", "anchor": { "source": "paper.tex", "line_start": 2635, "line_end": 2635, "char_start": 123226, "char_end": 123254 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:quadratic-large-sieve", "to": "sec:completed-reflection", "type": "internal_reference", "evidence": "t \\(\\lambda\\) is absent when \\(e(a)=0\\). This does not change any value on the indices in Equation~\\eqref{eq:quadratic-large-sieve}. In a fixed primary square class modulo \\(4\\), two indices have the same \\(e(a)\\); for coprime such indices the primitive product", "anchor": { "source": "paper.tex", "line_start": 2663, "line_end": 2663, "char_start": 124671, "char_end": 124703 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:reciprocity-four-class", "to": "sec:completed-reflection", "type": "internal_reference", "evidence": "exactly their product. The reciprocity factor is the fixed bicharacter \\(\\mathcal R\\) of Equation~\\eqref{eq:reciprocity-four-class}. These are precisely the family conditions of the cited theorem. A finite ray partition and transpose duality give Equation~\\eq", "anchor": { "source": "paper.tex", "line_start": 2669, "line_end": 2669, "char_start": 124947, "char_end": 124980 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:quadratic-large-sieve", "to": "sec:completed-reflection", "type": "internal_reference", "evidence": "family conditions of the cited theorem. A finite ray partition and transpose duality give Equation~\\eqref{eq:quadratic-large-sieve} in the displayed orientation. A fixed restriction on the row set decreases the positive outer sum. A fixed restriction on the co", "anchor": { "source": "paper.tex", "line_start": 2671, "line_end": 2671, "char_start": 125107, "char_end": 125139 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:completed-quadratic-reduction", "to": "lem:completed-quadratic-reduction", "type": "candidate_proof_dependency", "evidence": "om their fixed $S$-parts; the $S$-part of $c'$ is untouched. Combining the displays proves Equation~\\eqref{eq:completed-quadratic-reduction}. Suppose now $|\\beta|\\le1$. Ideal counting gives \\[ \\#\\mathcal D_r(t)\\ll \\frac C{q_r}\\frac NC\\frac GC. \\] For $q_r\\asym", "anchor": { "source": "paper.tex", "line_start": 2793, "line_end": 2793, "char_start": 130763, "char_end": 130803 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:completed-quadratic-reduction", "to": "lem:completed-quadratic-reduction", "type": "candidate_proof_dependency", "evidence": "le $r$ in this range and $O(T)$ possible $t$. Since $B\\asymp GT^2$, their contribution to Equation~\\eqref{eq:completed-quadratic-reduction} is at most \\[ (KNB)^\\epsilon\\frac{NB}{r_0} \\left(\\frac K{r_0}+\\frac{NG}{C^2}\\right) \\ll (KNB)^\\epsilon NB(K+NB). \\] Here", "anchor": { "source": "paper.tex", "line_start": 2803, "line_end": 2803, "char_start": 131131, "char_end": 131171 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:completed-quadratic-norm", "to": "lem:completed-quadratic-reduction", "type": "candidate_proof_dependency", "evidence": "bert-space triangle over the $C,G,T$ blocks, followed by a rescaling of $\\epsilon$, proves Equation~\\eqref{eq:completed-quadratic-norm}. \\subsection{The unmarked reflected energy} For a fixed finite-ray character $\\nu$, extended by zero away from prim", "anchor": { "source": "paper.tex", "line_start": 2812, "line_end": 2812, "char_start": 131548, "char_end": 131583 }, "origin": "source_reference", "context": "proof" }, { "from": "prop:completed-reflection", "to": "sec:completed-reflection", "type": "internal_reference", "evidence": "ng series is absolutely convergent for every $X>0$ when $V$ satisfies the hypotheses of Proposition~\\ref{prop:completed-reflection}. Complete multiplicativity, with the zeros retained, gives $\\chi_{cn^3}(m)=\\chi_c(m)\\chi_n(m)^3$. Thus for every $a>1$, \\begin{", "anchor": { "source": "paper.tex", "line_start": 2828, "line_end": 2828, "char_start": 132213, "char_end": 132244 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:unmarked-completed-moment", "to": "sec:completed-reflection", "type": "internal_reference", "evidence": "{A,J}(V)^2 \\] for every fixed $C_m\\ge1$ and $\\epsilon>0$, with suitable finite orders $A,J$. Lemma~\\ref{lem:unmarked-completed-moment} below gives the more general estimate that tracks the repeated prime factors of the row. The reason for the quadratic sieve is a", "anchor": { "source": "paper.tex", "line_start": 2845, "line_end": 2845, "char_start": 132923, "char_end": 132958 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:completed-quadratic-reduction", "to": "sec:completed-reflection", "type": "internal_reference", "evidence": "s the only arithmetic interaction between the row and the two column indices is the kernel of Lemma~\\ref{lem:completed-quadratic-reduction}. On dual dyads $q_n\\asymp U$, $q_b\\asymp B$, the reflected kernel has argument comparable to $UB^3/Z$. Its rapid decay permits", "anchor": { "source": "paper.tex", "line_start": 2863, "line_end": 2863, "char_start": 133935, "char_end": 133974 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:reflection-common-profile", "to": "sec:completed-reflection", "type": "internal_reference", "evidence": "on uniform when the ramified exponent and the other parameters vary. On the retained blocks, Lemma~\\ref{lem:reflection-common-profile} separates the kernel with one common coefficient measure while retaining the actual row annulus. Write $\\mathcal U_{U,B,k}(R)$ f", "anchor": { "source": "paper.tex", "line_start": 2869, "line_end": 2870, "char_start": 134319, "char_end": 134354 }, "origin": "source_reference", "context": "narrative" }, { "from": "prop:completed-reflection", "to": "def:unmarked-reflected-block", "type": "internal_reference", "evidence": "$Z\\ge2$, $X=Z^N$, one of the fixed characters $\\Psi_0$ and its common modulus $L$ from Proposition~\\ref{prop:completed-reflection}, and put $M_{\\rm ref}=\\lambda^{12}L^4$. For the current norm over $R$, freeze a finite set $\\mathcal F$ of good primes with expo", "anchor": { "source": "paper.tex", "line_start": 2891, "line_end": 2891, "char_start": 135467, "char_end": 135498 }, "origin": "source_reference", "context": "statement" }, { "from": "old-eq:4.3", "to": "def:unmarked-reflected-block", "type": "internal_reference", "evidence": "$v,\\ell_b\\ge0$. The squarefree $n$ and the primary $b$ retain the full source support of Equation~\\eqref{old-eq:4.3}, including permitted primes of $S$ and shared primes. Let $H\\ge0$, and choose a common smooth profile $W_1(y_R,y_n,y_b)$ on a fi", "anchor": { "source": "paper.tex", "line_start": 2932, "line_end": 2932, "char_start": 137477, "char_end": 137495 }, "origin": "source_reference", "context": "statement" }, { "from": "eq:reflection", "to": "def:unmarked-reflected-block", "type": "internal_reference", "evidence": "ck. Denote by $\\mathcal U_{v,\\ell_b,e_\\lambda}(R)$ the specified term of the right side of Equation~\\eqref{eq:reflection}, restricted to this dual representation and multiplied by $W_0(y_R,y_n,y_b)$. The original row set $\\mathcal I_H$ consists of th", "anchor": { "source": "paper.tex", "line_start": 2945, "line_end": 2945, "char_start": 138078, "char_end": 138099 }, "origin": "source_reference", "context": "statement" }, { "from": "prop:completed-reflection", "to": "lem:unmarked-reflected-block", "type": "candidate_proof_dependency", "evidence": "d the fixed class of $R$ modulo $M_{\\rm ref}^2$, the class of $F_{\\rm act}R$ is fixed. Proposition~\\ref{prop:completed-reflection} therefore gives one common $d$ and one common additive character $\\vartheta$ for the whole row set. Write $x_F=u\\lambda^{k+4}N_F", "anchor": { "source": "paper.tex", "line_start": 3005, "line_end": 3006, "char_start": 140660, "char_end": 140691 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:unmarked-structural-block", "to": "lem:unmarked-reflected-block", "type": "candidate_proof_dependency", "evidence": "F)^3$; it is bounded by one because $|\\zeta_R|\\le1/81$. The factor $27/81=1/3$ now proves Equation~\\eqref{eq:unmarked-structural-block}. Multiplicativity of the norm proves Equation~\\eqref{eq:unmarked-actual-annuli}. It does not use any coprimality between $N_F$", "anchor": { "source": "paper.tex", "line_start": 3046, "line_end": 3046, "char_start": 142602, "char_end": 142638 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:unmarked-actual-annuli", "to": "lem:unmarked-reflected-block", "type": "candidate_proof_dependency", "evidence": "proves Equation~\\eqref{eq:unmarked-structural-block}. Multiplicativity of the norm proves Equation~\\eqref{eq:unmarked-actual-annuli}. It does not use any coprimality between $N_F$ and $b$. By the definitions of the log-norms, \\[ Y=q_{c_F}^{-2}Z^{v+3\\ell_b+e_\\", "anchor": { "source": "paper.tex", "line_start": 3047, "line_end": 3047, "char_start": 142686, "char_end": 142719 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:reflection-common-profile", "to": "lem:unmarked-reflected-block", "type": "candidate_proof_dependency", "evidence": "only on $L$. Consequently $\\mathfrak m_A(Y)\\ll_L Z^{-(T_0-v-3\\ell_b-e_\\lambda)_+/4}$. Apply Lemma~\\ref{lem:reflection-common-profile} in dimension three and Equation~\\eqref{eq:reflection-common-minkowski} to the structural identity. The single density is common t", "anchor": { "source": "paper.tex", "line_start": 3056, "line_end": 3056, "char_start": 143005, "char_end": 143040 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:reflection-common-minkowski", "to": "lem:unmarked-reflected-block", "type": "candidate_proof_dependency", "evidence": "b-e_\\lambda)_+/4}$. Apply Lemma~\\ref{lem:reflection-common-profile} in dimension three and Equation~\\eqref{eq:reflection-common-minkowski} to the structural identity. The single density is common to all $R$ because $W$ uses the row and both dual norms as coordinates.", "anchor": { "source": "paper.tex", "line_start": 3057, "line_end": 3057, "char_start": 143073, "char_end": 143111 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:completed-quadratic-norm", "to": "lem:unmarked-reflected-block", "type": "candidate_proof_dependency", "evidence": "ll Z^H$, retaining any fixed restrictions. No kernel estimate is used on the added rows. Equation~\\eqref{eq:completed-quadratic-norm}, with $K=Z^H$, $N_{\\rm col}=Z^v$, and $B_{\\rm col}=Z^{\\ell_b}$, bounds the separated squared norm by \\[ Z^{\\epsilon+\\max(H,v+\\el", "anchor": { "source": "paper.tex", "line_start": 3065, "line_end": 3065, "char_start": 143577, "char_end": 143612 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:unmarked-reflected-energy", "to": "lem:unmarked-reflected-block", "type": "candidate_proof_dependency", "evidence": "e_\\lambda)_+/2$. Combining these exponents and the dimension-three separation norm proves Equation~\\eqref{eq:unmarked-reflected-energy}. For later summation, call a dual block retained when \\[ v+3\\ell_b+e_\\lambda\\le T_0+\\tau_{\\rm ref}, \\qquad \\tau_{\\", "anchor": { "source": "paper.tex", "line_start": 3075, "line_end": 3075, "char_start": 144040, "char_end": 144076 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:unmarked-actual-annuli", "to": "sec:completed-reflection", "type": "internal_reference", "evidence": "_b+e_\\lambda\\le T_0+\\tau_{\\rm ref}, \\qquad \\tau_{\\rm ref}>0. \\] On the fixed support box, Equation~\\eqref{eq:unmarked-actual-annuli} places the kernel argument in $[C_{\\rm ker}^{-1},C_{\\rm ker}]$ times $Z^{v+3\\ell_b+e_\\lambda-T_0}$, for one fixed $C_{\\rm ker}$.", "anchor": { "source": "paper.tex", "line_start": 3083, "line_end": 3083, "char_start": 144255, "char_end": 144288 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:lattice-kernel-tail", "to": "sec:completed-reflection", "type": "internal_reference", "evidence": "s not count discarded dual indices. The remaining dual sum is the unrestricted lattice sum in Lemma~\\ref{lem:lattice-kernel-tail}, with $U=Z^{\\tau_{\\rm ref}/2}$. For any $D>0$, choosing \\[ A>1+\\frac{2(B_{\\rm tail}+D)}{\\tau_{\\rm ref}} \\] makes the discarded c", "anchor": { "source": "paper.tex", "line_start": 3108, "line_end": 3108, "char_start": 145636, "char_end": 145665 }, "origin": "source_reference", "context": "narrative" }, { "from": "prop:completed-reflection", "to": "lem:unmarked-completed-moment", "type": "internal_reference", "evidence": "exist a finite $A\\ge0$ and an integer $J\\ge0$ such that, for every test $V$ satisfying Proposition~\\ref{prop:completed-reflection}, every actual $O$-dyad, and all sufficiently large $Z$, \\sum_{\\substack{00$ \\[ \\sum_{00$ terms require $j=m=0$ and give $W_{\\rm loc}/(1-W_", "anchor": { "source": "paper.tex", "line_start": 4130, "line_end": 4130, "char_start": 189677, "char_end": 189697 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:local-H-defect", "to": "sec:poisson-representation", "type": "internal_reference", "evidence": "he first region $4-6x_r-6z_r\\le-11/10$ and $\\vartheta\\le1/100$. The extra factor $1-W$ in Equation~\\eqref{eq:local-H-defect} therefore leaves this contribution at most $Q^{-27/25}$. The term $DW$ is at most $Q^{-1-\\epsilon_0}$; the other terms in that e", "anchor": { "source": "paper.tex", "line_start": 4155, "line_end": 4155, "char_start": 190667, "char_end": 190692 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:probe-high", "to": "sec:poisson-representation", "type": "internal_reference", "evidence": "imate for the product. Finally extracting $(1-V)^{-1}(1-W)^{-1}(1-D)$ at every prime gives Equation~\\eqref{eq:probe-high}. The normal convergence of the remaining product proves its claimed continuation. For $u=1$, Equation~\\eqref{eq:pro", "anchor": { "source": "paper.tex", "line_start": 4187, "line_end": 4187, "char_start": 191965, "char_end": 191986 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:probe-high", "to": "sec:poisson-representation", "type": "internal_reference", "evidence": "vergence of the remaining product proves its claimed continuation. For $u=1$, Equation~\\eqref{eq:probe-high} contains $1/L^S(x,\\eta)$, and its numerator has the two principal factors $\\zeta_F^S(6z)$ and $\\zeta_F^S(w)$. Their residues will", "anchor": { "source": "paper.tex", "line_start": 4191, "line_end": 4191, "char_start": 192103, "char_end": 192124 }, "origin": "source_reference", "context": "narrative" }, { "from": "sec:shared-analytic-estimates", "to": "sec:poisson-representation", "type": "internal_reference", "evidence": "d zero. The principal and bounded rows, together with the outer norm ranges, are treated in Section~\\ref{sec:shared-analytic-estimates}. \\section{A zero detector with saturated witnesses} The high expansion contains a sum over sixth-power-free", "anchor": { "source": "paper.tex", "line_start": 4199, "line_end": 4199, "char_start": 192583, "char_end": 192618 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:logarithmic-control", "to": "sec:detector", "type": "internal_reference", "evidence": "e witnesses can then be counted using different mean estimates. The analytic inputs here are Lemma~\\ref{lem:logarithmic-control}, the smooth calculus of Lemma~\\ref{lem:smooth-calculus}, and the global growth estimate in Equation~\\eqref{eq:hecke-growth}, toge", "anchor": { "source": "paper.tex", "line_start": 4209, "line_end": 4209, "char_start": 193152, "char_end": 193181 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:smooth-calculus", "to": "sec:detector", "type": "internal_reference", "evidence": "es. The analytic inputs here are Lemma~\\ref{lem:logarithmic-control}, the smooth calculus of Lemma~\\ref{lem:smooth-calculus}, and the global growth estimate in Equation~\\eqref{eq:hecke-growth}, together with the deleted Euler-factor bounds of Lemma~\\ref{", "anchor": { "source": "paper.tex", "line_start": 4210, "line_end": 4210, "char_start": 193212, "char_end": 193237 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:hecke-growth", "to": "sec:detector", "type": "internal_reference", "evidence": "the smooth calculus of Lemma~\\ref{lem:smooth-calculus}, and the global growth estimate in Equation~\\eqref{eq:hecke-growth}, together with the deleted Euler-factor bounds of Lemma~\\ref{lem:deleted-euler-factors}. Throughout this section, assume \\(\\beta", "anchor": { "source": "paper.tex", "line_start": 4211, "line_end": 4211, "char_start": 193282, "char_end": 193305 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:deleted-euler-factors", "to": "sec:detector", "type": "internal_reference", "evidence": "stimate in Equation~\\eqref{eq:hecke-growth}, together with the deleted Euler-factor bounds of Lemma~\\ref{lem:deleted-euler-factors}. Throughout this section, assume \\(\\beta_*>51/100\\), where \\(\\beta_*\\) is the global supremum in Equation~\\eqref{old-eq:1.1b}.", "anchor": { "source": "paper.tex", "line_start": 4212, "line_end": 4212, "char_start": 193362, "char_end": 193393 }, "origin": "source_reference", "context": "narrative" }, { "from": "old-eq:1.1b", "to": "sec:detector", "type": "internal_reference", "evidence": "ghout this section, assume \\(\\beta_*>51/100\\), where \\(\\beta_*\\) is the global supremum in Equation~\\eqref{old-eq:1.1b}. This setting does not select either of the two boundaries in the continuation criterion. \\subsection{Buffered rectangles and p", "anchor": { "source": "paper.tex", "line_start": 4215, "line_end": 4215, "char_start": 193501, "char_end": 193520 }, "origin": "source_reference", "context": "narrative" }, { "from": "sec:arithmetic", "to": "sec:detector", "type": "internal_reference", "evidence": "and pointwise bounds} Fix the arithmetic data \\(\\mathcal A\\) independently of \\(Z\\), as in Section~\\ref{sec:arithmetic}. In particular, the finite group \\[ \\Theta=\\langle\\eta,\\widehat T\\rangle \\] contains the target and the fixed ray twists used i", "anchor": { "source": "paper.tex", "line_start": 4222, "line_end": 4222, "char_start": 193745, "char_end": 193765 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:logarithmic-control", "to": "lem:buffered-bins", "type": "candidate_proof_dependency", "evidence": "\\(M_{i+1}(u)51/100\\), the zero in Lemma~\\ref{lem:buffered-bins} produces large polynomials to which the moments can be applied. [Pointwise dyadic estimates]\\label{lem:detector-dya", "anchor": { "source": "paper.tex", "line_start": 4382, "line_end": 4382, "char_start": 200170, "char_end": 200193 }, "origin": "source_reference", "context": "narrative" }, { "from": "old-eq:1.1a", "to": "lem:detector-dyads", "type": "internal_reference", "evidence": ",m\\), and let \\[ M_r=M_\\psi(r;W_M),\\qquad S_m=S_\\psi(m;W_S) \\] denote the polynomials of Equations~\\eqref{old-eq:1.1a} and \\eqref{old-eq:1.1}, now at base \\(U\\). Suppose \\(\\psi\\in\\mathcal X_u\\), the untwisted profiles have uniformly bounded smooth", "anchor": { "source": "paper.tex", "line_start": 4390, "line_end": 4390, "char_start": 200465, "char_end": 200484 }, "origin": "source_reference", "context": "statement" }, { "from": "old-eq:1.1", "to": "lem:detector-dyads", "type": "internal_reference", "evidence": "\\psi(r;W_M),\\qquad S_m=S_\\psi(m;W_S) \\] denote the polynomials of Equations~\\eqref{old-eq:1.1a} and \\eqref{old-eq:1.1}, now at base \\(U\\). Suppose \\(\\psi\\in\\mathcal X_u\\), the untwisted profiles have uniformly bounded smooth seminorms on a fixed a", "anchor": { "source": "paper.tex", "line_start": 4391, "line_end": 4391, "char_start": 200489, "char_end": 200507 }, "origin": "source_reference", "context": "statement" }, { "from": "lem:smooth-calculus", "to": "lem:detector-dyads", "type": "candidate_proof_dependency", "evidence": "omega,\\psi)^{-1}\\,ds. \\] Here \\(\\mathcal MW\\) is the Mellin transform defined in Lemma~\\ref{lem:smooth-calculus}. The twist occurs in the \\(L\\)-argument, not in the transform whose tails are estimated. Shift only the portion with added frequ", "anchor": { "source": "paper.tex", "line_start": 4435, "line_end": 4435, "char_start": 202101, "char_end": 202126 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:buffered-bins", "to": "lem:detector-dyads", "type": "candidate_proof_dependency", "evidence": "ma)=a+6e\\). The horizontal joins and the shifted portion remain in the zero-free rectangle of Lemma~\\ref{lem:buffered-bins}. Its bound gives \\(U^{(a-1/2+6e)r+\\epsilon_1}\\) for the central portion. Leave the remaining tails on the absolute line, which", "anchor": { "source": "paper.tex", "line_start": 4440, "line_end": 4440, "char_start": 202417, "char_end": 202440 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:pointwise-mellin-tail", "to": "lem:detector-dyads", "type": "candidate_proof_dependency", "evidence": "\\) for fixed \\(B,J\\) chosen before the external tail order. The pointwise Mellin bound in Equation~\\eqref{eq:pointwise-mellin-tail}, applied on that compact real interval with derivative order \\(N+\\lceil J\\rceil+1\\), bounds each horizontal join by \\(O(U^B T_1^{", "anchor": { "source": "paper.tex", "line_start": 4450, "line_end": 4450, "char_start": 202986, "char_end": 203018 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:late-fourier-tail", "to": "lem:detector-dyads", "type": "candidate_proof_dependency", "evidence": "orizontal join by \\(O(U^B T_1^{-N})\\). The separate vertical tails have the same bound by Equation~\\eqref{eq:late-fourier-tail}. Both estimates apply to the untwisted profile, uniformly in the stated family. Because \\(T_1=Z^\\tau\\), \\(\\tau>0\\), and all leng", "anchor": { "source": "paper.tex", "line_start": 4453, "line_end": 4453, "char_start": 203217, "char_end": 203245 }, "origin": "source_reference", "context": "proof" }, { "from": "old-eq:6.2", "to": "lem:detector-dyads", "type": "candidate_proof_dependency", "evidence": "claimed bound. Taking \\(e\\) and \\(\\epsilon_1\\) small in the prescribed \\(\\epsilon\\) proves Equation~\\eqref{old-eq:6.2}. For the plain polynomial the same calculation uses \\(L_{\\rm orig}\\) instead of its reciprocal. Shifting to \\(\\Re(s+\\sigma)=a+6", "anchor": { "source": "paper.tex", "line_start": 4459, "line_end": 4459, "char_start": 203581, "char_end": 203599 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:buffered-bins", "to": "lem:detector-dyads", "type": "candidate_proof_dependency", "evidence": "1-a-6e\\). The function is entire because the row is nonprincipal. The reflected estimate of Lemma~\\ref{lem:buffered-bins} gives \\[ |S_m|\\ll U^{a-1/2+12e+\\epsilon_1} U^{(1/2-a-6e)m}(3+T_1)^C +O(U^B T_1^{-N}). \\] The joins here need only the global", "anchor": { "source": "paper.tex", "line_start": 4471, "line_end": 4471, "char_start": 203969, "char_end": 203992 }, "origin": "source_reference", "context": "proof" }, { "from": "old-eq:6.3", "to": "lem:detector-dyads", "type": "candidate_proof_dependency", "evidence": "possibly negative exponent when \\(m>1\\), and using the displayed height hypothesis proves Equation~\\eqref{old-eq:6.3}. A primitive conductor smaller than \\(U\\) reduces the reflected conductor factor. The original deleted Euler factors have alrea", "anchor": { "source": "paper.tex", "line_start": 4486, "line_end": 4486, "char_start": 204549, "char_end": 204567 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:buffered-bins", "to": "lem:detector-dyads", "type": "candidate_proof_dependency", "evidence": "reflected conductor factor. The original deleted Euler factors have already been included in Lemma~\\ref{lem:buffered-bins}. There are only finitely many separated variables in any one \\(L\\)-argument. Assign each a fixed fraction of the single \\(T_1/2", "anchor": { "source": "paper.tex", "line_start": 4489, "line_end": 4489, "char_start": 204723, "char_end": 204746 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:smooth-calculus", "to": "lem:detector-dyads", "type": "candidate_proof_dependency", "evidence": "ofile windows and the finite height orders used in the row estimates are fixed before \\(N\\). Lemma~\\ref{lem:smooth-calculus} differentiates only the untwisted separating profiles when increasing \\(N\\), which proves the last assertion. \\subse", "anchor": { "source": "paper.tex", "line_start": 4495, "line_end": 4495, "char_start": 205053, "char_end": 205078 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:detector-dyads", "to": "prop:detector-witness", "type": "internal_reference", "evidence": "et \\(u\\) have a bin \\((i,a)\\) with \\(a>51/100\\), and assume the loss and height hypotheses of Lemma~\\ref{lem:detector-dyads}. For every \\(t\\in[1,3/2]\\), and every prescribed \\(\\epsilon>0\\), after reducing its preliminary losses, there are \\(\\psi\\in\\mathc", "anchor": { "source": "paper.tex", "line_start": 4512, "line_end": 4512, "char_start": 205853, "char_end": 205877 }, "origin": "source_reference", "context": "statement" }, { "from": "lem:buffered-bins", "to": "prop:detector-witness", "type": "internal_reference", "evidence": "s preliminary losses, there are \\(\\psi\\in\\mathcal X_u\\), a zero \\(\\rho=\\sigma+i\\gamma\\) as in Lemma~\\ref{lem:buffered-bins}, dyadic lengths \\(D=U^r\\), \\(N=U^m\\), and two polynomials \\(M_r,S_m\\) for the common row character \\(\\psi\\) such that \\begin{equa", "anchor": { "source": "paper.tex", "line_start": 4517, "line_end": 4517, "char_start": 206060, "char_end": 206083 }, "origin": "source_reference", "context": "statement" }, { "from": "eq:hecke-growth", "to": "prop:detector-witness", "type": "candidate_proof_dependency", "evidence": "and its residue vanishes because \\(L_{\\rm orig}(\\rho,\\psi)=0\\). The global strip bound in Equation~\\eqref{eq:hecke-growth}, together with the deleted Euler factors, bounds the new line by \\[ U^{-5}U^2D_*^{\\,1-\\sigma+1/4+o(1)}T_1^2. \\] Indeed \\(\\Re(\\rh", "anchor": { "source": "paper.tex", "line_start": 4571, "line_end": 4571, "char_start": 208271, "char_end": 208294 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:detector-note", "to": "prop:detector-witness", "type": "candidate_proof_dependency", "evidence": "} \\] for sufficiently large \\(U\\). The support inequalities give the two length bounds in Equation~\\eqref{eq:detector-note}. Apply Lemma~\\ref{lem:detector-dyads} to these profiles, choosing its loss smaller than the present \\(\\epsilon\\). Comparison of", "anchor": { "source": "paper.tex", "line_start": 4666, "line_end": 4666, "char_start": 211955, "char_end": 211979 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:detector-dyads", "to": "prop:detector-witness", "type": "candidate_proof_dependency", "evidence": "support inequalities give the two length bounds in Equation~\\eqref{eq:detector-note}. Apply Lemma~\\ref{lem:detector-dyads} to these profiles, choosing its loss smaller than the present \\(\\epsilon\\). Comparison of the product lower bound with the two u", "anchor": { "source": "paper.tex", "line_start": 4668, "line_end": 4668, "char_start": 211994, "char_end": 212018 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:saturated-witness", "to": "prop:detector-witness", "type": "candidate_proof_dependency", "evidence": "ower bound in turn by each individual upper bound gives the two individual lower bounds in Equation~\\eqref{eq:saturated-witness}. Only the dyadic support errors are \\(O(1/\\log U)\\); all other losses are arbitrarily small fixed powers. Lemma~\\ref{lem:smooth", "anchor": { "source": "paper.tex", "line_start": 4681, "line_end": 4681, "char_start": 212503, "char_end": 212531 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:smooth-calculus", "to": "prop:detector-witness", "type": "candidate_proof_dependency", "evidence": "dic support errors are \\(O(1/\\log U)\\); all other losses are arbitrarily small fixed powers. Lemma~\\ref{lem:smooth-calculus} permits the stated rowwise choices with a fixed polynomial height factor. \\section{A sextic-sieve row count}\\label{s", "anchor": { "source": "paper.tex", "line_start": 4683, "line_end": 4683, "char_start": 212646, "char_end": 212671 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:fixed-gauss-phase", "to": "lem:sextic-large-sieve", "type": "candidate_proof_dependency", "evidence": "also have the same residue modulo $4$, which controls the reciprocity factor $\\mathcal R$ of Lemma~\\ref{lem:fixed-gauss-phase}. Let $b,c$ be coprime squarefree good ideals with $t(b)=t(c)$ and let $j\\in J$. The raw character on elements \\[ \\Theta_{b,c}^", "anchor": { "source": "paper.tex", "line_start": 4791, "line_end": 4791, "char_start": 217833, "char_end": 217860 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:sextic-sieve-unit-class", "to": "lem:sextic-large-sieve", "type": "candidate_proof_dependency", "evidence": "chi_b(z)^j\\overline{\\chi_c(z)^j} \\quad (z\\bmod bc) \\] is trivial on global units by Equation~\\eqref{eq:sextic-sieve-unit-class}. It therefore defines a finite-order ray character on ideals by \\[ \\Psi_{b,c}^{(j)}((z))=\\Theta_{b,c}^{(j)}(z). \\] For fractiona", "anchor": { "source": "paper.tex", "line_start": 4799, "line_end": 4799, "char_start": 218100, "char_end": 218134 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:sextic-sieve-pair-gauss", "to": "lem:sextic-large-sieve", "type": "candidate_proof_dependency", "evidence": "n class. Every $\\gamma_{\\pm j}$ here has modulus one by prime Gauss orthogonality and CRT. Formula~\\eqref{eq:sextic-sieve-pair-gauss} retains $\\gamma_{-j}$ itself, including its factor at $-1$ under conjugation. Choose once a nonnegative $W\\in C_c^\\infty((0,\\inf", "anchor": { "source": "paper.tex", "line_start": 4832, "line_end": 4832, "char_start": 219743, "char_end": 219777 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:sextic-sieve-pair-poisson", "to": "lem:sextic-large-sieve", "type": "candidate_proof_dependency", "evidence": "$\\|\\lambda\\|_2=1$. Then every pair in $Q_j$ is nontrivial and $q_{bc}\\asymp N^2$. Apply Equation~\\eqref{eq:sextic-sieve-pair-poisson} and take the triangle inequality over the dual ideals. For a small $\\delta>0$ put $Z=(2+MN)^\\delta$. Cauchy gives $(\\sum_b|\\lam", "anchor": { "source": "paper.tex", "line_start": 4907, "line_end": 4907, "char_start": 223068, "char_end": 223104 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:sextic-sieve-transform-bounds", "to": "lem:sextic-large-sieve", "type": "candidate_proof_dependency", "evidence": "r a small $\\delta>0$ put $Z=(2+MN)^\\delta$. Cauchy gives $(\\sum_b|\\lambda_b|)^2\\ll N$. By Equation~\\eqref{eq:sextic-sieve-transform-bounds}, the part with $q_{\\mathfrak a}>ZN^2/M$ is, for any $A>1$, at most \\[ M\\sum_{q_{\\mathfrak a}>ZN^2/M}(Mq_{\\mathfrak a}/N^2)^{-A}", "anchor": { "source": "paper.tex", "line_start": 4910, "line_end": 4910, "char_start": 223263, "char_end": 223303 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:sextic-sieve-pair-values", "to": "lem:sextic-large-sieve", "type": "candidate_proof_dependency", "evidence": "thfrak s$ supported on $S$, and choose one generator $s_0$ for each fixed $\\mathfrak s$. Equations~\\eqref{eq:sextic-sieve-pair-values} and~\\eqref{eq:sextic-sieve-pair-gauss} give the separated factors \\[ \\overline{\\Psi_{b,c}^{(j)}(a_0\\mathfrak s)} =\\overline{F_b", "anchor": { "source": "paper.tex", "line_start": 4921, "line_end": 4922, "char_start": 223748, "char_end": 223783 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:sextic-sieve-pair-gauss", "to": "lem:sextic-large-sieve", "type": "candidate_proof_dependency", "evidence": "ne generator $s_0$ for each fixed $\\mathfrak s$. Equations~\\eqref{eq:sextic-sieve-pair-values} and~\\eqref{eq:sextic-sieve-pair-gauss} give the separated factors \\[ \\overline{\\Psi_{b,c}^{(j)}(a_0\\mathfrak s)} =\\overline{F_b(a_0)^j}F_c(a_0)^j \\overline{\\ch", "anchor": { "source": "paper.tex", "line_start": 4922, "line_end": 4922, "char_start": 223788, "char_end": 223822 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:sextic-sieve-transform-bounds", "to": "lem:sextic-large-sieve", "type": "candidate_proof_dependency", "evidence": "and of the dyadic shell, because $S$ is fixed and a nonzero $Q_j$ has $M$ bounded below. Equation~\\eqref{eq:sextic-sieve-transform-bounds} integrates the $t$ variable. Choosing $\\delta$ sufficiently small in terms of $\\varepsilon$ proves Equation~\\eqref{eq:sextic-sie", "anchor": { "source": "paper.tex", "line_start": 4956, "line_end": 4957, "char_start": 225368, "char_end": 225408 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:sextic-sieve-pair-norm", "to": "lem:sextic-large-sieve", "type": "candidate_proof_dependency", "evidence": "s the $t$ variable. Choosing $\\delta$ sufficiently small in terms of $\\varepsilon$ proves Equation~\\eqref{eq:sextic-sieve-pair-norm}, including the allowed global loss on its $M$ term. We now apply this paired estimate to $E_j$. Partition its inner sum by $t$,", "anchor": { "source": "paper.tex", "line_start": 4958, "line_end": 4958, "char_start": 225518, "char_end": 225551 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:sextic-sieve-pair-values", "to": "lem:sextic-large-sieve", "type": "candidate_proof_dependency", "evidence": ". Their residual classes agree because $t$ is multiplicative. Index multiplicativity and Equation~\\eqref{eq:sextic-sieve-pair-values} give on every good $a$ \\[ F_{db}(a)^j\\overline{F_{dc}(a)^j} =1_{(a,d)=1}\\Psi_{b,c}^{(j)}(a). \\] Let $\\mathfrak s_S=\\prod_{\\ma", "anchor": { "source": "paper.tex", "line_start": 4965, "line_end": 4965, "char_start": 225910, "char_end": 225945 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:sextic-sieve-pair-norm", "to": "lem:sextic-large-sieve", "type": "candidate_proof_dependency", "evidence": "mbda'_b|\\le|\\lambda_{db}|. \\] It is supported on $\\mathcal A_{N/q_d}$ in one class. Apply Equation~\\eqref{eq:sextic-sieve-pair-norm} with lengths $M/q_{\\mathfrak r}$ and $N/q_d$. The summed coefficient mass is at most \\[ \\sum_d\\sum_{\\mathfrak r\\mid\\mathfrak s_", "anchor": { "source": "paper.tex", "line_start": 4980, "line_end": 4980, "char_start": 226571, "char_end": 226604 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:planar-additive-sieve", "to": "lem:sextic-large-sieve", "type": "candidate_proof_dependency", "evidence": "$; equality modulo $\\mathcal O$ forces the same primary denominator and residue. The dual of Lemma~\\ref{lem:planar-additive-sieve}, with the coefficient generators in the ball $q_b\\le2N$, proves Equation~\\eqref{eq:sextic-sieve-norm-initial}. We need the follo", "anchor": { "source": "paper.tex", "line_start": 5033, "line_end": 5033, "char_start": 228969, "char_end": 229000 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:sextic-sieve-norm-initial", "to": "lem:sextic-large-sieve", "type": "candidate_proof_dependency", "evidence": "lem:planar-additive-sieve}, with the coefficient generators in the ball $q_b\\le2N$, proves Equation~\\eqref{eq:sextic-sieve-norm-initial}. We need the following near-monotonicity, also used in the large-sieve recursions of \\cite[Lemma~4.4 and Remark~5]{GL} and \\cite", "anchor": { "source": "paper.tex", "line_start": 5034, "line_end": 5034, "char_start": 229074, "char_end": 229110 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:sextic-sieve-norm-monotone", "to": "lem:sextic-large-sieve", "type": "candidate_proof_dependency", "evidence": "). \\] For the stated threshold with $C$ sufficiently large, $d_{M_1}+4d_N\\le P/2$, proving Equation~\\eqref{eq:sextic-sieve-norm-monotone}. We finish with the power decomposition and recursion of \\cite[Lemmas~3.4--3.5]{BGL}. Suppose, simultaneously for $j\\in J$, tha", "anchor": { "source": "paper.tex", "line_start": 5064, "line_end": 5064, "char_start": 230486, "char_end": 230523 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:sextic-sieve-inductive-bound", "to": "lem:sextic-large-sieve", "type": "candidate_proof_dependency", "evidence": "{1/3}\\ll(M/X_2)^{2/3}$. The number of boxes is a fixed power of $\\log(2+M)$. Substituting Equation~\\eqref{eq:sextic-sieve-inductive-bound} into Equation~\\eqref{eq:sextic-sieve-norm-power} yields \\[ E_j(M,N)\\ll_{S,\\varepsilon}(MN)^\\varepsilon (M^\\a", "anchor": { "source": "paper.tex", "line_start": 5106, "line_end": 5106, "char_start": 232337, "char_end": 232376 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:sextic-sieve-norm-power", "to": "lem:sextic-large-sieve", "type": "candidate_proof_dependency", "evidence": "power of $\\log(2+M)$. Substituting Equation~\\eqref{eq:sextic-sieve-inductive-bound} into Equation~\\eqref{eq:sextic-sieve-norm-power} yields \\[ E_j(M,N)\\ll_{S,\\varepsilon}(MN)^\\varepsilon (M^\\alpha+M^{1/3}N). \\] The three terms before simplif", "anchor": { "source": "paper.tex", "line_start": 5107, "line_end": 5107, "char_start": 232391, "char_end": 232425 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:sextic-sieve-norm-reflection", "to": "lem:sextic-large-sieve", "type": "candidate_proof_dependency", "evidence": "s bounded by one of the first two according as $N\\le M$ or $N\\ge M$. Use this estimate in Equation~\\eqref{eq:sextic-sieve-norm-reflection}. All auxiliary nonempty scales are bounded by fixed powers of $2+MN$: here $N/\\Delta_{\\mathrm d}\\ge1/2$ and $H(N/\\Delta_{\\mathrm", "anchor": { "source": "paper.tex", "line_start": 5115, "line_end": 5115, "char_start": 232730, "char_end": 232769 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:sextic-sieve-norm-recurrence", "to": "lem:sextic-large-sieve", "type": "candidate_proof_dependency", "evidence": "2\\alpha-5/3)/(\\alpha-1/3)$ and $\\beta=2-2/(3\\alpha-1)$. If $M\\ge N^h$, the middle term in Equation~\\eqref{eq:sextic-sieve-norm-recurrence} is at most $(MN)^{2/3}$. If $M0$, after reducing the preliminary losses in Proposition~\\ref{prop:detector-witness}, for all sufficiently large $Z$, \\#\\mathcal B\\ll_{\\mathcal A,\\epsilon} U^{R(\\delta)", "anchor": { "source": "paper.tex", "line_start": 5191, "line_end": 5191, "char_start": 236647, "char_end": 236674 }, "origin": "source_reference", "context": "statement" }, { "from": "prop:detector-witness", "to": "prop:sextic-row-count", "type": "internal_reference", "evidence": "shared profile seminorms. The rowwise profile choices are precisely those permitted by Proposition~\\ref{prop:detector-witness} and Lemma~\\ref{lem:smooth-calculus}; no arbitrary row-dependent coefficients are allowed. The floor bin $a=51/100$ is excluded f", "anchor": { "source": "paper.tex", "line_start": 5206, "line_end": 5206, "char_start": 237428, "char_end": 237455 }, "origin": "source_reference", "context": "statement" }, { "from": "lem:smooth-calculus", "to": "prop:sextic-row-count", "type": "internal_reference", "evidence": "profile choices are precisely those permitted by Proposition~\\ref{prop:detector-witness} and Lemma~\\ref{lem:smooth-calculus}; no arbitrary row-dependent coefficients are allowed. The floor bin $a=51/100$ is excluded from the witness assertion. For that", "anchor": { "source": "paper.tex", "line_start": 5207, "line_end": 5207, "char_start": 237466, "char_end": 237491 }, "origin": "source_reference", "context": "statement" }, { "from": "prop:detector-witness", "to": "prop:sextic-row-count", "type": "candidate_proof_dependency", "evidence": "$. Fix a small preliminary loss $\\epsilon_0>0$. Apply Proposition~\\ref{prop:detector-witness} with $t=1$ to every row in $\\mathcal B$. Subdivide by its presentation labels in $\\psi_{u,\\nu,\\varsigma}(n)=\\nu(n)\\chi_n(u)^{\\va", "anchor": { "source": "paper.tex", "line_start": 5217, "line_end": 5217, "char_start": 237833, "char_end": 237860 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:sextic-actual-row-annulus", "to": "prop:sextic-row-count", "type": "candidate_proof_dependency", "evidence": "es by the same ideal count. For each fixed $\\mathfrak w$ in its dyad, the ball containing Equation~\\eqref{eq:sextic-actual-row-annulus} has $O_{\\mathcal A}(U/V)$ ideals. A nonempty annulus has $U/V$ bounded below by a fixed positive constant, which also covers the", "anchor": { "source": "paper.tex", "line_start": 5284, "line_end": 5284, "char_start": 241021, "char_end": 241057 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:sextic-large-sieve", "to": "prop:sextic-row-count", "type": "candidate_proof_dependency", "evidence": "plicativity. In particular, the inverse polynomial for a fixed profile $W$ is the row sum in Lemma~\\ref{lem:sextic-large-sieve} with coefficients \\[ c_n=D^{-1/2}\\mu(n)\\nu(n)\\chi_n(v)^{\\varsigma}W(q_n/D). \\] The factor $\\mu(n)$ keeps the columns squarefree.", "anchor": { "source": "paper.tex", "line_start": 5305, "line_end": 5305, "char_start": 242024, "char_end": 242052 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:sextic-large-sieve", "to": "prop:sextic-row-count", "type": "candidate_proof_dependency", "evidence": "e finitely many profile derivatives used below, with the corresponding fixed seminorm. Apply Lemma~\\ref{lem:sextic-large-sieve} after enlarging the positive residual row sum from its actual annulus to $q_k\\ll_{\\mathcal A}U/V$. A fixed enlargement makes the", "anchor": { "source": "paper.tex", "line_start": 5327, "line_end": 5327, "char_start": 243062, "char_end": 243090 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:sextic-witness-spike", "to": "prop:sextic-row-count", "type": "candidate_proof_dependency", "evidence": "re summed only after those positive bounds. We now restore the rowwise choice of $W_u$ in Equation~\\eqref{eq:sextic-witness-spike}. Here the detector was used with $t=1$, so $D_*=U$. After the presentation and dyadic pair are fixed, its inverse profiles have", "anchor": { "source": "paper.tex", "line_start": 5343, "line_end": 5343, "char_start": 243943, "char_end": 243974 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:smooth-calculus", "to": "prop:sextic-row-count", "type": "candidate_proof_dependency", "evidence": "There are at most a fixed power of $1+T_1$ such boxes. The parameter Sobolev inequality in Lemma~\\ref{lem:smooth-calculus} bounds the supremum on a box by a finite sum of integrals of squared profile derivatives. Sum over the rows before these integra", "anchor": { "source": "paper.tex", "line_start": 5357, "line_end": 5357, "char_start": 244692, "char_end": 244717 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:sextic-stratum-fixed-moment", "to": "prop:sextic-row-count", "type": "candidate_proof_dependency", "evidence": "ed profile coefficient, inserting the allowed logarithmic weights or profile derivatives. Equation~\\eqref{eq:sextic-stratum-fixed-moment} therefore applies to every such integral. This proves \\sum_{u\\in\\mathcal U_V}|", "anchor": { "source": "paper.tex", "line_start": 5361, "line_end": 5362, "char_start": 245019, "char_end": 245057 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:sextic-stratum-cardinality", "to": "prop:sextic-row-count", "type": "candidate_proof_dependency", "evidence": "ted separately for each $k$. It remains to combine this moment with the independent count Equation~\\eqref{eq:sextic-stratum-cardinality}. Write \\[ V=U^o,\\qquad D=U^r,\\qquad e_o(r)=\\frac o2+\\max\\left\\{1-o,r,\\frac{2(1-o+r)}3\\right\\}. \\] These are the parameters of", "anchor": { "source": "paper.tex", "line_start": 5376, "line_end": 5376, "char_start": 245792, "char_end": 245829 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:sextic-stratum-moment", "to": "prop:sextic-row-count", "type": "candidate_proof_dependency", "evidence": "has $o=0$. Since the witness lengths stay bounded, the arbitrarily small power of $UD$ in Equation~\\eqref{eq:sextic-stratum-moment} can be written as an arbitrarily small power of $U$. Dividing that equation by the spike in Equation~\\eqref{eq:sextic-witness-sp", "anchor": { "source": "paper.tex", "line_start": 5384, "line_end": 5384, "char_start": 246150, "char_end": 246182 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:sextic-witness-spike", "to": "prop:sextic-row-count", "type": "candidate_proof_dependency", "evidence": "n be written as an arbitrarily small power of $U$. Dividing that equation by the spike in Equation~\\eqref{eq:sextic-witness-spike}, and also using Equation~\\eqref{eq:sextic-stratum-cardinality}, gives for this subdivision \\label{eq:sextic-strat", "anchor": { "source": "paper.tex", "line_start": 5386, "line_end": 5386, "char_start": 246285, "char_end": 246316 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:sextic-stratum-cardinality", "to": "prop:sextic-row-count", "type": "candidate_proof_dependency", "evidence": "ing that equation by the spike in Equation~\\eqref{eq:sextic-witness-spike}, and also using Equation~\\eqref{eq:sextic-stratum-cardinality}, gives for this subdivision \\#\\mathcal U_V \\ll U^{\\min\\{1-o/2,e_o(r)-\\delta", "anchor": { "source": "paper.tex", "line_start": 5387, "line_end": 5387, "char_start": 246342, "char_end": 246379 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:sextic-witness-spike", "to": "prop:sextic-row-count", "type": "candidate_proof_dependency", "evidence": "nt in $[1/2,1]$, changes the exponent by at most $O_{\\mathcal A}(\\epsilon_0+1/\\log U)$, by Equation~\\eqref{eq:sextic-witness-spike} and the actual dyad bounds. Thus this replacement is made only in the final optimization, not in the row sum or its sieve applica", "anchor": { "source": "paper.tex", "line_start": 5433, "line_end": 5433, "char_start": 248254, "char_end": 248285 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:sextic-stratum-envelope", "to": "prop:sextic-row-count", "type": "candidate_proof_dependency", "evidence": "rs and $O(\\log U)$ powerful-part dyads cost only another arbitrarily small power. Summing Equation~\\eqref{eq:sextic-stratum-envelope} proves Equation~\\eqref{eq:sextic-row-count}. Finally, the bin $a=51/100$ need not contain an actual zero, so no use of Propositi", "anchor": { "source": "paper.tex", "line_start": 5439, "line_end": 5439, "char_start": 248657, "char_end": 248691 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:sextic-row-count", "to": "prop:sextic-row-count", "type": "candidate_proof_dependency", "evidence": "other arbitrarily small power. Summing Equation~\\eqref{eq:sextic-stratum-envelope} proves Equation~\\eqref{eq:sextic-row-count}. Finally, the bin $a=51/100$ need not contain an actual zero, so no use of Proposition~\\ref{prop:detector-witness} is made there", "anchor": { "source": "paper.tex", "line_start": 5440, "line_end": 5440, "char_start": 248708, "char_end": 248735 }, "origin": "source_reference", "context": "proof" }, { "from": "prop:detector-witness", "to": "prop:sextic-row-count", "type": "candidate_proof_dependency", "evidence": "-row-count}. Finally, the bin $a=51/100$ need not contain an actual zero, so no use of Proposition~\\ref{prop:detector-witness} is made there. Counting all elements in its physical annulus gives $O_{\\mathcal A}(U)$ rows. Since $1/50<1/3$, the displayed fo", "anchor": { "source": "paper.tex", "line_start": 5443, "line_end": 5443, "char_start": 248824, "char_end": 248851 }, "origin": "source_reference", "context": "proof" }, { "from": "sec:local-euler", "to": "sec:shared-analytic-estimates", "type": "internal_reference", "evidence": "ta_*-11/12>0$ and set $X=Y=Z^{1/2}$. Write $s$ for the first Mellin variable called $x$ in Section~\\ref{sec:local-euler}. Equations~\\eqref{eq:probe-poisson-identity} and~\\eqref{eq:probe-high} give the actual integral to be estimated: \\[ \\begin{split", "anchor": { "source": "paper.tex", "line_start": 5467, "line_end": 5467, "char_start": 249705, "char_end": 249726 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:probe-poisson-identity", "to": "sec:shared-analytic-estimates", "type": "internal_reference", "evidence": "$. Write $s$ for the first Mellin variable called $x$ in Section~\\ref{sec:local-euler}. Equations~\\eqref{eq:probe-poisson-identity} and~\\eqref{eq:probe-high} give the actual integral to be estimated: \\[ I_\\eta(Z^{1/2},Z^{1/2},Z) =\\frac1{(2\\pi i)", "anchor": { "source": "paper.tex", "line_start": 5467, "line_end": 5468, "char_start": 249739, "char_end": 249772 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:probe-high", "to": "sec:shared-analytic-estimates", "type": "internal_reference", "evidence": "iable called $x$ in Section~\\ref{sec:local-euler}. Equations~\\eqref{eq:probe-poisson-identity} and~\\eqref{eq:probe-high} give the actual integral to be estimated: \\[ I_\\eta(Z^{1/2},Z^{1/2},Z) =\\frac1{(2\\pi i)^3}\\int_{(3)}\\int_{(3)}\\in", "anchor": { "source": "paper.tex", "line_start": 5468, "line_end": 5468, "char_start": 249777, "char_end": 249798 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:fixed-numerator-ray", "to": "sec:shared-analytic-estimates", "type": "internal_reference", "evidence": "sociated sextic character is nontrivial \\cite[Theorem 1.1]{TZ}. By the Kummer description in Lemma~\\ref{lem:fixed-numerator-ray}, this is $\\chi_\\bullet(u)$. Therefore $u=1$ is the sole principal numerator row. A denominator attached to a bounded unit row ma", "anchor": { "source": "paper.tex", "line_start": 5513, "line_end": 5513, "char_start": 252027, "char_end": 252056 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:local-euler", "to": "sec:shared-analytic-estimates", "type": "internal_reference", "evidence": "atorname*{Res}_{v=1}\\zeta_F^S(v)\\right)^2/6. Lemma~\\ref{lem:local-euler} makes $H_\\eta$ holomorphic on $\\Re s>7/8$. Its uniform estimate $H_\\eta=1+O(P_0^{-4/5})$ permits a choice of $P_0$ before the tar", "anchor": { "source": "paper.tex", "line_start": 5525, "line_end": 5525, "char_start": 252493, "char_end": 252514 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:radial-mellin-positive", "to": "sec:shared-analytic-estimates", "type": "internal_reference", "evidence": "e. Indeed $\\widehat W_1(1)>0$ because $W_1$ is nonnegative and nonzero, and $M(1/6)>0$ by Equation~\\eqref{eq:radial-mellin-positive}. The simple pole of $\\zeta_F^S$ has positive residue: deletion multiplies the positive residue of $\\zeta_F$ by $\\prod_{p\\in S}(1", "anchor": { "source": "paper.tex", "line_start": 5541, "line_end": 5541, "char_start": 253311, "char_end": 253344 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:common-residue", "to": "sec:shared-analytic-estimates", "type": "internal_reference", "evidence": "}(Z)=\\frac{I_\\eta(Z^{1/2},Z^{1/2},Z)}{c_S}. \\] Let $f_{\\mathrm I,\\eta}$ be the integral in Equation~\\eqref{eq:common-residue} with $C=C_{\\mathrm I}$, $\\mathcal S=S$, and the function $H_\\eta$ just defined. The two scalar poles at $w=1$ and $z=1/6$ will g", "anchor": { "source": "paper.tex", "line_start": 5551, "line_end": 5551, "char_start": 253670, "char_end": 253695 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:probe-high", "to": "sec:shared-analytic-estimates", "type": "internal_reference", "evidence": "of $Z$, and the parameters below are $l_x=l_y=h=1/2$ and $\\ell=0$. The scalar quotient in Equation~\\eqref{eq:probe-high} is the part of the high expansion that determines both the contour move and the principal residues. We keep that quotient fixed", "anchor": { "source": "paper.tex", "line_start": 5565, "line_end": 5565, "char_start": 254397, "char_end": 254418 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:local-euler", "to": "sec:shared-analytic-estimates", "type": "internal_reference", "evidence": "uad \\Re z\\ge33/200,\\quad \\Re w\\ge19/20\\}. \\] These are the two regions in Lemma~\\ref{lem:local-euler}. A real box below means a compact rectangular subset of $\\mathbb R^3$ for $(\\Re s,\\Re w,\\Re z)$. [Data for an", "anchor": { "source": "paper.tex", "line_start": 5578, "line_end": 5578, "char_start": 254926, "char_end": 254947 }, "origin": "source_reference", "context": "narrative" }, { "from": "sec:probe", "to": "def:stage-high-data", "type": "internal_reference", "evidence": "character. For each primitive finite-order target $\\eta$, fix the data $S,\\xi,W_0,W_1$ of Section~\\ref{sec:probe}, independently of $Z$. Data for an exact high representation consist of the following objects and assertions. First, $\\mathscr", "anchor": { "source": "paper.tex", "line_start": 5594, "line_end": 5594, "char_start": 255533, "char_end": 255548 }, "origin": "source_reference", "context": "statement" }, { "from": "eq:general-probe", "to": "def:stage-high-data", "type": "internal_reference", "evidence": "ication it is given by a finite linear combination, for that $Z$, of the completed sums in Equation~\\eqref{eq:general-probe}, allowing specified restrictions on the completed index $A$ and specified rescalings of its three positive scales. The number of", "anchor": { "source": "paper.tex", "line_start": 5600, "line_end": 5600, "char_start": 255895, "char_end": 255919 }, "origin": "source_reference", "context": "statement" }, { "from": "eq:probe-poisson-identity", "to": "def:stage-high-data", "type": "internal_reference", "evidence": "abel{eq:stage-high-identity} where the sum has the physical restrictions of Equation~\\eqref{eq:probe-poisson-identity}, and \\[ \\mathcal W(X,Y,Z;s,w,z) =X^{1/2-z}Z^{s+z-1}Y^{w-1} e^{(s+z-1)^2}M(z)\\widehat W_1(w). \\] Thus Equation~\\eqref{eq:", "anchor": { "source": "paper.tex", "line_start": 5642, "line_end": 5642, "char_start": 257727, "char_end": 257760 }, "origin": "source_reference", "context": "statement" }, { "from": "eq:stage-high-identity", "to": "def:stage-high-data", "type": "internal_reference", "evidence": "(X,Y,Z;s,w,z) =X^{1/2-z}Z^{s+z-1}Y^{w-1} e^{(s+z-1)^2}M(z)\\widehat W_1(w). \\] Thus Equation~\\eqref{eq:stage-high-identity} is a hypothesis to be proved for the physical expression, not its definition. For the base probe, the physical", "anchor": { "source": "paper.tex", "line_start": 5648, "line_end": 5648, "char_start": 257880, "char_end": 257910 }, "origin": "source_reference", "context": "statement" }, { "from": "eq:probe-poisson-identity", "to": "sec:shared-analytic-estimates", "type": "internal_reference", "evidence": "$I_\\eta(Z^{l_x},Z^{l_y},Z)$, $\\ell=0$, and $\\mathfrak H_{\\eta,u,Z}=\\mathcal H_{\\eta,u}$. Equations~\\eqref{eq:probe-poisson-identity} and~\\eqref{eq:probe-high} prove the required identity, and Lemma~\\ref{lem:local-euler} gives the holomorphy and Equation~\\eqref{e", "anchor": { "source": "paper.tex", "line_start": 5655, "line_end": 5655, "char_start": 258153, "char_end": 258186 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:probe-high", "to": "sec:shared-analytic-estimates", "type": "internal_reference", "evidence": ", and $\\mathfrak H_{\\eta,u,Z}=\\mathcal H_{\\eta,u}$. Equations~\\eqref{eq:probe-poisson-identity} and~\\eqref{eq:probe-high} prove the required identity, and Lemma~\\ref{lem:local-euler} gives the holomorphy and Equation~\\eqref{eq:stage-all-height-majoran", "anchor": { "source": "paper.tex", "line_start": 5655, "line_end": 5656, "char_start": 258191, "char_end": 258212 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:local-euler", "to": "sec:shared-analytic-estimates", "type": "internal_reference", "evidence": "~\\eqref{eq:probe-poisson-identity} and~\\eqref{eq:probe-high} prove the required identity, and Lemma~\\ref{lem:local-euler} gives the holomorphy and Equation~\\eqref{eq:stage-all-height-majorant}. More generally, the function $\\mathfrak H$ in the defini", "anchor": { "source": "paper.tex", "line_start": 5656, "line_end": 5656, "char_start": 258252, "char_end": 258273 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:stage-all-height-majorant", "to": "sec:shared-analytic-estimates", "type": "internal_reference", "evidence": "igh} prove the required identity, and Lemma~\\ref{lem:local-euler} gives the holomorphy and Equation~\\eqref{eq:stage-all-height-majorant}. More generally, the function $\\mathfrak H$ in the definition always means the \\emph{full correction after the displayed scalar", "anchor": { "source": "paper.tex", "line_start": 5657, "line_end": 5657, "char_start": 258308, "char_end": 258344 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:external-arithmetic-majorant", "to": "lem:external-contour-tails", "type": "candidate_proof_dependency", "evidence": "ter and the remaining coordinates. Multiply the kernel bound by Equation~\\eqref{eq:external-arithmetic-majorant}. Integrating $(1+|\\boldsymbol t|)^{J-L}$ outside the box proves Equation~\\eqref{eq:external-integrated-tail} when $L>N+J+m$. On", "anchor": { "source": "paper.tex", "line_start": 5715, "line_end": 5715, "char_start": 260937, "char_end": 260976 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:external-integrated-tail", "to": "lem:external-contour-tails", "type": "candidate_proof_dependency", "evidence": "rnal-arithmetic-majorant}. Integrating $(1+|\\boldsymbol t|)^{J-L}$ outside the box proves Equation~\\eqref{eq:external-integrated-tail} when $L>N+J+m$. On $t_j=T$, integration in the other $m-1$ coordinates gives \\[ \\int_{\\mathbb R^{m-1}} (1+|T|+|\\widehat{\\boldsy", "anchor": { "source": "paper.tex", "line_start": 5717, "line_end": 5717, "char_start": 261051, "char_end": 261086 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:external-trace-tail", "to": "lem:external-contour-tails", "type": "candidate_proof_dependency", "evidence": "_j|)^{J-L} d\\widehat{\\boldsymbol t}_j \\ll(1+|T|)^{J-L+m-1}. \\] Taking $L>N+J+m-1$ proves Equation~\\eqref{eq:external-trace-tail}. Restricting to $\\mathcal D$ decreases these positive integrals, and a bounded real interval contributes only its length. \\end{pr", "anchor": { "source": "paper.tex", "line_start": 5725, "line_end": 5725, "char_start": 261323, "char_end": 261353 }, "origin": "source_reference", "context": "proof" }, { "from": "sec:poisson-representation", "to": "sec:shared-analytic-estimates", "type": "internal_reference", "evidence": "fixed real box with $r_z$ in a compact subinterval of $(0,\\infty)$, the Mellin estimates of Section~\\ref{sec:poisson-representation} give arbitrary polynomial decay of $M$ and $\\widehat W_1$. The Gaussian has the same property. Their product is therefore $O_L(", "anchor": { "source": "paper.tex", "line_start": 5738, "line_end": 5738, "char_start": 261817, "char_end": 261849 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:external-density-contract", "to": "sec:shared-analytic-estimates", "type": "internal_reference", "evidence": "bounds above. Uniform annular profiles separated by logarithmic Fourier inversion satisfy Equation~\\eqref{eq:external-density-contract} by repeated integration by parts, as in Lemma~\\ref{lem:smooth-calculus}. Translate every pure norm twist into its Mellin argumen", "anchor": { "source": "paper.tex", "line_start": 5772, "line_end": 5772, "char_start": 263448, "char_end": 263484 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:smooth-calculus", "to": "sec:shared-analytic-estimates", "type": "internal_reference", "evidence": "satisfy Equation~\\eqref{eq:external-density-contract} by repeated integration by parts, as in Lemma~\\ref{lem:smooth-calculus}. Translate every pure norm twist into its Mellin argument first. The additional height coordinates then append a block triangul", "anchor": { "source": "paper.tex", "line_start": 5773, "line_end": 5773, "char_start": 263531, "char_end": 263556 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:external-density-contract", "to": "sec:shared-analytic-estimates", "type": "internal_reference", "evidence": "y additional external coordinate on which a trace estimate is used must separately satisfy Equation~\\eqref{eq:external-density-contract}. The domain $\\mathcal D$ in the lemma is important. Suppose a reciprocal is bounded only while the imaginary part of its argume", "anchor": { "source": "paper.tex", "line_start": 5783, "line_end": 5783, "char_start": 264125, "char_end": 264161 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:external-arithmetic-majorant", "to": "sec:shared-analytic-estimates", "type": "internal_reference", "evidence": "l bound beyond it. Coordinates with $a_j=0$ may be extended when the other factors satisfy Equation~\\eqref{eq:external-arithmetic-majorant} there. All factors on an extended axis must use their global or absolute estimates, not an estimate valid only on a retained int", "anchor": { "source": "paper.tex", "line_start": 5791, "line_end": 5791, "char_start": 264583, "char_end": 264622 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:external-trace-tail", "to": "sec:shared-analytic-estimates", "type": "internal_reference", "evidence": "In particular, an integrated tail alone does not justify a horizontal join: that join uses Equation~\\eqref{eq:external-trace-tail}. Increasing $N$ in this lemma increases only the decay orders of the external smooth tests. It does not differentiate $F_Z$. T", "anchor": { "source": "paper.tex", "line_start": 5795, "line_end": 5795, "char_start": 264860, "char_end": 264890 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:smooth-calculus", "to": "sec:shared-analytic-estimates", "type": "internal_reference", "evidence": "t in $F_Z$ is unchanged when $N$ is chosen later. This is the same order distinction made in Lemma~\\ref{lem:smooth-calculus}. \\subsection{Moving a fixed bin} We first move the full row integral. The row envelope will be applied only after this step, o", "anchor": { "source": "paper.tex", "line_start": 5801, "line_end": 5801, "char_start": 265189, "char_end": 265214 }, "origin": "source_reference", "context": "narrative" }, { "from": "def:stage-high-data", "to": "lem:fixed-bin-contour", "type": "internal_reference", "evidence": "egin{lemma}[Contour transformation for a fixed bin] Assume Definition~\\ref{def:stage-high-data}. Fix $\\sigma_0\\in[7/8,1)$ with $\\beta_*>\\sigma_0$, put $z_0=17/50$, and let $U=Z^d$ with $0\\beta_*$. Its global bound follows from Lemmas~\\ref{lem:logarithmic-control} and~\\ref{lem:deleted-euler-factors}, using the conductor and deletion radical bounds in Section~\\ref{sec:detector}. The correcti", "anchor": { "source": "paper.tex", "line_start": 5845, "line_end": 5846, "char_start": 267231, "char_end": 267260 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:deleted-euler-factors", "to": "lem:fixed-bin-contour", "type": "candidate_proof_dependency", "evidence": "l stays in $\\Re s>\\beta_*$. Its global bound follows from Lemmas~\\ref{lem:logarithmic-control} and~\\ref{lem:deleted-euler-factors}, using the conductor and deletion radical bounds in Section~\\ref{sec:detector}. The correction obeys Equation~\\eqref{eq:stage-al", "anchor": { "source": "paper.tex", "line_start": 5846, "line_end": 5846, "char_start": 267265, "char_end": 267296 }, "origin": "source_reference", "context": "proof" }, { "from": "sec:detector", "to": "lem:fixed-bin-contour", "type": "candidate_proof_dependency", "evidence": "ol} and~\\ref{lem:deleted-euler-factors}, using the conductor and deletion radical bounds in Section~\\ref{sec:detector}. The correction obeys Equation~\\eqref{eq:stage-all-height-majorant}. Lemma~\\ref{lem:external-contour-tails} with $T\\to\\infty$ t", "anchor": { "source": "paper.tex", "line_start": 5847, "line_end": 5847, "char_start": 267357, "char_end": 267375 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:stage-all-height-majorant", "to": "lem:fixed-bin-contour", "type": "candidate_proof_dependency", "evidence": "conductor and deletion radical bounds in Section~\\ref{sec:detector}. The correction obeys Equation~\\eqref{eq:stage-all-height-majorant}. Lemma~\\ref{lem:external-contour-tails} with $T\\to\\infty$ therefore justifies these moves. Now move $w$ from $3$ to $1-a-6e$ wh", "anchor": { "source": "paper.tex", "line_start": 5848, "line_end": 5848, "char_start": 267408, "char_end": 267444 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:external-contour-tails", "to": "lem:fixed-bin-contour", "type": "candidate_proof_dependency", "evidence": "ion~\\ref{sec:detector}. The correction obeys Equation~\\eqref{eq:stage-all-height-majorant}. Lemma~\\ref{lem:external-contour-tails} with $T\\to\\infty$ therefore justifies these moves. Now move $w$ from $3$ to $1-a-6e$ while $s$ stays on that global line. Throug", "anchor": { "source": "paper.tex", "line_start": 5848, "line_end": 5849, "char_start": 267453, "char_end": 267485 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:stage-all-height-majorant", "to": "lem:fixed-bin-contour", "type": "candidate_proof_dependency", "evidence": "till global. The global upper strip bound for the numerator, the deleted-factor bound and Equation~\\eqref{eq:stage-all-height-majorant} give a majorant of the form Equation~\\eqref{eq:external-arithmetic-majorant} on the axes integrated there. The trace estimate ma", "anchor": { "source": "paper.tex", "line_start": 5862, "line_end": 5862, "char_start": 268078, "char_end": 268114 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:external-arithmetic-majorant", "to": "lem:fixed-bin-contour", "type": "candidate_proof_dependency", "evidence": "factor bound and Equation~\\eqref{eq:stage-all-height-majorant} give a majorant of the form Equation~\\eqref{eq:external-arithmetic-majorant} on the axes integrated there. The trace estimate makes the joins tend to zero. Before moving $s$ farther, restrict the three or", "anchor": { "source": "paper.tex", "line_start": 5863, "line_end": 5863, "char_start": 268152, "char_end": 268191 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:external-contour-tails", "to": "lem:fixed-bin-contour", "type": "candidate_proof_dependency", "evidence": "s bounded by $q_u\\ll Z^{d_{\\max}}$, and the number of rows is $O(U)$. The integrated tail in Lemma~\\ref{lem:external-contour-tails} gives $O_{\\eta,N}(Z^{B_\\eta}T_1^{-N})$. Move only the retained $s$ segment to $a+16e$. On this rectangle, \\[ \\Re(s+w)\\ge1+10e,", "anchor": { "source": "paper.tex", "line_start": 5872, "line_end": 5872, "char_start": 268712, "char_end": 268744 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:buffered-bins", "to": "lem:fixed-bin-contour", "type": "candidate_proof_dependency", "evidence": "laces both scalar arguments strictly inside $|\\Im v|\\le(3i+2)T_1$ for sufficiently large $Z$. Lemma~\\ref{lem:buffered-bins} therefore excludes its zeros throughout the retained rectangle. Since $6z_0>1$, the scalar zeta factor has no pole on this move.", "anchor": { "source": "paper.tex", "line_start": 5883, "line_end": 5883, "char_start": 269136, "char_end": 269159 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:stage-high-identity", "to": "lem:fixed-bin-contour", "type": "candidate_proof_dependency", "evidence": "imaginary coordinate is fixed at a constant multiple of $T_1$. The scalar denominator in Equation~\\eqref{eq:stage-high-identity} depends only on $s$, so its buffered estimate remains valid when the $\\Im z$ and $\\Im w$ integrations are extended to their whole", "anchor": { "source": "paper.tex", "line_start": 5888, "line_end": 5889, "char_start": 269437, "char_end": 269467 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:stage-all-height-majorant", "to": "lem:fixed-bin-contour", "type": "candidate_proof_dependency", "evidence": "extended to their whole axes. On those extended axes use the global numerator estimate and Equation~\\eqref{eq:stage-all-height-majorant}. The trace estimate then gives $O_{\\eta,N}(Z^{B_\\eta}T_1^{-N})$. No $w$- or $z$-join in this argument extends an $s$-axis that", "anchor": { "source": "paper.tex", "line_start": 5892, "line_end": 5892, "char_start": 269674, "char_end": 269710 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:local-euler", "to": "sec:shared-analytic-estimates", "type": "internal_reference", "evidence": "ntour identity just proved. \\subsection{Applying the row envelope} For the base correction, Lemma~\\ref{lem:local-euler} verifies the hypotheses of Lemma~\\ref{lem:fixed-bin-contour}. Indeed $|\\mathcal H_{\\eta,u}|\\ll_\\epsilon q_u^\\epsilon$ uniformly", "anchor": { "source": "paper.tex", "line_start": 5907, "line_end": 5907, "char_start": 270359, "char_end": 270380 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:fixed-bin-contour", "to": "sec:shared-analytic-estimates", "type": "internal_reference", "evidence": "ow envelope} For the base correction, Lemma~\\ref{lem:local-euler} verifies the hypotheses of Lemma~\\ref{lem:fixed-bin-contour}. Indeed $|\\mathcal H_{\\eta,u}|\\ll_\\epsilon q_u^\\epsilon$ uniformly in all three heights in both Euler regions; for $q_u\\le Z^D$,", "anchor": { "source": "paper.tex", "line_start": 5908, "line_end": 5908, "char_start": 270414, "char_end": 270441 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:stage-all-height-majorant", "to": "sec:shared-analytic-estimates", "type": "internal_reference", "evidence": "ly in all three heights in both Euler regions; for $q_u\\le Z^D$, taking $\\epsilon=1$ gives Equation~\\eqref{eq:stage-all-height-majorant} with $B=D$ and height order zero. We now estimate the retained integral, using $l_x=l_y=h=1/2$ and $\\ell=0$ throughout this subs", "anchor": { "source": "paper.tex", "line_start": 5911, "line_end": 5911, "char_start": 270607, "char_end": 270643 }, "origin": "source_reference", "context": "narrative" }, { "from": "prop:sextic-row-count", "to": "sec:shared-analytic-estimates", "type": "internal_reference", "evidence": "fix one dynamic bin $(i,a)$ before moving any Mellin variable, and put $\\delta=2a-1$. Proposition~\\ref{prop:sextic-row-count}, with requested loss $\\epsilon_r>0$, gives its cardinality at most $U^{R(\\delta)+\\epsilon_r}(1+T_1)^{A_\\eta}$ up to a fixed const", "anchor": { "source": "paper.tex", "line_start": 5921, "line_end": 5921, "char_start": 271015, "char_end": 271042 }, "origin": "source_reference", "context": "narrative" }, { "from": "sec:detector", "to": "sec:shared-analytic-estimates", "type": "internal_reference", "evidence": "^S(w,\\chi_\\bullet(u))$ is precisely the original zero-extended presentation with $\\nu=1$ in Section~\\ref{sec:detector}. On the retained line $\\Re w=1-a-6e$, Lemma~\\ref{lem:buffered-bins} bounds it by $U^{\\delta/2+12e+\\epsilon_n}(3+T_1)^C$ for any $", "anchor": { "source": "paper.tex", "line_start": 5928, "line_end": 5928, "char_start": 271385, "char_end": 271403 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:buffered-bins", "to": "sec:shared-analytic-estimates", "type": "internal_reference", "evidence": "presentation with $\\nu=1$ in Section~\\ref{sec:detector}. On the retained line $\\Re w=1-a-6e$, Lemma~\\ref{lem:buffered-bins} bounds it by $U^{\\delta/2+12e+\\epsilon_n}(3+T_1)^C$ for any $\\epsilon_n>0$. This also holds in the floor bin. The central point", "anchor": { "source": "paper.tex", "line_start": 5929, "line_end": 5929, "char_start": 271448, "char_end": 271471 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:local-euler", "to": "sec:shared-analytic-estimates", "type": "internal_reference", "evidence": "floor bin. The central point lies in $\\mathcal D_1(10e)$, so the stronger local estimate in Lemma~\\ref{lem:local-euler} gives $|\\mathcal H_{\\eta,u}|\\ll_{e,\\epsilon_H}U^{\\epsilon_H}$ there, for any $\\epsilon_H>0$. Consequently \\[ \\sum_{u\\in\\mathcal", "anchor": { "source": "paper.tex", "line_start": 5934, "line_end": 5934, "char_start": 271670, "char_end": 271691 }, "origin": "source_reference", "context": "narrative" }, { "from": "sec:detector", "to": "sec:shared-analytic-estimates", "type": "internal_reference", "evidence": "ses} \\] The first two pieces are at most $-1/12$. In the third, the exact bin ceiling from Section~\\ref{sec:detector} is \\[ \\delta\\le2\\beta_*-1=\\frac56+2\\Delta_1. \\] It follows in every piece that \\[ E_{\\mathrm I}(h)-\\Delta_1 \\le-\\frac1{24}-\\fr", "anchor": { "source": "paper.tex", "line_start": 5985, "line_end": 5985, "char_start": 273579, "char_end": 273597 }, "origin": "source_reference", "context": "narrative" }, { "from": "def:stage-high-data", "to": "lem:bin-contour-accounting", "type": "internal_reference", "evidence": "etained integral and its exponent] Assume the data of Definition~\\ref{def:stage-high-data}. Fix $\\sigma_0\\in[7/8,1)$ with $\\beta_*>\\sigma_0$, and put $z_0=17/50$. Let $0\\sigma_0$ and $0\\sigma_0$ and $00$, uniformly in all imagi", "anchor": { "source": "paper.tex", "line_start": 6177, "line_end": 6177, "char_start": 282651, "char_end": 282691 }, "origin": "source_reference", "context": "statement" }, { "from": "eq:stage-high-identity", "to": "lem:principal-residue-interface", "type": "internal_reference", "evidence": "(s)|\\ll_\\eta Z^{-\\mu}$ on that entire line. Let $\\mathscr P_\\eta(Z)$ be the $u=1$ term of Equation~\\eqref{eq:stage-high-identity}, and define \\[ f_\\eta(Z)=\\frac1{2\\pi i}\\int_{(2)} Z^{C(s)}e^{(s-5/6)^2}\\frac{H_\\eta(s)}{L_F^S(s,\\eta)}\\,ds. \\] Then $c_S>0$,", "anchor": { "source": "paper.tex", "line_start": 6205, "line_end": 6205, "char_start": 283816, "char_end": 283846 }, "origin": "source_reference", "context": "statement" }, { "from": "lem:external-contour-tails", "to": "lem:principal-residue-interface", "type": "candidate_proof_dependency", "evidence": "e paths lie in $\\mathcal D_2$, so the correction is holomorphic. The all-height majorant and Lemma~\\ref{lem:external-contour-tails} justify the horizontal limits. Move $w$ to $19/20$, crossing its simple pole at $w=1$. In that residue move $z$ to $33/200=1/6-", "anchor": { "source": "paper.tex", "line_start": 6234, "line_end": 6234, "char_start": 284878, "char_end": 284910 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:external-contour-tails", "to": "lem:principal-residue-interface", "type": "candidate_proof_dependency", "evidence": "a functions have fixed polynomial bounds. Consequently the integrated and trace estimates of Lemma~\\ref{lem:external-contour-tails} apply without an unremoved pole in their majorants. After the $w$ residue, its Mellin factor has become the constant $\\widehat W_", "anchor": { "source": "paper.tex", "line_start": 6246, "line_end": 6246, "char_start": 285686, "char_end": 285718 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:external-two-dimensional-kernel", "to": "lem:principal-residue-interface", "type": "candidate_proof_dependency", "evidence": "e the constant $\\widehat W_1(1)$. The remaining heights use the two-dimensional kernel in Equation~\\eqref{eq:external-two-dimensional-kernel}, with $(y_1,y_2)=(\\Im(s+z),\\Im z)$; its trace estimate justifies the subsequent $z$-join. On the principal contours, Equation~\\e", "anchor": { "source": "paper.tex", "line_start": 6250, "line_end": 6250, "char_start": 285921, "char_end": 285963 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:principal-correction-bound", "to": "lem:principal-residue-interface", "type": "candidate_proof_dependency", "evidence": "\\Im z)$; its trace estimate justifies the subsequent $z$-join. On the principal contours, Equation~\\eqref{eq:principal-correction-bound} and the outside powers give the raw exponent \\[ \\frac{l_x}{2}+\\Re s-1+h\\Re z+l_y(\\Re w-1). \\] For the unresidued $w$ integral th", "anchor": { "source": "paper.tex", "line_start": 6254, "line_end": 6255, "char_start": 286091, "char_end": 286128 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:logarithmic-control", "to": "lem:principal-residue-interface", "type": "candidate_proof_dependency", "evidence": "an arbitrarily small power of its fixed target conductor and a fixed polynomial in height by Lemmas~\\ref{lem:logarithmic-control} and~\\ref{lem:deleted-euler-factors}. The tests integrate those height powers. Requesting the small powers in the correction and", "anchor": { "source": "paper.tex", "line_start": 6264, "line_end": 6264, "char_start": 286510, "char_end": 286539 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:deleted-euler-factors", "to": "lem:principal-residue-interface", "type": "candidate_proof_dependency", "evidence": "fixed target conductor and a fixed polynomial in height by Lemmas~\\ref{lem:logarithmic-control} and~\\ref{lem:deleted-euler-factors}. The tests integrate those height powers. Requesting the small powers in the correction and in $A_\\eta^{-1}$ to be sufficiently", "anchor": { "source": "paper.tex", "line_start": 6264, "line_end": 6264, "char_start": 286544, "char_end": 286575 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:principal-remainder-bound", "to": "lem:principal-residue-interface", "type": "candidate_proof_dependency", "evidence": "the correction and in $A_\\eta^{-1}$ to be sufficiently small gives the first two terms of Equation~\\eqref{eq:principal-remainder-bound}. The product of the scalar residues is $c_S$: the residue of $\\zeta_F^S(6z)$ at $1/6$ is one sixth of the residue of $\\zeta_F^S(", "anchor": { "source": "paper.tex", "line_start": 6267, "line_end": 6267, "char_start": 286749, "char_end": 286785 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:principal-correction-residue", "to": "lem:principal-residue-interface", "type": "candidate_proof_dependency", "evidence": "power is \\[ X^{1/3}Z^{s-5/6}Z^{\\ell/6}=Z^{C(s)}, \\qquad \\Phi(s+1/6-1)=e^{(s-5/6)^2}. \\] Equation~\\eqref{eq:principal-correction-residue} therefore makes the normalized double residue \\[ \\frac1{2\\pi i}\\int_{(\\beta_*+e)} Z^{C(s)}e^{(s-5/6)^2}\\frac{H_\\eta(s)}{L_F^S(s", "anchor": { "source": "paper.tex", "line_start": 6278, "line_end": 6278, "char_start": 287129, "char_end": 287168 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:local-euler", "to": "lem:principal-residue-interface", "type": "candidate_proof_dependency", "evidence": "ocal has a zero, so that case adds no residue. The function $H_\\eta$ is holomorphic there by Lemma~\\ref{lem:local-euler}, and is bounded there by Equation~\\eqref{eq:shared-principal-nonvanishing}. The global reciprocal estimate is polynomial in heig", "anchor": { "source": "paper.tex", "line_start": 6294, "line_end": 6294, "char_start": 287864, "char_end": 287885 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:shared-principal-nonvanishing", "to": "lem:principal-residue-interface", "type": "candidate_proof_dependency", "evidence": "tion $H_\\eta$ is holomorphic there by Lemma~\\ref{lem:local-euler}, and is bounded there by Equation~\\eqref{eq:shared-principal-nonvanishing}. The global reciprocal estimate is polynomial in height uniformly on the fixed real strip, while the Gaussian is $O(e^{-(\\Im s)^", "anchor": { "source": "paper.tex", "line_start": 6295, "line_end": 6295, "char_start": 287920, "char_end": 287960 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:principal-correction-residue", "to": "sec:shared-analytic-estimates", "type": "internal_reference", "evidence": "uct at the double residue. A different full correction is permitted only when it verifies Equation~\\eqref{eq:principal-correction-residue} with a nonzero normalizer. For the base correction that equation holds with $A_\\eta=1$ and $\\mathcal R_{\\eta,Z}=0$. The normaliz", "anchor": { "source": "paper.tex", "line_start": 6306, "line_end": 6306, "char_start": 288489, "char_end": 288528 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:principal-residue-interface", "to": "sec:shared-analytic-estimates", "type": "internal_reference", "evidence": "raph{The base correction.} Take $l_x=l_y=h=1/2$ and $\\ell=0$. On the principal rectangle of Lemma~\\ref{lem:principal-residue-interface}, the local correction is bounded uniformly in every height because that rectangle lies in $\\mathcal D_2$. Thus Equation~\\eqref{e", "anchor": { "source": "paper.tex", "line_start": 6316, "line_end": 6316, "char_start": 289077, "char_end": 289114 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:principal-correction-bound", "to": "sec:shared-analytic-estimates", "type": "internal_reference", "evidence": "is bounded uniformly in every height because that rectangle lies in $\\mathcal D_2$. Thus Equation~\\eqref{eq:principal-correction-bound} holds with $\\ell=0$ and height order zero. At the double residue it satisfies exactly \\[ \\mathfrak H_{\\eta,1,Z}(s,1,1/6)=H_\\eta", "anchor": { "source": "paper.tex", "line_start": 6319, "line_end": 6319, "char_start": 289236, "char_end": 289273 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:principal-residue-interface", "to": "sec:shared-analytic-estimates", "type": "internal_reference", "evidence": "i(u)}$ is one; the sole factor $1/6$ is already in $c_S$ from the residue of $\\zeta_F^S(6z)$. Lemma~\\ref{lem:principal-residue-interface} therefore has just its two remainder terms. Their savings before losses are \\[ m_w=\\frac{l_y}{20}=\\frac1{40},\\qquad m_z=\\frac", "anchor": { "source": "paper.tex", "line_start": 6328, "line_end": 6328, "char_start": 289674, "char_end": 289711 }, "origin": "source_reference", "context": "narrative" }, { "from": "def:stage-high-data", "to": "lem:outer-row-tails", "type": "internal_reference", "evidence": "entral decomposition. [Outer row norms] Assume Definition~\\ref{def:stage-high-data}. Fix $\\sigma_0\\in[7/8,1)$ with $\\beta_*>\\sigma_0$, $00$. For a dyadic number $U\\ge1$, l", "anchor": { "source": "paper.tex", "line_start": 6344, "line_end": 6344, "char_start": 290236, "char_end": 290261 }, "origin": "source_reference", "context": "statement" }, { "from": "eq:stage-high-identity", "to": "lem:outer-row-tails", "type": "internal_reference", "evidence": "n}>0$. For a dyadic number $U\\ge1$, let $\\mathscr R_\\eta(U;Z)$ denote the contribution in Equation~\\eqref{eq:stage-high-identity} of physical rows $U\\le q_u<2U$, omitting $u=1$. Suppose that for every $\\epsilon>0$, for all such rows with $q_u\\le2Z^{d_{\\min}}", "anchor": { "source": "paper.tex", "line_start": 6348, "line_end": 6348, "char_start": 290453, "char_end": 290483 }, "origin": "source_reference", "context": "statement" }, { "from": "lem:principal-residue-interface", "to": "lem:outer-row-tails", "type": "candidate_proof_dependency", "evidence": "oof} Every row under consideration has nonprincipal numerator by the classification preceding Lemma~\\ref{lem:principal-residue-interface}. The denominator may be principal for a bounded unit row; its reciprocal is nevertheless holomorphic and bounded on $\\Re s=\\beta_", "anchor": { "source": "paper.tex", "line_start": 6405, "line_end": 6405, "char_start": 292332, "char_end": 292369 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:logarithmic-control", "to": "lem:outer-row-tails", "type": "candidate_proof_dependency", "evidence": "nevertheless holomorphic and bounded on $\\Re s=\\beta_*+e$ by the principal specialization of Lemma~\\ref{lem:logarithmic-control}. For all rows, the conductor and deletion radical bounds in Section~\\ref{sec:detector}, together with Lemma~\\ref{lem:deleted-eule", "anchor": { "source": "paper.tex", "line_start": 6408, "line_end": 6408, "char_start": 292545, "char_end": 292574 }, "origin": "source_reference", "context": "proof" }, { "from": "sec:detector", "to": "lem:outer-row-tails", "type": "candidate_proof_dependency", "evidence": "a~\\ref{lem:logarithmic-control}. For all rows, the conductor and deletion radical bounds in Section~\\ref{sec:detector}, together with Lemma~\\ref{lem:deleted-euler-factors}, give an arbitrarily small power of $q_u$ and a fixed height polynomial for", "anchor": { "source": "paper.tex", "line_start": 6410, "line_end": 6410, "char_start": 292643, "char_end": 292661 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:deleted-euler-factors", "to": "lem:outer-row-tails", "type": "candidate_proof_dependency", "evidence": "rows, the conductor and deletion radical bounds in Section~\\ref{sec:detector}, together with Lemma~\\ref{lem:deleted-euler-factors}, give an arbitrarily small power of $q_u$ and a fixed height polynomial for that reciprocal. For a small dyad move its finite su", "anchor": { "source": "paper.tex", "line_start": 6411, "line_end": 6411, "char_start": 292683, "char_end": 292714 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:external-contour-tails", "to": "lem:outer-row-tails", "type": "candidate_proof_dependency", "evidence": "*+e+1/2>1+3/8$, and the other inequalities are immediate. The all-height correction bound and Lemma~\\ref{lem:external-contour-tails} justify the moves. On the final lines, Equation~\\eqref{eq:hecke-growth} and the deletion bound give $U^{3/5+\\epsilon}$ times a f", "anchor": { "source": "paper.tex", "line_start": 6421, "line_end": 6421, "char_start": 293188, "char_end": 293220 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:hecke-growth", "to": "lem:outer-row-tails", "type": "candidate_proof_dependency", "evidence": "n bound and Lemma~\\ref{lem:external-contour-tails} justify the moves. On the final lines, Equation~\\eqref{eq:hecke-growth} and the deletion bound give $U^{3/5+\\epsilon}$ times a fixed height polynomial for the nonprincipal numerator. There are $O(U)$", "anchor": { "source": "paper.tex", "line_start": 6422, "line_end": 6422, "char_start": 293270, "char_end": 293293 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:small-correction-bound", "to": "lem:outer-row-tails", "type": "candidate_proof_dependency", "evidence": "dyad, including unit factors, and $q_u^{-z_0}\\ll U^{-z_0}$. The correction is bounded by Equation~\\eqref{eq:small-correction-bound}. The tests integrate all fixed height powers. The resulting exponent outside the row power, relative to $C(\\beta_*)$, is \\[ h(", "anchor": { "source": "paper.tex", "line_start": 6426, "line_end": 6426, "char_start": 293542, "char_end": 293575 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:small-row-bound", "to": "lem:outer-row-tails", "type": "candidate_proof_dependency", "evidence": "0=63/50$. Requesting the component small powers to sum to the displayed $\\epsilon$ proves Equation~\\eqref{eq:small-row-bound}. The row exponent is positive, so dyadic summation up to $Z^{d_{\\min}}$ gives Equation~\\eqref{eq:small-row-sum}, after decreasin", "anchor": { "source": "paper.tex", "line_start": 6434, "line_end": 6434, "char_start": 293856, "char_end": 293882 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:small-row-sum", "to": "lem:outer-row-tails", "type": "candidate_proof_dependency", "evidence": "-row-bound}. The row exponent is positive, so dyadic summation up to $Z^{d_{\\min}}$ gives Equation~\\eqref{eq:small-row-sum}, after decreasing the preliminary losses. For each fixed $Z$ and large dyad, move its finite row sum to the absolute lines $(2,2", "anchor": { "source": "paper.tex", "line_start": 6436, "line_end": 6436, "char_start": 293971, "char_end": 293995 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:stage-all-height-majorant", "to": "lem:outer-row-tails", "type": "candidate_proof_dependency", "evidence": "c there. For this fixed dyad choose $D$ large enough to contain its rows in $q_u\\le Z^D$; Equation~\\eqref{eq:stage-all-height-majorant} and the external trace bound make its horizontal limits vanish. Constants used only to justify this equality may depend on the f", "anchor": { "source": "paper.tex", "line_start": 6442, "line_end": 6442, "char_start": 294338, "char_end": 294374 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:large-correction-bound", "to": "lem:outer-row-tails", "type": "candidate_proof_dependency", "evidence": "y may depend on the fixed dyad. The estimate on the final lines is uniform in the dyad by Equation~\\eqref{eq:large-correction-bound}. The scalar $L$-factors are absolutely bounded there, as is $\\zeta_F^S(6z)$, and Equation~\\eqref{eq:large-correction-bound} appli", "anchor": { "source": "paper.tex", "line_start": 6445, "line_end": 6445, "char_start": 294583, "char_end": 294616 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:large-correction-bound", "to": "lem:outer-row-tails", "type": "candidate_proof_dependency", "evidence": "on-bound}. The scalar $L$-factors are absolutely bounded there, as is $\\zeta_F^S(6z)$, and Equation~\\eqref{eq:large-correction-bound} applies. The outside power, including the correction but not the row count, is \\[ l_x(1/2-v)+1+v+l_y+\\ell v=B_0+hv. \\] The $O(U", "anchor": { "source": "paper.tex", "line_start": 6448, "line_end": 6448, "char_start": 294707, "char_end": 294740 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:large-row-bound", "to": "lem:outer-row-tails", "type": "candidate_proof_dependency", "evidence": "d $q_u^{-v}$ give $U^{1-v}$; the tests integrate the fixed height polynomial. This proves Equation~\\eqref{eq:large-row-bound}. Because $1+\\epsilon-v<0$, summing its geometric dyadic tail gives Equation~\\eqref{eq:large-row-sum}. For fixed $\\zeta>0$, the c", "anchor": { "source": "paper.tex", "line_start": 6454, "line_end": 6454, "char_start": 294980, "char_end": 295006 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:large-row-sum", "to": "lem:outer-row-tails", "type": "candidate_proof_dependency", "evidence": "ref{eq:large-row-bound}. Because $1+\\epsilon-v<0$, summing its geometric dyadic tail gives Equation~\\eqref{eq:large-row-sum}. For fixed $\\zeta>0$, the coefficient of $v$ in that exponent is $-\\zeta$, proving the last assertion. The original row series i", "anchor": { "source": "paper.tex", "line_start": 6456, "line_end": 6456, "char_start": 295083, "char_end": 295107 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:small-correction-bound", "to": "sec:shared-analytic-estimates", "type": "internal_reference", "evidence": "cal bound $\\mathcal H_{\\eta,u}\\ll_\\epsilon q_u^\\epsilon$, uniform in all heights, verifies Equation~\\eqref{eq:small-correction-bound} for all the prescribed rows, with height order zero. Equation~\\eqref{eq:small-row-sum} then has the following relative exponent", "anchor": { "source": "paper.tex", "line_start": 6471, "line_end": 6471, "char_start": 295873, "char_end": 295906 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:small-row-sum", "to": "sec:shared-analytic-estimates", "type": "internal_reference", "evidence": "on~\\eqref{eq:small-correction-bound} for all the prescribed rows, with height order zero. Equation~\\eqref{eq:small-row-sum} then has the following relative exponent before its losses: \\[ h(z_0-1/6)-\\frac{l_y}{2}+\\frac{63}{50}d_{\\min} =\\frac{13}{150}-\\", "anchor": { "source": "paper.tex", "line_start": 6472, "line_end": 6472, "char_start": 295970, "char_end": 295994 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:large-correction-bound", "to": "sec:shared-analytic-estimates", "type": "internal_reference", "evidence": "ws, $(2,2,v)$ with $v>2$ lies in $\\mathcal D_2$. The same all-height local bound verifies Equation~\\eqref{eq:large-correction-bound} for every physical row. Here $B_0=l_x/2+1+l_y=7/4$, so Equation~\\eqref{eq:large-row-sum} has exponent \\[ \\frac74+\\frac{501}{100", "anchor": { "source": "paper.tex", "line_start": 6480, "line_end": 6481, "char_start": 296280, "char_end": 296313 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:large-row-sum", "to": "sec:shared-analytic-estimates", "type": "internal_reference", "evidence": "~\\eqref{eq:large-correction-bound} for every physical row. Here $B_0=l_x/2+1+l_y=7/4$, so Equation~\\eqref{eq:large-row-sum} has exponent \\[ \\frac74+\\frac{501}{1000}(1+\\epsilon)-\\frac v{1000}+\\epsilon. \\] It tends to $-\\infty$ as the fixed number $v$ in", "anchor": { "source": "paper.tex", "line_start": 6482, "line_end": 6482, "char_start": 296379, "char_end": 296403 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:detector-dyads", "to": "lem:late-height-closure", "type": "internal_reference", "evidence": "additional ceiling may encode a height condition in a preceding estimate, such as the one in Lemma~\\ref{lem:detector-dyads}. The lower threshold may depend on $\\eta,N,\\tau$. Increasing $N$ is assumed to change only external test seminorms, not $A_\\eta$", "anchor": { "source": "paper.tex", "line_start": 6546, "line_end": 6546, "char_start": 299176, "char_end": 299200 }, "origin": "source_reference", "context": "statement" }, { "from": "eq:late-height-hypothesis", "to": "lem:late-height-closure", "type": "internal_reference", "evidence": "d to change only external test seminorms, not $A_\\eta$ or the already fixed real powers in Equation~\\eqref{eq:late-height-hypothesis}. Then $\\tau$ and $N$ can be chosen after $\\eta$ so that \\[ |J_\\eta(Z)-f_\\eta(Z)|\\ll_\\eta Z^{C(\\beta_*)-m/2}. \\] In particular t", "anchor": { "source": "paper.tex", "line_start": 6549, "line_end": 6549, "char_start": 299376, "char_end": 299409 }, "origin": "source_reference", "context": "statement" }, { "from": "lem:continuation-criterion", "to": "lem:late-height-closure", "type": "internal_reference", "evidence": "a=m/2$, together with the displayed low estimate, has the target quantifier required by Proposition~\\ref{lem:continuation-criterion}. For the fixed target choose \\[ 0<\\tau_\\eta\\le \\min\\left\\{\\frac{d_{\\min}}{100},\\tau_{0,\\eta},", "anchor": { "source": "paper.tex", "line_start": 6557, "line_end": 6557, "char_start": 299662, "char_end": 299694 }, "origin": "source_reference", "context": "statement" }, { "from": "eq:late-height-hypothesis", "to": "lem:late-height-closure", "type": "candidate_proof_dependency", "evidence": "\\qquad T_1=Z^{\\tau_\\eta}. \\] Then $(1+T_1)^{A_\\eta}\\ll_\\eta Z^{m/4}$, so the first term in Equation~\\eqref{eq:late-height-hypothesis} is $O_\\eta(Z^{C(\\beta_*)-3m/4})$. Next choose a fixed integer $N$ such that \\[ B_\\eta-N\\tau_\\eta0. \\] For $0<\\tau\\le\\tau_{0,\\eta}$", "anchor": { "source": "paper.tex", "line_start": 6651, "line_end": 6651, "char_start": 303967, "char_end": 303991 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:fixed-bin-contour", "to": "prop:balanced-high", "type": "candidate_proof_dependency", "evidence": "the lower threshold, which may depend on $\\eta$ and $\\tau$. For every final integer $N\\ge0$, Lemma~\\ref{lem:fixed-bin-contour} bounds the discarded high portions and joins by $O_{\\eta,N}(Z^{B_\\eta}T_1^{-N})$, with $B_\\eta$ fixed before $N$. The bounded rea", "anchor": { "source": "paper.tex", "line_start": 6674, "line_end": 6675, "char_start": 304935, "char_end": 304962 }, "origin": "source_reference", "context": "proof" }, { "from": "prop:balanced-low", "to": "prop:balanced-high", "type": "candidate_proof_dependency", "evidence": "au_{0,\\eta}\\}$ and $Z$ is sufficiently large. Both functions are independent of $T_1$. Proposition~\\ref{prop:balanced-low}, applied with loss $\\omega$ and divided by the same fixed $c_S$, supplies \\[ |J_{\\mathrm I,\\eta}(Z)|\\ll_\\eta Z^{1/4+\\omega}=Z^{", "anchor": { "source": "paper.tex", "line_start": 6689, "line_end": 6689, "char_start": 305635, "char_end": 305658 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:late-height-closure", "to": "prop:balanced-high", "type": "candidate_proof_dependency", "evidence": "{1/4+\\omega}=Z^{C_{\\mathrm I}(11/12)+\\omega}, \\qquad 0<\\omega<\\Delta_1. \\] All hypotheses of Lemma~\\ref{lem:late-height-closure} are now verified. It gives the saving $m/2=1/4800$ asserted in the proposition. \\subsection{Transfer to Dirichlet \\t", "anchor": { "source": "paper.tex", "line_start": 6696, "line_end": 6696, "char_start": 305872, "char_end": 305901 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:local-euler", "to": "thm:eleven-twelfths", "type": "candidate_proof_dependency", "evidence": "of} [Proof of Theorem~\\ref{thm:eleven-twelfths}] Suppose that $\\beta_*>11/12$. Lemma~\\ref{lem:local-euler} and Equation~\\eqref{eq:shared-principal-nonvanishing} give the required holomorphic, nonzero $H_\\eta$ on $\\Re s>11/12$ for every", "anchor": { "source": "paper.tex", "line_start": 6784, "line_end": 6784, "char_start": 309865, "char_end": 309886 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:shared-principal-nonvanishing", "to": "thm:eleven-twelfths", "type": "candidate_proof_dependency", "evidence": "~\\ref{thm:eleven-twelfths}] Suppose that $\\beta_*>11/12$. Lemma~\\ref{lem:local-euler} and Equation~\\eqref{eq:shared-principal-nonvanishing} give the required holomorphic, nonzero $H_\\eta$ on $\\Re s>11/12$ for every primitive target. The low and high bounds established", "anchor": { "source": "paper.tex", "line_start": 6785, "line_end": 6785, "char_start": 309900, "char_end": 309940 }, "origin": "source_reference", "context": "proof" }, { "from": "prop:balanced-high", "to": "thm:eleven-twelfths", "type": "candidate_proof_dependency", "evidence": "a$ on $\\Re s>11/12$ for every primitive target. The low and high bounds established in Proposition~\\ref{prop:balanced-high} and its proof verify Proposition~\\ref{lem:continuation-criterion} with \\[ \\sigma_0=\\frac{11}{12},\\qquad C=C_{\\mathrm I},\\qquad", "anchor": { "source": "paper.tex", "line_start": 6788, "line_end": 6788, "char_start": 310086, "char_end": 310110 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:continuation-criterion", "to": "thm:eleven-twelfths", "type": "candidate_proof_dependency", "evidence": "nd high bounds established in Proposition~\\ref{prop:balanced-high} and its proof verify Proposition~\\ref{lem:continuation-criterion} with \\[ \\sigma_0=\\frac{11}{12},\\qquad C=C_{\\mathrm I},\\qquad \\omega=\\frac{\\Delta_1}{2},\\qquad \\sigma=\\frac1{4800}. \\] These two", "anchor": { "source": "paper.tex", "line_start": 6789, "line_end": 6789, "char_start": 310144, "char_end": 310176 }, "origin": "source_reference", "context": "proof" }, { "from": "prop:hecke-dirichlet-transfer", "to": "thm:eleven-twelfths", "type": "candidate_proof_dependency", "evidence": "is zero-free in the strict half-plane $\\Re s>11/12$, with the principal pole allowed. Proposition~\\ref{prop:hecke-dirichlet-transfer}, applied at the same boundary, extends this assertion to all finite-order Hecke characters and all Dirichlet $L$-functions, inclu", "anchor": { "source": "paper.tex", "line_start": 6801, "line_end": 6801, "char_start": 310737, "char_end": 310772 }, "origin": "source_reference", "context": "proof" }, { "from": "thm:eleven-twelfths", "to": "sec:compensation", "type": "internal_reference", "evidence": "plane} \\section{The compensated probe} Theorem~\\ref{thm:eleven-twelfths} applies to every primitive character entering the supremum in Equation~\\eqref{old-eq:1.1b}, and therefore gives $\\beta_*\\le11/12$", "anchor": { "source": "paper.tex", "line_start": 6812, "line_end": 6812, "char_start": 311165, "char_end": 311190 }, "origin": "source_reference", "context": "narrative" }, { "from": "old-eq:1.1b", "to": "sec:compensation", "type": "internal_reference", "evidence": "em~\\ref{thm:eleven-twelfths} applies to every primitive character entering the supremum in Equation~\\eqref{old-eq:1.1b}, and therefore gives $\\beta_*\\le11/12$. Suppose for contradiction throughout Part~II that $\\beta_*>7/8$, and put \\begin{equation", "anchor": { "source": "paper.tex", "line_start": 6813, "line_end": 6813, "char_start": 311262, "char_end": 311281 }, "origin": "source_reference", "context": "narrative" }, { "from": "sec:detector", "to": "sec:compensation", "type": "internal_reference", "evidence": "quad \\kappa:=2\\beta_*-1=\\frac34+2\\Delta\\le\\frac56. The exact bin ceiling in Section~\\ref{sec:detector} applies, since $\\beta_*>7/8>51/100$. For every retained nonprincipal row and its bin $(i,a)$ it gives \\label{eq:", "anchor": { "source": "paper.tex", "line_start": 6821, "line_end": 6821, "char_start": 311599, "char_end": 311617 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:continuation-criterion", "to": "sec:compensation", "type": "internal_reference", "evidence": "boundary case $\\delta=5/6$ remains part of the argument. For the second application of Proposition~\\ref{lem:continuation-criterion}, write throughout Part~II C(s)=C_{\\mathrm{II}}(s):=s-\\frac{11}{16},\\qquad C(7/", "anchor": { "source": "paper.tex", "line_start": 6832, "line_end": 6832, "char_start": 312135, "char_end": 312167 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:high-probe-contract", "to": "sec:compensation", "type": "internal_reference", "evidence": "\\omega}$ for a common $0<\\omega<\\Delta$; its high side must satisfy the signal estimate in Equation~\\eqref{eq:high-probe-contract} with this $C$. The nonzero scalar normalization will be specified after the principal term is evaluated. For each primitive tar", "anchor": { "source": "paper.tex", "line_start": 6843, "line_end": 6843, "char_start": 312657, "char_end": 312687 }, "origin": "source_reference", "context": "narrative" }, { "from": "sec:probe", "to": "sec:compensation", "type": "internal_reference", "evidence": "et $\\eta$, use an admissible fixed instance of the data $T,S,b_*,\\xi,\\tau,\\Xi,W_0,W_1$ from Section~\\ref{sec:probe}. The permitted choice of $P_0$ will be made in the parameter order below. These data may differ from those used in Part~I, but", "anchor": { "source": "paper.tex", "line_start": 6847, "line_end": 6847, "char_start": 312918, "char_end": 312933 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:general-probe", "to": "sec:compensation", "type": "internal_reference", "evidence": "ular support. For a squarefree product $D$ of slot primes, let $I_{\\eta;D}(X,Y,Z)$ denote Equation~\\eqref{eq:general-probe} with the indicator $1_{D\\mid cn^3}$ inserted in its completed row. Thus $I_{\\eta;1}=I_\\eta$. For a tuple $(p_1,\\ldots,p_K)\\in\\pr", "anchor": { "source": "paper.tex", "line_start": 6886, "line_end": 6886, "char_start": 314577, "char_end": 314601 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:compensated-probe-definition", "to": "sec:compensation", "type": "internal_reference", "evidence": "e{\\eta(p)}I_{\\eta;p}(X,Y,Zq_p)$ minus the rescaled term $q_p^{-3/2}I_\\eta(X/q_p,Y/q_p,Z)$. Formula \\eqref{eq:compensated-probe-definition} specifies their composition: every slot window stays at its original scale $P_i$, including when another slot changes the complet", "anchor": { "source": "paper.tex", "line_start": 6907, "line_end": 6907, "char_start": 315577, "char_end": 315616 }, "origin": "source_reference", "context": "narrative" }, { "from": "sec:local-compensation", "to": "sec:compensation", "type": "internal_reference", "evidence": "n on the high side, leaving the sextic-character prime factor used by the moment estimates. Section~\\ref{sec:local-compensation} proves the exact identity and bounds the remaining local errors. \\section{Coefficient conventions and finite correlations} \\labe", "anchor": { "source": "paper.tex", "line_start": 6915, "line_end": 6915, "char_start": 316127, "char_end": 316155 }, "origin": "source_reference", "context": "narrative" }, { "from": "sec:arithmetic", "to": "sec:additional-arithmetic", "type": "internal_reference", "evidence": "ain and inverse polynomials, annular profiles, and fixed arithmetic datum $\\mathcal A$ from Section~\\ref{sec:arithmetic}. The following conditions specify the extra prime factors and moving zero supports used in this part. A prime slot of log-lengt", "anchor": { "source": "paper.tex", "line_start": 6930, "line_end": 6930, "char_start": 316829, "char_end": 316849 }, "origin": "source_reference", "context": "narrative" }, { "from": "sec:plain", "to": "sec:additional-arithmetic", "type": "internal_reference", "evidence": "supports before any common mask or row zero extension is imposed. For the fourth moment of Section~\\ref{sec:plain}, a fixed finite group $\\Theta$ of ray characters is part of the data, and every character component with nonzero coefficient in e", "anchor": { "source": "paper.tex", "line_start": 6943, "line_end": 6943, "char_start": 317439, "char_end": 317454 }, "origin": "source_reference", "context": "narrative" }, { "from": "sec:inverse", "to": "sec:additional-arithmetic", "type": "internal_reference", "evidence": "ro coefficient in every positive-length slot must belong to $\\Theta$. The inverse moment of Section~\\ref{sec:inverse} also allows a bounded coefficient $a_i(p)$ chosen independently for each slot and independently of the row and all other columns;", "anchor": { "source": "paper.tex", "line_start": 6946, "line_end": 6946, "char_start": 317665, "char_end": 317682 }, "origin": "source_reference", "context": "narrative" }, { "from": "sec:plain", "to": "sec:additional-arithmetic", "type": "internal_reference", "evidence": "isplayed moving factor, and the common puncture is deleted before the natural reflection in Section~\\ref{sec:plain}. No zero prime may be omitted from both supports. If a moving character still ramified at that prime remains, the prime stays in", "anchor": { "source": "paper.tex", "line_start": 6975, "line_end": 6975, "char_start": 319386, "char_end": 319401 }, "origin": "source_reference", "context": "narrative" }, { "from": "sec:plain", "to": "sec:additional-arithmetic", "type": "internal_reference", "evidence": "part of this class. This full-support convention governs every fourth-moment invocation in Section~\\ref{sec:plain}. The common uniformity convention has the following additional clauses for these coefficient classes. Any required slot mesh de", "anchor": { "source": "paper.tex", "line_start": 6988, "line_end": 6988, "char_start": 320397, "char_end": 320412 }, "origin": "source_reference", "context": "narrative" }, { "from": "sec:arithmetic", "to": "sec:additional-arithmetic", "type": "internal_reference", "evidence": "labels, and any separate requirement that the multiplicity depend only on $f$, remain as in Section~\\ref{sec:arithmetic}. The slot count is a separate fixed parameter, and moving labels remain outside $\\mathcal A$. \\subsection{Prime counting in a f", "anchor": { "source": "paper.tex", "line_start": 7001, "line_end": 7001, "char_start": 321273, "char_end": 321293 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:ray-prime-normalizer", "to": "lem:ray-prime-normalizer", "type": "candidate_proof_dependency", "evidence": "\\le bP$ as $P\\to\\infty$. Stieltjes integration by parts therefore changes the left side of Equation~\\eqref{eq:ray-prime-normalizer} by $o(P^{1/6}/\\log P)$ when $d\\pi_{1_T}(x)$ is replaced by $dx/(|T|\\log x)$. The resulting integral is \\[ \\frac{P^{1/6}}{|T|\\lo", "anchor": { "source": "paper.tex", "line_start": 7031, "line_end": 7031, "char_start": 322592, "char_end": 322623 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:fixed-gauss-phase", "to": "sec:additional-arithmetic", "type": "internal_reference", "evidence": "w is a finite Fourier sum; it is distinct from the one-argument finite-ray function $G(a)$ of Lemma~\\ref{lem:fixed-gauss-phase}. Define G(a,k)=q_a^{-1/2}\\sum_{x\\bmod a}\\chi_a(x)e(kx/a),\\qquad G(1,k)=1. A", "anchor": { "source": "paper.tex", "line_start": 7046, "line_end": 7046, "char_start": 323295, "char_end": 323322 }, "origin": "source_reference", "context": "narrative" }, { "from": "sec:arithmetic", "to": "sec:additional-arithmetic", "type": "internal_reference", "evidence": "2.4} All characters in this subsection have the zero extensions specified in Section~\\ref{sec:arithmetic}, and we put $v_p(0)=+\\infty$. [Prime-power Fourier sums] For $P=q_p$ and every integ", "anchor": { "source": "paper.tex", "line_start": 7052, "line_end": 7052, "char_start": 323531, "char_end": 323551 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:correlation-lift", "to": "lem:full-correlation", "type": "internal_reference", "evidence": "\\nmid k}$. On the genuine residual locus $(n_1,n_2)=1$, $L_C$ means the congruence sum in Equation~\\eqref{eq:correlation-lift}. When $L_C$ is used on all residual pairs, it instead denotes the artificial product extension of these local formulas, with the", "anchor": { "source": "paper.tex", "line_start": 7125, "line_end": 7125, "char_start": 325959, "char_end": 325986 }, "origin": "source_reference", "context": "statement" }, { "from": "eq:correlation-lift", "to": "lem:full-correlation", "type": "candidate_proof_dependency", "evidence": "ines $x\\bmod p^{c+d}$ from it. Reduction modulo $p^c$ is exactly the common congruence in Equation~\\eqref{eq:correlation-lift}. The case $p\\mid n_2$ is symmetric, and this argument includes $c=0$. The Chinese remainder theorem thus gives a bijection with t", "anchor": { "source": "paper.tex", "line_start": 7164, "line_end": 7164, "char_start": 327888, "char_end": 327915 }, "origin": "source_reference", "context": "proof" }, { "from": "old-eq:2.11", "to": "lem:complete-support-correlation", "type": "candidate_proof_dependency", "evidence": "me factors. At primes of $a$ and $b$, solving the congruence in Equation~\\eqref{old-eq:2.11} gives respectively $\\chi_a(j)\\overline{\\chi_a(Eb)}$ and $\\overline{\\chi_b(-j)}\\chi_b(Da)$. These statements remain valid when a", "anchor": { "source": "paper.tex", "line_start": 7214, "line_end": 7214, "char_start": 330169, "char_end": 330188 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:correlation-child-character", "to": "sec:additional-arithmetic", "type": "internal_reference", "evidence": "heorem and the definition of $F(D,E;j)$ complete the proof. The hypotheses of Equation~\\eqref{eq:correlation-child-character} are not removed by extending $\\mathcal R$ as a bicharacter. To state exactly the extension used with M\\\"obius inversion, fix $D,", "anchor": { "source": "paper.tex", "line_start": 7231, "line_end": 7231, "char_start": 330945, "char_end": 330983 }, "origin": "source_reference", "context": "narrative" }, { "from": "sec:arithmetic", "to": "sec:marked", "type": "internal_reference", "evidence": "asks introduced by the inverse moment's Poisson transformations. We retain the notation of Section~\\ref{sec:arithmetic}. In particular, ideal variables have their multiplicative primary generators, the fixed excluded set \\(S\\) contains the primes o", "anchor": { "source": "paper.tex", "line_start": 7258, "line_end": 7258, "char_start": 332375, "char_end": 332395 }, "origin": "source_reference", "context": "narrative" }, { "from": "old-eq:4.2", "to": "sec:marked", "type": "internal_reference", "evidence": ", the number of tuples dividing \\(A\\) is \\(O_\\epsilon(q_A^\\epsilon)\\). The coefficient in Equation~\\eqref{old-eq:4.2} is a product of functions of the individual primes; this property, not merely the pointwise divisor bound, will be used when some", "anchor": { "source": "paper.tex", "line_start": 7313, "line_end": 7313, "char_start": 334918, "char_end": 334936 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:reflection", "to": "sec:marked", "type": "internal_reference", "evidence": "hole index \\(nb^3\\). With \\(\\mathfrak d=1\\), this is the Mellin completion on the left of Equation~\\eqref{eq:reflection}. The Gaussian Mellin test from that Equation is also permitted when only the completed estimate is used. For the reflection for", "anchor": { "source": "paper.tex", "line_start": 7327, "line_end": 7327, "char_start": 335639, "char_end": 335660 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:reflection-row-sectors", "to": "sec:marked", "type": "internal_reference", "evidence": "s also permitted when only the completed estimate is used. For the reflection formula we use Lemma~\\ref{lem:reflection-row-sectors}, only for a nonzero row. Write \\(k=u k_S k_{\\rm good}\\) as in that Lemma. After fixing the unit, the \\(S\\)-valuations modulo six", "anchor": { "source": "paper.tex", "line_start": 7331, "line_end": 7331, "char_start": 335808, "char_end": 335840 }, "origin": "source_reference", "context": "narrative" }, { "from": "prop:completed-reflection", "to": "cor:marked-reflection", "type": "internal_reference", "evidence": "d reflection with whole-index marks] Fix an invocation of Proposition~\\ref{prop:completed-reflection}, with base local prime set \\(\\mathcal P_0\\), powers \\(j_p\\), fixed multiplier \\(\\phi\\), and test \\(V\\). Put \\(M_{\\rm ref}=\\lambd", "anchor": { "source": "paper.tex", "line_start": 7363, "line_end": 7363, "char_start": 337416, "char_end": 337447 }, "origin": "source_reference", "context": "statement" }, { "from": "eq:marked-completion", "to": "cor:marked-reflection", "type": "internal_reference", "evidence": "enote this sum by \\(\\mathcal T_{\\mathcal Q}(X;\\Psi,V)\\). It has the same normalization as Equation~\\eqref{eq:marked-completion}, with \\(\\Psi_k,V_*\\) replaced by \\(\\Psi,V\\). The primal squarefree and cube ideals may share primes. There is an exact finite e", "anchor": { "source": "paper.tex", "line_start": 7370, "line_end": 7370, "char_start": 337904, "char_end": 337932 }, "origin": "source_reference", "context": "statement" }, { "from": "prop:completed-reflection", "to": "cor:marked-reflection", "type": "internal_reference", "evidence": "r. \\] The base factors \\(B_p\\) and the canonical functions \\(d,\\vartheta\\) are those of Proposition~\\ref{prop:completed-reflection} for this active radical. With every \\(\\sigma_p,\\epsilon_p,\\omega_{p,j}\\) computed using this branch's whole \\(c\\), the scalar is", "anchor": { "source": "paper.tex", "line_start": 7395, "line_end": 7395, "char_start": 338948, "char_end": 338979 }, "origin": "source_reference", "context": "statement" }, { "from": "prop:completed-reflection", "to": "cor:marked-reflection", "type": "candidate_proof_dependency", "evidence": "ntity \\(1_{q\\mid A}=1-\\chi_q(A)^0\\). Expand its product over \\(\\mathcal Q\\), and apply Proposition~\\ref{prop:completed-reflection} to each of the resulting finitely many completed sums. For a fixed \\(h_0\\) and active radical, the Proposition uses the same \\(c", "anchor": { "source": "paper.tex", "line_start": 7420, "line_end": 7420, "char_start": 340203, "char_end": 340234 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:marked-reflection", "to": "cor:marked-reflection", "type": "candidate_proof_dependency", "evidence": "_{q,2}^{+}\\chi_q(\\epsilon_q)^{-2}\\). Collecting these choices at all marked primes proves Equations~\\eqref{eq:marked-reflection} and \\eqref{eq:marked-reflection-phase}. The inactive prime is absent from \\(r\\), hence from that branch's conductor and kernel s", "anchor": { "source": "paper.tex", "line_start": 7431, "line_end": 7431, "char_start": 340901, "char_end": 340929 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:marked-reflection-phase", "to": "cor:marked-reflection", "type": "candidate_proof_dependency", "evidence": "\\). Collecting these choices at all marked primes proves Equations~\\eqref{eq:marked-reflection} and \\eqref{eq:marked-reflection-phase}. The inactive prime is absent from \\(r\\), hence from that branch's conductor and kernel scale. This is finite inclusion-exclusi", "anchor": { "source": "paper.tex", "line_start": 7432, "line_end": 7432, "char_start": 340934, "char_end": 340968 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:reflection-cross-prime-phase", "to": "sec:marked", "type": "internal_reference", "evidence": "s corollary must be stated with its exact scope. For distinct active good primes \\(p,q\\), Equation~\\eqref{eq:reflection-cross-prime-phase} and cubic reciprocity give \\chi_p(q)^{2j_p+2}\\chi_q(p)^{2j_q+2} =\\left(\\", "anchor": { "source": "paper.tex", "line_start": 7445, "line_end": 7445, "char_start": 341654, "char_end": 341693 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:completed-quadratic-reduction", "to": "sec:marked", "type": "internal_reference", "evidence": "dmakher--Louvel's quadratic large sieve and proved the zero-preserving completed reduction in Lemma~\\ref{lem:completed-quadratic-reduction}. We add Heath--Brown's cubic large sieve: \\sum_{\\substack{a\\equiv1\\ (3)\\\\q_a\\le U}", "anchor": { "source": "paper.tex", "line_start": 7489, "line_end": 7489, "char_start": 344070, "char_end": 344109 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:quadratic-large-sieve", "to": "sec:marked", "type": "internal_reference", "evidence": "ors. Conjugation and cubic reciprocity allow the opposite orientation. In this sieve and Equation~\\eqref{eq:quadratic-large-sieve}, a fixed restriction on the row set decreases the positive outer sum. A fixed restriction on the coefficient support is instead", "anchor": { "source": "paper.tex", "line_start": 7502, "line_end": 7502, "char_start": 344758, "char_end": 344790 }, "origin": "source_reference", "context": "narrative" }, { "from": "old-eq:4.6b", "to": "lem:hybrid-energy", "type": "candidate_proof_dependency", "evidence": "alities in the proof. Write \\(\\mathcal H\\) for the left side of Equation~\\eqref{old-eq:4.6b}. First remove the complete moving row-mark mask: \\[ 1_{(P,k)=1}=\\sum_{D\\mid(P,k)}\\mu(D). \\] For each row the number of such divi", "anchor": { "source": "paper.tex", "line_start": 7543, "line_end": 7543, "char_start": 346525, "char_end": 346544 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:completed-quadratic-reduction", "to": "lem:hybrid-energy", "type": "candidate_proof_dependency", "evidence": "and the inner \\(P'\\)-sum is an arbitrary complex coefficient independent of \\(k'\\). Apply Equation~\\eqref{eq:completed-quadratic-reduction} with row length \\(K/q_D\\). Use its notation \\[ b=gt^2,\\qquad c=(n,g),\\qquad n=cm,\\qquad g=ch, \\qquad q_c\\asymp C,\\quad q_g\\asym", "anchor": { "source": "paper.tex", "line_start": 7579, "line_end": 7579, "char_start": 348025, "char_end": 348065 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:hybrid-mark-mask-reduction", "to": "lem:hybrid-energy", "type": "candidate_proof_dependency", "evidence": "en the remaining row and \\(c'\\). For one \\(C,G,T\\) block its positive output, inserted in Equation~\\eqref{eq:hybrid-mark-mask-reduction}, is at most (KNBL)^\\epsilon\\sum_D\\frac1{q_D}T\\sum_{q_", "anchor": { "source": "paper.tex", "line_start": 7593, "line_end": 7593, "char_start": 348681, "char_end": 348718 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:hybrid-after-quadratic-reduction", "to": "lem:hybrid-energy", "type": "candidate_proof_dependency", "evidence": "itely many squarefree \\(S\\)-parts of \\(m,h\\) before sieving their good product; \\(m,h\\) in Equation~\\eqref{eq:hybrid-after-quadratic-reduction} are the original ideals. We may now discard \\(|\\beta_D|^2\\le1\\). For fixed \\(h,t,D,r\\), the \\(P'\\)-coefficients are independent", "anchor": { "source": "paper.tex", "line_start": 7620, "line_end": 7620, "char_start": 349807, "char_end": 349850 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:cubic-large-sieve", "to": "lem:hybrid-energy", "type": "candidate_proof_dependency", "evidence": ". Enlarge only this positive squarefree \\(j\\)-range. Cubic reciprocity, conjugation, and Equation~\\eqref{eq:cubic-large-sieve}, followed by the \\(O(G/C)\\) choices of \\(h\\), give, on \\(q_D\\asymp D_0,\\ q_r\\asymp r_0\\), (K", "anchor": { "source": "paper.tex", "line_start": 7633, "line_end": 7633, "char_start": 350482, "char_end": 350510 }, "origin": "source_reference", "context": "proof" }, { "from": "old-eq:4.6b", "to": "lem:hybrid-energy", "type": "candidate_proof_dependency", "evidence": "re replaced by \\(1\\), and \\(NG/C^2\\ll NB\\). Summing the logarithmically many dyads proves Equation~\\eqref{old-eq:4.6b}. Subunit quotient ranges are empty, and bounded ranges are included by changing the fixed annular constants. This argument has", "anchor": { "source": "paper.tex", "line_start": 7652, "line_end": 7652, "char_start": 351327, "char_end": 351346 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:marked-completion", "to": "sec:marked", "type": "internal_reference", "evidence": ". \\subsection{The reflected energy bound} We next quantify the reflection of Equation~\\eqref{eq:marked-completion}. The following description also specifies the dyadic contribution used in the statement. \\paragraph{Reflected block data.}\\label", "anchor": { "source": "paper.tex", "line_start": 7659, "line_end": 7659, "char_start": 351648, "char_end": 351676 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:actual-residual-row-dyad", "to": "sec:marked", "type": "internal_reference", "evidence": "q_{k_{\\rm res}}\\ge Z^H/C_H. \\] Since \\(q_{k_{\\rm sup}}\\ge1\\), the general norm inequality Equation~\\eqref{eq:actual-residual-row-dyad} gives here \\[ H\\le M-O+\\frac{\\log C_{\\rm res}}{\\log Z}, \\qquad C_{\\rm res}=C_HC_kC_O. \\] The inequality need not be an equality", "anchor": { "source": "paper.tex", "line_start": 7689, "line_end": 7689, "char_start": 352985, "char_end": 353020 }, "origin": "source_reference", "context": "narrative" }, { "from": "old-eq:4.3", "to": "sec:marked", "type": "internal_reference", "evidence": "nonnegative. These are source dual ideals: they retain every prime of \\(S\\) permitted by Equation~\\eqref{old-eq:4.3}; no extra \\(S\\)-mask is placed on them. Only the displayed \\(\\lambda\\)-valuation support is imposed. The unit and the integer \\(", "anchor": { "source": "paper.tex", "line_start": 7733, "line_end": 7733, "char_start": 354966, "char_end": 354984 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:marked-reflection", "to": "sec:marked", "type": "internal_reference", "evidence": "hese choices, denote by \\(\\mathcal U_{v,\\ell_b,e_\\lambda}(k_{\\rm res})\\) the right side of Equation~\\eqref{eq:marked-reflection}, summed with the product-form tuple coefficients, with the specified local choices and dual restrictions, and the actual factor \\", "anchor": { "source": "paper.tex", "line_start": 7738, "line_end": 7738, "char_start": 355260, "char_end": 355288 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:marked-completion", "to": "lem:reflected-energy", "type": "internal_reference", "evidence": "e2\\), and suppose their log-lengths range over fixed bounded sets. In the completed sum of Equation~\\eqref{eq:marked-completion}, the fixed character, puncture, and slot coefficients are independent of \\(k,f\\), the slot supports are disjoint, and the slot co", "anchor": { "source": "paper.tex", "line_start": 7750, "line_end": 7750, "char_start": 356003, "char_end": 356031 }, "origin": "source_reference", "context": "statement" }, { "from": "old-eq:4.2", "to": "lem:reflected-energy", "type": "internal_reference", "evidence": "t of \\(k,f\\), the slot supports are disjoint, and the slot coefficients are products as in Equation~\\eqref{old-eq:4.2}. The fixed row restrictions after freezing \\(k_{\\rm pow},k_{\\rm sup}\\) must be independent of the active slots and dual variable", "anchor": { "source": "paper.tex", "line_start": 7753, "line_end": 7753, "char_start": 356200, "char_end": 356218 }, "origin": "source_reference", "context": "statement" }, { "from": "eq:actual-residual-row-dyad", "to": "lem:reflected-energy", "type": "internal_reference", "evidence": "log Z)\\). In particular, the formula is used only on actual residual-row dyads satisfying Equation~\\eqref{eq:actual-residual-row-dyad}. First remove the inactive slots by triangle inequality in the row Hilbert space. At such a slot the", "anchor": { "source": "paper.tex", "line_start": 7803, "line_end": 7803, "char_start": 358121, "char_end": 358156 }, "origin": "source_reference", "context": "statement" }, { "from": "eq:unmarked-structural-block", "to": "lem:reflected-energy", "type": "candidate_proof_dependency", "evidence": "-slot \\(j=4\\) label remains averaged. We use only the algebraic frozen-base extraction in Equation~\\eqref{eq:unmarked-structural-block}, inside the original sum over active tuples. Corollary~\\ref{cor:marked-reflection} and the subsequent phase separation make that", "anchor": { "source": "paper.tex", "line_start": 7823, "line_end": 7823, "char_start": 359075, "char_end": 359111 }, "origin": "source_reference", "context": "proof" }, { "from": "cor:marked-reflection", "to": "lem:reflected-energy", "type": "candidate_proof_dependency", "evidence": "uation~\\eqref{eq:unmarked-structural-block}, inside the original sum over active tuples. Corollary~\\ref{cor:marked-reflection} and the subsequent phase separation make that extraction simultaneous across the tuples: the base coefficient is common to the re", "anchor": { "source": "paper.tex", "line_start": 7824, "line_end": 7824, "char_start": 359168, "char_end": 359195 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:marked-reflection", "to": "lem:reflected-energy", "type": "candidate_proof_dependency", "evidence": "ll source support, including the permitted primes of \\(S\\). Every active marked column in Equation~\\eqref{eq:marked-reflection} contributes \\(q_p^{-1/2}\\). Since \\(q_P/Z^{z_a}\\) lies in a fixed compact interval, this proves the per-tuple coefficient exponen", "anchor": { "source": "paper.tex", "line_start": 7842, "line_end": 7842, "char_start": 360153, "char_end": 360181 }, "origin": "source_reference", "context": "proof" }, { "from": "old-eq:4.5", "to": "lem:reflected-energy", "type": "candidate_proof_dependency", "evidence": "z_a}\\) lies in a fixed compact interval, this proves the per-tuple coefficient exponent in Equation~\\eqref{old-eq:4.5}. We now record the precise marked instance of the common profile. Let \\(F_{\\rm act},N_F,B_F\\) be the actual frozen products wit", "anchor": { "source": "paper.tex", "line_start": 7844, "line_end": 7844, "char_start": 360325, "char_end": 360343 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:reflection-common-profile", "to": "lem:reflected-energy", "type": "candidate_proof_dependency", "evidence": "tivity. Put \\(D_{\\rm ker}=T_d-v-3\\ell_b-e_\\lambda\\). On this full product of actual annuli, Lemma~\\ref{lem:reflection-common-profile} permits \\[ m_{\\rm ker}=Z^{-(D_{\\rm ker})_+/4} \\] to be factored from the entire homogeneous linear row vector, at a factor at mo", "anchor": { "source": "paper.tex", "line_start": 7882, "line_end": 7882, "char_start": 361817, "char_end": 361852 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:lattice-kernel-tail", "to": "lem:reflected-energy", "type": "candidate_proof_dependency", "evidence": "n \\(\\tau_{\\rm ref}\\) has actual kernel argument greater than \\(Z^{\\tau_{\\rm ref}/2}\\). Apply Lemma~\\ref{lem:lattice-kernel-tail} with lattice \\(\\lambda^{-4}\\mathcal O\\), \\(a=X/q_c^2\\), and \\(U=Z^{\\tau_{\\rm ref}/2}\\). On the full source support, the coeffici", "anchor": { "source": "paper.tex", "line_start": 7895, "line_end": 7896, "char_start": 362537, "char_end": 362566 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:reflection-common-minkowski", "to": "lem:reflected-energy", "type": "candidate_proof_dependency", "evidence": "ed before Fourier absolutization. For a retained dyad, apply the common-profile lemma and Equation~\\eqref{eq:reflection-common-minkowski} to the one joint profile in Equation~\\eqref{eq:reflection-actual-annuli}. Its density is common to the entire current row Hilber", "anchor": { "source": "paper.tex", "line_start": 7917, "line_end": 7917, "char_start": 363577, "char_end": 363615 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:reflection-actual-annuli", "to": "lem:reflected-energy", "type": "candidate_proof_dependency", "evidence": "file lemma and Equation~\\eqref{eq:reflection-common-minkowski} to the one joint profile in Equation~\\eqref{eq:reflection-actual-annuli}. Its density is common to the entire current row Hilbert space, including the active-tuple sum: the actual row, dual, and indivi", "anchor": { "source": "paper.tex", "line_start": 7918, "line_end": 7918, "char_start": 363653, "char_end": 363688 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:reflection-pair-phase", "to": "lem:reflected-energy", "type": "candidate_proof_dependency", "evidence": "The separate full \\(M_{\\rm ref}^2\\) classes fix the cusp coefficient and additive factor. Equation~\\eqref{eq:reflection-pair-phase} cancels every row-slot phase, while the preceding marked-reflection discussion assigns all row-row, slot-slot, and frozen interac", "anchor": { "source": "paper.tex", "line_start": 7956, "line_end": 7956, "char_start": 365414, "char_end": 365446 }, "origin": "source_reference", "context": "proof" }, { "from": "old-eq:4.6a", "to": "lem:reflected-energy", "type": "candidate_proof_dependency", "evidence": "b\\), and \\(\\beta\\) is independent of \\(k_{\\rm res},\\boldsymbol p\\), exactly as required in Equation~\\eqref{old-eq:4.6a}. The function \\(a\\) need not retain product form at this norm step. Apply Lemma~\\ref{lem:hybrid-energy} with \\[ K=Z^H,\\qquad N", "anchor": { "source": "paper.tex", "line_start": 7966, "line_end": 7966, "char_start": 366139, "char_end": 366158 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:hybrid-energy", "to": "lem:reflected-energy", "type": "candidate_proof_dependency", "evidence": "qref{old-eq:4.6a}. The function \\(a\\) need not retain product form at this norm step. Apply Lemma~\\ref{lem:hybrid-energy} with \\[ K=Z^H,\\qquad N=Z^v,\\qquad B=Z^{\\ell_b},\\qquad L=Z^{z_a}. \\] Aggregating tuples with the same \\(P\\) costs only a divisor", "anchor": { "source": "paper.tex", "line_start": 7969, "line_end": 7969, "char_start": 366241, "char_end": 366264 }, "origin": "source_reference", "context": "proof" }, { "from": "old-eq:4.6", "to": "lem:reflected-energy", "type": "candidate_proof_dependency", "evidence": "e_\\lambda/3. \\] Only now does the squared scalar \\(m_{\\rm ker}^2\\) supply the last term in Equation~\\eqref{old-eq:4.6}. This proves the fixed-part assertion. Every powerful ideal is uniquely \\(x^2y^3\\) with \\(y\\) squarefree. Ideal counting and \\", "anchor": { "source": "paper.tex", "line_start": 7982, "line_end": 7982, "char_start": 366652, "char_end": 366670 }, "origin": "source_reference", "context": "proof" }, { "from": "old-eq:4.6", "to": "sec:marked", "type": "internal_reference", "evidence": "ge-\\eta\\); the latter follows from \\(\\log Z\\ge4\\log q_\\lambda/\\eta\\). On a retained dyad, Equation~\\eqref{old-eq:4.6} and \\(v+3\\ell_b+e_\\lambda\\le T_d+\\tau_{\\rm ref}\\) give \\[ E_{\\rm ref} \\le O/2+\\max\\{H,T_d-S_0-B_0\\}+\\tfrac53\\eta+\\tau_{\\rm ref}", "anchor": { "source": "paper.tex", "line_start": 8038, "line_end": 8038, "char_start": 369045, "char_end": 369063 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:actual-residual-row-dyad", "to": "sec:marked", "type": "internal_reference", "evidence": "lambda+\\tau_{\\rm ref}\\). The maximum is increasing in each argument. We may therefore use Equation~\\eqref{eq:actual-residual-row-dyad} for its first argument and Equation~\\eqref{eq:common-width-charge} for its second, then sum the logarithmically many actual \\(H\\)", "anchor": { "source": "paper.tex", "line_start": 8048, "line_end": 8048, "char_start": 369436, "char_end": 369471 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:common-width-charge", "to": "sec:marked", "type": "internal_reference", "evidence": "may therefore use Equation~\\eqref{eq:actual-residual-row-dyad} for its first argument and Equation~\\eqref{eq:common-width-charge} for its second, then sum the logarithmically many actual \\(H\\)-dyads. Taking \\(\\eta,\\tau_{\\rm ref}\\) and the local small-power l", "anchor": { "source": "paper.tex", "line_start": 8049, "line_end": 8049, "char_start": 369508, "char_end": 369538 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:compensated-probe-definition", "to": "sec:compensated-low", "type": "internal_reference", "evidence": "estimate} We now bound the finite compensated probe defined in Equation~\\eqref{eq:compensated-probe-definition}. Its exact low separation has two factors. We first bound the completed row after summing the marked slots by applying Lemma~\\r", "anchor": { "source": "paper.tex", "line_start": 8064, "line_end": 8064, "char_start": 370201, "char_end": 370240 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:reflected-energy", "to": "sec:compensated-low", "type": "internal_reference", "evidence": "has two factors. We first bound the completed row after summing the marked slots by applying Lemma~\\ref{lem:reflected-energy} directly. We then prove a quantitative Gram bound for the additive polynomial from Section~\\ref{sec:probe}. Cauchy--Schwarz wil", "anchor": { "source": "paper.tex", "line_start": 8066, "line_end": 8066, "char_start": 370368, "char_end": 370394 }, "origin": "source_reference", "context": "narrative" }, { "from": "sec:probe", "to": "sec:compensated-low", "type": "internal_reference", "evidence": "energy} directly. We then prove a quantitative Gram bound for the additive polynomial from Section~\\ref{sec:probe}. Cauchy--Schwarz will combine these estimates to give the exponent $3/16$. \\subsection{The marked completed row} For a fixed r", "anchor": { "source": "paper.tex", "line_start": 8068, "line_end": 8068, "char_start": 370487, "char_end": 370502 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:compensated-probe-definition", "to": "sec:compensated-low", "type": "internal_reference", "evidence": "\\,dt. \\] Thus $B^J$ is exactly the completed-row factor for the $J$ summand of Equation~\\eqref{eq:compensated-probe-definition} after the marked slots have been summed. The fixed tuple $p_J$ affects only $X',Y'$ in its low separation. [The co", "anchor": { "source": "paper.tex", "line_start": 8107, "line_end": 8107, "char_start": 371827, "char_end": 371866 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:gaussian-annular", "to": "lem:probe-row-norm", "type": "candidate_proof_dependency", "evidence": "),\\\\ r=\\frac{q_A}{Z^{1+\\ell'}}. \\] Here $V=W_{\\mathrm G}$ is the Gaussian of Lemma~\\ref{lem:gaussian-annular}. Empty products are one. Use the fixed translate partition $\\chi$ from that lemma on $r$. On the annulus $r=e^kx$, retain the s", "anchor": { "source": "paper.tex", "line_start": 8137, "line_end": 8137, "char_start": 372949, "char_end": 372975 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:gaussian-annular", "to": "lem:probe-row-norm", "type": "candidate_proof_dependency", "evidence": "s the logarithm of the Gaussian argument is $k+O_{\\rm fixed}(1)$. The derivative estimate in Lemma~\\ref{lem:gaussian-annular}, with this bounded shift and the fixed slot windows, gives for every fixed $\\kappa_{\\mathrm G}>0$, $A_{\\rm cnt},B_{\\rm cnt},C_{\\r", "anchor": { "source": "paper.tex", "line_start": 8150, "line_end": 8150, "char_start": 373488, "char_end": 373514 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:smooth-calculus", "to": "lem:probe-row-norm", "type": "candidate_proof_dependency", "evidence": "lus, keep fixed individual annular cutoffs equal to one on the support of $w_{k,J}$ and apply Lemma~\\ref{lem:smooth-calculus} to this whole profile. Its logarithmic Fourier inversion uses one coefficient density common to every row and every surviving sl", "anchor": { "source": "paper.tex", "line_start": 8176, "line_end": 8176, "char_start": 374678, "char_end": 374703 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:reflection-row-sectors", "to": "lem:probe-row-norm", "type": "candidate_proof_dependency", "evidence": "Choose $\\kappa_{\\mathrm G}$ within the final $\\epsilon$ allowance. For each fixed $\\theta$, Lemma~\\ref{lem:reflection-row-sectors} supplies the row-sector reduction of $\\nu_\\sigma\\theta\\chi_\\bullet(m)$ with every original zero mask. It covers all nonzero elem", "anchor": { "source": "paper.tex", "line_start": 8199, "line_end": 8199, "char_start": 375999, "char_end": 376031 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:reflected-energy", "to": "lem:probe-row-norm", "type": "candidate_proof_dependency", "evidence": "isjoint and their separated coefficients are product-form. Finite triangle over $\\theta$ and Lemma~\\ref{lem:reflected-energy} therefore give the exponent in Equation~\\eqref{old-eq:4.6} at this actual $N_*$, taking the retention parameter $\\tau_{\\rm ref}$", "anchor": { "source": "paper.tex", "line_start": 8206, "line_end": 8206, "char_start": 376533, "char_end": 376559 }, "origin": "source_reference", "context": "proof" }, { "from": "old-eq:4.6", "to": "lem:probe-row-norm", "type": "candidate_proof_dependency", "evidence": "triangle over $\\theta$ and Lemma~\\ref{lem:reflected-energy} therefore give the exponent in Equation~\\eqref{old-eq:4.6} at this actual $N_*$, taking the retention parameter $\\tau_{\\rm ref}$ and the independently prescribed output and separation loss", "anchor": { "source": "paper.tex", "line_start": 8207, "line_end": 8207, "char_start": 376600, "char_end": 376618 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:reflected-energy", "to": "lem:probe-row-norm", "type": "candidate_proof_dependency", "evidence": "non-slot primes are those of the row. After the finite unit and reflection-sector splits in Lemma~\\ref{lem:reflected-energy}, freeze its powerful part of norm length $O$ and its valuation-one part supported on $S$, and split its residual squarefree good", "anchor": { "source": "paper.tex", "line_start": 8216, "line_end": 8216, "char_start": 377138, "char_end": 377164 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:reflection-common-minkowski", "to": "lem:probe-row-norm", "type": "candidate_proof_dependency", "evidence": "efore the row norm is squared, and only then may the positive sieve sum be enlarged, as in Equation~\\eqref{eq:reflection-common-minkowski}. In particular the kernel saving in Equation~\\eqref{old-eq:4.6} uses this $H$; it is not reevaluated on any shorter rows added by", "anchor": { "source": "paper.tex", "line_start": 8230, "line_end": 8230, "char_start": 377898, "char_end": 377936 }, "origin": "source_reference", "context": "proof" }, { "from": "old-eq:4.6", "to": "lem:probe-row-norm", "type": "candidate_proof_dependency", "evidence": "as in Equation~\\eqref{eq:reflection-common-minkowski}. In particular the kernel saving in Equation~\\eqref{old-eq:4.6} uses this $H$; it is not reevaluated on any shorter rows added by the positive enlargement. In the notation of that energy estim", "anchor": { "source": "paper.tex", "line_start": 8232, "line_end": 8232, "char_start": 377982, "char_end": 378000 }, "origin": "source_reference", "context": "proof" }, { "from": "old-eq:4.3", "to": "lem:probe-row-norm", "type": "candidate_proof_dependency", "evidence": "arbitrarily small power. The source dual ideals $n,b$ in these dyads retain the support of Equation~\\eqref{old-eq:4.3}: they may share primes and may contain the permitted primes of $S$; no additional $S$-mask or coprimality condition between $n$ a", "anchor": { "source": "paper.tex", "line_start": 8256, "line_end": 8256, "char_start": 378932, "char_end": 378950 }, "origin": "source_reference", "context": "proof" }, { "from": "old-eq:4.7", "to": "lem:probe-row-norm", "type": "candidate_proof_dependency", "evidence": "epsilon_Z). \\] Write $s_{\\rm hyb}=\\min\\{v,z_a,(v+z_a)/3\\}$ for the quantity called $u$ in Equation~\\eqref{old-eq:4.7}. If $s_{\\rm hyb}=z_a$, then $-S_0-B_0+z_a-s_{\\rm hyb}\\le0$, the kernel term is nonpositive, and \\[ -\\ell_b-\\frac{2e_\\lambda}{3}", "anchor": { "source": "paper.tex", "line_start": 8268, "line_end": 8268, "char_start": 379477, "char_end": 379495 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:full-correlation", "to": "sec:compensated-low", "type": "internal_reference", "evidence": "nd} We next estimate the additive factor in the range $Y'^2/Q\\ge1$. The full correlation of Lemma~\\ref{lem:full-correlation} retains the collision zeros that determine both its cancellation and its exceptional frequencies. [A quantita", "anchor": { "source": "paper.tex", "line_start": 8337, "line_end": 8337, "char_start": 382059, "char_end": 382085 }, "origin": "source_reference", "context": "narrative" }, { "from": "sec:probe", "to": "prop:probe-gram", "type": "internal_reference", "evidence": "[ P_a=Y'^2/Q\\ge1. \\] For a real height $\\nu$, put $A_m=A_{m,\\sigma,\\nu}(Y')$ as defined in Section~\\ref{sec:probe}. For every fixed row ball $q_m\\ll Q$ and every $\\epsilon>0$, one has \\sum_{q_m\\ll Q}|A_m|^2 \\ll (1+|\\nu|)^{J_", "anchor": { "source": "paper.tex", "line_start": 8348, "line_end": 8348, "char_start": 382527, "char_end": 382542 }, "origin": "source_reference", "context": "statement" }, { "from": "eq:correlation-lift", "to": "prop:probe-gram", "type": "candidate_proof_dependency", "evidence": "It therefore gives the exact prefactor $Q/Y'^3$ and the correlation $F(Cn_1,Cn_2;Ck)$ of Equation~\\eqref{eq:correlation-lift}. Here $F$ again denotes the full correlation of Equation~\\eqref{old-eq:2.11}, not the local coefficient of Equation~\\eqref{eq:loc", "anchor": { "source": "paper.tex", "line_start": 8381, "line_end": 8381, "char_start": 384019, "char_end": 384046 }, "origin": "source_reference", "context": "proof" }, { "from": "old-eq:2.11", "to": "prop:probe-gram", "type": "candidate_proof_dependency", "evidence": ")$ of Equation~\\eqref{eq:correlation-lift}. Here $F$ again denotes the full correlation of Equation~\\eqref{old-eq:2.11}, not the local coefficient of Equation~\\eqref{eq:local-F}. The first Fourier kernel has argument comparable to $q_Cq_k/P_a$. On", "anchor": { "source": "paper.tex", "line_start": 8382, "line_end": 8382, "char_start": 384104, "char_end": 384123 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:local-F", "to": "prop:probe-gram", "type": "candidate_proof_dependency", "evidence": "denotes the full correlation of Equation~\\eqref{old-eq:2.11}, not the local coefficient of Equation~\\eqref{eq:local-F}. The first Fourier kernel has argument comparable to $q_Cq_k/P_a$. On coprime residuals the factor $L_C$ in that correlation has", "anchor": { "source": "paper.tex", "line_start": 8383, "line_end": 8383, "char_start": 384163, "char_end": 384181 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:fixed-numerator-ray", "to": "prop:probe-gram", "type": "candidate_proof_dependency", "evidence": "k=\\zeta k_Sk_{\\mathrm{good}}$ using the unit convention and the fixed $S$-prime generators of Lemma~\\ref{lem:fixed-numerator-ray}, with $k_S=\\prod_{\\mathfrak p\\in S}\\pi_{\\mathfrak p}^{e_{\\mathfrak p}}$ and $k_{\\mathrm{good}}=\\prod_{p\\notin S}p^{e_p}$. For a", "anchor": { "source": "paper.tex", "line_start": 8404, "line_end": 8404, "char_start": 385216, "char_end": 385245 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:fixed-numerator-ray", "to": "prop:probe-gram", "type": "candidate_proof_dependency", "evidence": "y displayed good local power retains its zero on a shared prime, including when $6\\mid e_p$. Lemma~\\ref{lem:fixed-numerator-ray} places the finitely many $\\kappa_{\\zeta,\\boldsymbol e}$ in a fixed ray group supported on $S$. Also $\\mathcal R(k_{\\mathrm{good}", "anchor": { "source": "paper.tex", "line_start": 8419, "line_end": 8420, "char_start": 385855, "char_end": 385884 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:gram-periodic-poisson", "to": "prop:probe-gram", "type": "candidate_proof_dependency", "evidence": "so it still contains $O(N)$ lattice points; below that constant it is empty. This proves Equation~\\eqref{eq:gram-periodic-poisson} without any primitivity or tensor-product assumption on $a_{C,d,k}$. Put $K_R=RP_a/C_0$. The fixed shell comparison constants g", "anchor": { "source": "paper.tex", "line_start": 8500, "line_end": 8501, "char_start": 389990, "char_end": 390022 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:gram-periodic-poisson", "to": "prop:probe-gram", "type": "candidate_proof_dependency", "evidence": "K_R. \\] There are therefore $O(C_0D_0K_R)$ choices of $C,d,k$ on these dyads. Multiplying Equation~\\eqref{eq:gram-periodic-poisson} by this count and by the common prefactor gives at most \\[ (1+|\\nu|)^{J_{\\mathrm{Gr}}} R^{-B}\\frac Q{Y'}\\frac{RP_a}{C_0D_0}", "anchor": { "source": "paper.tex", "line_start": 8513, "line_end": 8513, "char_start": 390523, "char_end": 390555 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:compensated-probe-definition", "to": "prop:probe-low", "type": "internal_reference", "evidence": "[The compensated low estimate] For the probe in Equation~\\eqref{eq:compensated-probe-definition}, the geometry in Equation~\\eqref{old-eq:5.1}, and every $\\epsilon>0$, |I_{\\eta,\\mathrm{modified}}(Z)| \\l", "anchor": { "source": "paper.tex", "line_start": 8565, "line_end": 8565, "char_start": 392749, "char_end": 392788 }, "origin": "source_reference", "context": "statement" }, { "from": "old-eq:5.1", "to": "prop:probe-low", "type": "internal_reference", "evidence": "be-low} For the probe in Equation~\\eqref{eq:compensated-probe-definition}, the geometry in Equation~\\eqref{old-eq:5.1}, and every $\\epsilon>0$, |I_{\\eta,\\mathrm{modified}}(Z)| \\ll Z^{l_x/2+b/12+\\epsilon}=Z^{3/16+\\epsilon}.", "anchor": { "source": "paper.tex", "line_start": 8566, "line_end": 8566, "char_start": 392815, "char_end": 392833 }, "origin": "source_reference", "context": "statement" }, { "from": "eq:low-separated", "to": "prop:probe-low", "type": "candidate_proof_dependency", "evidence": "begin{proof} For a fixed rescaled subset and tuple, put $Q=q_{b_*}X'Y'$. Its separation is Equation~\\eqref{eq:low-separated} with $A_{m,\\sigma,v}(Y')$ and $B^J_{m,\\sigma}(Z)$. Put $r_J=q_{p_J}/Z^d$; the fixed annular supports give $00$. The primewise defect bounds in Lemma~\\ref{lem:local-euler} give, uniformly in the sel", "anchor": { "source": "paper.tex", "line_start": 8735, "line_end": 8735, "char_start": 399793, "char_end": 399820 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:local-euler", "to": "sec:local-compensation", "type": "internal_reference", "evidence": ":local-region-one}, put $\\epsilon_H=\\min(\\epsilon_0,1/50)>0$. The primewise defect bounds in Lemma~\\ref{lem:local-euler} give, uniformly in the selected tuple, \\prod_{\\substack{p\\notin S\\\\p\\notin\\m", "anchor": { "source": "paper.tex", "line_start": 8737, "line_end": 8737, "char_start": 399902, "char_end": 399923 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:unselected-local-product", "to": "sec:local-compensation", "type": "internal_reference", "evidence": "egion the same argument uses the respective defect exponents $-363/200$ and $-33/40$. Thus Equation~\\eqref{eq:unselected-local-product} holds there as well, with the corresponding positive majorants. No nonvanishing of $H_p$ is asserted by these upper bounds. For", "anchor": { "source": "paper.tex", "line_start": 8750, "line_end": 8750, "char_start": 400538, "char_end": 400573 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:holomorphic-selected-tuple", "to": "sec:local-compensation", "type": "internal_reference", "evidence": "jorants. No nonvanishing of $H_p$ is asserted by these upper bounds. For each fixed $Z$, Equation~\\eqref{eq:holomorphic-selected-tuple} is a finite sum of products of holomorphic selected factors and normally convergent unselected products. It is therefore holomor", "anchor": { "source": "paper.tex", "line_start": 8754, "line_end": 8754, "char_start": 400729, "char_end": 400766 }, "origin": "source_reference", "context": "narrative" }, { "from": "def:stage-high-data", "to": "sec:local-compensation", "type": "internal_reference", "evidence": "d of every point in both Euler regions. It also satisfies the all-height requirement in Definition~\\ref{def:stage-high-data}. Indeed, on a fixed real box in either region, the formula for $P_p^*$ and the denominators bounded away from zero give $|G_p|\\ll", "anchor": { "source": "paper.tex", "line_start": 8758, "line_end": 8758, "char_start": 401017, "char_end": 401042 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:unselected-local-product", "to": "sec:local-compensation", "type": "internal_reference", "evidence": "There are $O(P_i)$ ideals in a slot and $q_p\\asymp P_i$ there. The finite tuple sum and Equation~\\eqref{eq:unselected-local-product} consequently give |\\mathfrak H_{\\eta,u,Z}(x,w,z)| \\ll_\\epsilon q_u^\\", "anchor": { "source": "paper.tex", "line_start": 8763, "line_end": 8763, "char_start": 401347, "char_end": 401382 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:stage-all-height-majorant", "to": "sec:local-compensation", "type": "internal_reference", "evidence": "_u^\\epsilon Z^{B'}. for a fixed $B'$ on that box. For $q_u\\le Z^D$ this is Equation~\\eqref{eq:stage-all-height-majorant}, even with height exponent zero. All constants here are fixed before any later order of integration by parts. Combining the exa", "anchor": { "source": "paper.tex", "line_start": 8770, "line_end": 8770, "char_start": 401653, "char_end": 401689 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:probe-poisson-identity", "to": "sec:local-compensation", "type": "internal_reference", "evidence": "efore any later order of integration by parts. Combining the exact finite operation with Equations~\\eqref{eq:probe-poisson-identity} and \\eqref{eq:probe-high} now gives I_{\\eta,\\mathrm{modifi", "anchor": { "source": "paper.tex", "line_start": 8775, "line_end": 8775, "char_start": 401854, "char_end": 401887 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:probe-high", "to": "sec:local-compensation", "type": "internal_reference", "evidence": "y parts. Combining the exact finite operation with Equations~\\eqref{eq:probe-poisson-identity} and \\eqref{eq:probe-high} now gives I_{\\eta,\\mathrm{modified}}(Z) ={}&\\frac1{(2\\pi", "anchor": { "source": "paper.tex", "line_start": 8775, "line_end": 8776, "char_start": 401892, "char_end": 401913 }, "origin": "source_reference", "context": "narrative" }, { "from": "def:stage-high-data", "to": "sec:local-compensation", "type": "internal_reference", "evidence": "It has no additional outside Euler factor. It verifies the exact high representation of Definition~\\ref{def:stage-high-data} for the physical expression $I_{\\eta,\\mathrm{modified}}$, with the geometry in Equation~\\eqref{old-eq:5.1}. In particular, $l_x/", "anchor": { "source": "paper.tex", "line_start": 8791, "line_end": 8792, "char_start": 402610, "char_end": 402635 }, "origin": "source_reference", "context": "narrative" }, { "from": "old-eq:5.1", "to": "sec:local-compensation", "type": "internal_reference", "evidence": "-high-data} for the physical expression $I_{\\eta,\\mathrm{modified}}$, with the geometry in Equation~\\eqref{old-eq:5.1}. In particular, $l_x/2-1+h/6=17/96-1+13/96=-11/16$, as required by Equation~\\eqref{eq:part-II-mellin-target}. It is an identity", "anchor": { "source": "paper.tex", "line_start": 8793, "line_end": 8793, "char_start": 402724, "char_end": 402742 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:part-II-mellin-target", "to": "sec:local-compensation", "type": "internal_reference", "evidence": "ion~\\eqref{old-eq:5.1}. In particular, $l_x/2-1+h/6=17/96-1+13/96=-11/16$, as required by Equation~\\eqref{eq:part-II-mellin-target}. It is an identity for the very probe bounded in Proposition~\\ref{prop:probe-low}. Every continuation uses the full correction", "anchor": { "source": "paper.tex", "line_start": 8795, "line_end": 8795, "char_start": 402820, "char_end": 402852 }, "origin": "source_reference", "context": "narrative" }, { "from": "prop:probe-low", "to": "sec:local-compensation", "type": "internal_reference", "evidence": "tion~\\eqref{eq:part-II-mellin-target}. It is an identity for the very probe bounded in Proposition~\\ref{prop:probe-low}. Every continuation uses the full correction in Equation~\\eqref{eq:holomorphic-selected-tuple}, not a globally defined product o", "anchor": { "source": "paper.tex", "line_start": 8796, "line_end": 8796, "char_start": 402915, "char_end": 402935 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:holomorphic-selected-tuple", "to": "sec:local-compensation", "type": "internal_reference", "evidence": "unded in Proposition~\\ref{prop:probe-low}. Every continuation uses the full correction in Equation~\\eqref{eq:holomorphic-selected-tuple}, not a globally defined product of individual quotients. \\subsection{Dynamic local errors} At an identity-ray prime every fixed", "anchor": { "source": "paper.tex", "line_start": 8797, "line_end": 8797, "char_start": 402994, "char_end": 403031 }, "origin": "source_reference", "context": "narrative" }, { "from": "old-eq:5.13c", "to": "sec:local-compensation", "type": "internal_reference", "evidence": "c identity there. For $p\\mid u$, the main term is zero by the original zero extension, and Equation~\\eqref{old-eq:5.13c} instead becomes \\[ G_p= \\overline{\\eta(p)}Q^x(1-V)\\mathcal E_p-Q^{-w}H_p. \\] We now give the bound on the remaining error slot", "anchor": { "source": "paper.tex", "line_start": 8829, "line_end": 8829, "char_start": 404368, "char_end": 404388 }, "origin": "source_reference", "context": "narrative" }, { "from": "sec:detector", "to": "sec:local-compensation", "type": "internal_reference", "evidence": "citly: the reflected primitive numerator must be small at the retained height. Instantiate Section~\\ref{sec:detector} with the current fixed data and $\\Theta=\\langle\\eta,\\widehat T\\rangle$, using its zero-extended presentations $\\nu(n)\\chi_n(u)^\\v", "anchor": { "source": "paper.tex", "line_start": 8837, "line_end": 8838, "char_start": 404656, "char_end": 404674 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:deleted-euler-factors", "to": "sec:local-compensation", "type": "internal_reference", "evidence": "a\\in\\Theta$. The buffered estimates there, together with the inverse deleted-factor bound of Lemma~\\ref{lem:deleted-euler-factors}, therefore supply the reflected-numerator hypothesis below at the retained heights: for the points used below, $\\Re(1-w)=a+6e$ an", "anchor": { "source": "paper.tex", "line_start": 8845, "line_end": 8845, "char_start": 405179, "char_end": 405210 }, "origin": "source_reference", "context": "narrative" }, { "from": "old-eq:5.13c", "to": "prop:probe-errors", "type": "internal_reference", "evidence": "s preceding slot sum, interpreting each \\(\\mathcal B_p\\) as \\(G_p/H_p\\), with \\(G_p\\) from Equation~\\eqref{old-eq:5.13c}. The stated choice of \\(P_0\\) makes \\(H_p\\ne0\\) there for every slot prime, as proved at the start of the proof. This statement", "anchor": { "source": "paper.tex", "line_start": 8871, "line_end": 8871, "char_start": 406358, "char_end": 406378 }, "origin": "source_reference", "context": "statement" }, { "from": "lem:local-euler", "to": "prop:probe-errors", "type": "candidate_proof_dependency", "evidence": "tion} In this region $\\vartheta=(-w_r)_+\\le6e$. The estimates in the proof of Lemma~\\ref{lem:local-euler} sharpen to $H_p-1=O(Q^{-1-10e})$ for $p\\nmid u$ and $H_p-1=O(Q^{-10e})$ for $p\\mid u$. Increase the fixed $P_0$ so that $|H_p-1|", "anchor": { "source": "paper.tex", "line_start": 8909, "line_end": 8909, "char_start": 407883, "char_end": 407904 }, "origin": "source_reference", "context": "proof" }, { "from": "old-eq:5.13c", "to": "prop:probe-errors", "type": "candidate_proof_dependency", "evidence": "e fixed $P_0$ so that $|H_p-1|<1/2$ for all these primes. The holomorphic factor $G_p$ in Equation~\\eqref{old-eq:5.13c}, divided by $H_p$, now defines $\\mathcal B_p$ throughout this region. It is $H_p$, not the possibly singular raw factor $P_p$, t", "anchor": { "source": "paper.tex", "line_start": 8913, "line_end": 8913, "char_start": 408104, "char_end": 408124 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:holomorphic-selected-tuple", "to": "prop:probe-errors", "type": "candidate_proof_dependency", "evidence": "factor $P_p$, that is bounded away from zero when $w_r<0$. Consequently the tuple sum in Equation~\\eqref{eq:holomorphic-selected-tuple} factors as $\\mathcal H_{\\eta,u}\\prod_i\\mathcal B_i$ on this dynamic region. Substituting $\\mathcal B_i=\\mathcal Q_i+\\mathcal D_i", "anchor": { "source": "paper.tex", "line_start": 8918, "line_end": 8919, "char_start": 408338, "char_end": 408375 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:dynamic-compensated-decomposition", "to": "prop:probe-errors", "type": "candidate_proof_dependency", "evidence": "B_i$ on this dynamic region. Substituting $\\mathcal B_i=\\mathcal Q_i+\\mathcal D_i$ proves Equation~\\eqref{eq:dynamic-compensated-decomposition}. This factorization is not used to continue the correction outside the present region. For $p\\nmid u$, Equation~\\eqref{old-eq:5", "anchor": { "source": "paper.tex", "line_start": 8921, "line_end": 8921, "char_start": 408523, "char_end": 408567 }, "origin": "source_reference", "context": "proof" }, { "from": "old-eq:5.13b", "to": "prop:probe-errors", "type": "candidate_proof_dependency", "evidence": "zation is not used to continue the correction outside the present region. For $p\\nmid u$, Equation~\\eqref{old-eq:5.13b} and the estimate for $\\mathcal E_p$ give \\[ |H_p(\\mathcal B_p+\\overline{\\chi_p(u)})| \\ll Q^{-x_r+2\\vartheta}+Q^{-6z_r+2\\varthet", "anchor": { "source": "paper.tex", "line_start": 8924, "line_end": 8924, "char_start": 408682, "char_end": 408702 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:fixed-numerator-ray", "to": "prop:probe-errors", "type": "candidate_proof_dependency", "evidence": "ulus times $\\operatorname{rad}(u)$. It agrees with the Kummer character $\\chi_\\bullet(u)$ of Lemma~\\ref{lem:fixed-numerator-ray} on ideals avoiding $S$. These characters induce the same primitive character: in the principal ideal ring $\\mathcal O$, the Chin", "anchor": { "source": "paper.tex", "line_start": 8990, "line_end": 8990, "char_start": 411550, "char_end": 411579 }, "origin": "source_reference", "context": "proof" }, { "from": "old-eq:5.13d", "to": "prop:probe-errors", "type": "candidate_proof_dependency", "evidence": "s. They are distinct because the slot supports are disjoint. Apply the single inequality Equation~\\eqref{old-eq:5.13d} to this entire $J_0$ before summing the labels. The numerator together with these local factors, including $q_p^{z-1}$ at each s", "anchor": { "source": "paper.tex", "line_start": 9028, "line_end": 9028, "char_start": 413378, "char_end": 413398 }, "origin": "source_reference", "context": "proof" }, { "from": "old-eq:5.13e", "to": "prop:probe-errors", "type": "candidate_proof_dependency", "evidence": "^\\epsilon$, while the coprime labels have the stronger exponent $z_r-51/100$. This proves Equation~\\eqref{old-eq:5.13e}. This argument uses triangle only on error labels. It never changes the physical row or the coefficients and masks in any main", "anchor": { "source": "paper.tex", "line_start": 9041, "line_end": 9041, "char_start": 414020, "char_end": 414040 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:local-euler", "to": "prop:probe-errors", "type": "candidate_proof_dependency", "evidence": "ain slot. The tuple-independent factor $\\mathcal H_{\\eta,u}$ separately costs $U^\\epsilon$ by Lemma~\\ref{lem:local-euler}. Relative to the central scale $P_i^{z_r-1/2}$ of a main slot, an error slot thus has no positive amplitude exponent in the late", "anchor": { "source": "paper.tex", "line_start": 9046, "line_end": 9046, "char_start": 414266, "char_end": 414287 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:local-euler", "to": "sec:local-compensation", "type": "internal_reference", "evidence": "row count. \\paragraph{The principal local factor.} In the second Euler region of Lemma~\\ref{lem:local-euler}, take $u=1$ and $p\\in1_T$. Then $v=1$ in Equation~\\eqref{old-eq:5.13b}, and $H_p$ is bounded away from zero after the same fixed", "anchor": { "source": "paper.tex", "line_start": 9052, "line_end": 9052, "char_start": 414519, "char_end": 414540 }, "origin": "source_reference", "context": "narrative" }, { "from": "old-eq:5.13b", "to": "sec:local-compensation", "type": "internal_reference", "evidence": "cond Euler region of Lemma~\\ref{lem:local-euler}, take $u=1$ and $p\\in1_T$. Then $v=1$ in Equation~\\eqref{old-eq:5.13b}, and $H_p$ is bounded away from zero after the same fixed enlargement of $P_0$. For this principal row on this region, define $\\", "anchor": { "source": "paper.tex", "line_start": 9054, "line_end": 9054, "char_start": 414592, "char_end": 414612 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:unselected-local-product", "to": "sec:local-compensation", "type": "internal_reference", "evidence": "and the unselected product has a bounded positive majorant by the second-region version of Equation~\\eqref{eq:unselected-local-product}. Ideal counting in each annular slot therefore gives, on every fixed real box in the second Euler region and uniformly in all im", "anchor": { "source": "paper.tex", "line_start": 9082, "line_end": 9082, "char_start": 415852, "char_end": 415887 }, "origin": "source_reference", "context": "narrative" }, { "from": "sec:compensation", "to": "lem:compensated-absolute-local-bounds", "type": "internal_reference", "evidence": "_u\\asymp U\\ge1$, with fixed comparison constants. Retain the fixed physical slot system of Section~\\ref{sec:compensation}. The following bounds are uniform in all imaginary parts and unit phases. \\item On \\[ x_r=\\beta_*+e,\\qquad w_", "anchor": { "source": "paper.tex", "line_start": 9105, "line_end": 9105, "char_start": 417013, "char_end": 417035 }, "origin": "source_reference", "context": "statement" }, { "from": "eq:unselected-local-product", "to": "lem:compensated-absolute-local-bounds", "type": "candidate_proof_dependency", "evidence": "Euler region with $\\epsilon_0=3/8$. Its primewise defect estimates give $H_p=O(1)$, while Equation~\\eqref{eq:unselected-local-product} bounds the positive product of all unselected factors by $O_\\epsilon(U^\\epsilon)$. Consider first $x_r=\\beta_*+e$, $w_r=1/2$, an", "anchor": { "source": "paper.tex", "line_start": 9136, "line_end": 9136, "char_start": 418169, "char_end": 418204 }, "origin": "source_reference", "context": "proof" }, { "from": "old-eq:5.13b", "to": "lem:compensated-absolute-local-bounds", "type": "candidate_proof_dependency", "evidence": "$. Consider first $x_r=\\beta_*+e$, $w_r=1/2$, and $z_r=17/50$. For a selected $p\\nmid u$, Equation~\\eqref{old-eq:5.13b} gives \\[ |G_p+\\overline{\\chi_p(u)}H_p| \\ll Q^{-x_r}+Q^{-6z_r} +Q^{4-5x_r-6z_r}+Q^{1-w_r-6z_r}. \\] The four exponents are", "anchor": { "source": "paper.tex", "line_start": 9140, "line_end": 9140, "char_start": 418385, "char_end": 418405 }, "origin": "source_reference", "context": "proof" }, { "from": "old-eq:5.13c", "to": "lem:compensated-absolute-local-bounds", "type": "candidate_proof_dependency", "evidence": "t division by $H_p$. For a selected $p\\mid u$, one has $W=D=0$ and $\\mathcal E_p=P_p^*$. Equation~\\eqref{old-eq:5.13c} becomes \\[ G_p=\\overline{\\eta(p)}Q^x(1-V)\\mathcal E_p-Q^{-w}H_p. \\] After multiplication by $Q^{x_r}$, the strict term of the lo", "anchor": { "source": "paper.tex", "line_start": 9151, "line_end": 9151, "char_start": 418733, "char_end": 418753 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:absolute-local-tuple-bound", "to": "lem:compensated-absolute-local-bounds", "type": "candidate_proof_dependency", "evidence": "d product and distributing the requested $\\epsilon$ among the fixed number of slots proves Equation~\\eqref{eq:absolute-local-tuple-bound} on the first lines. Now fix $x_r=w_r=2$ and $z_r=z_\\infty>2$. For a selected prime off $u$, the four exponents in Equation~\\eqr", "anchor": { "source": "paper.tex", "line_start": 9183, "line_end": 9183, "char_start": 419971, "char_end": 420008 }, "origin": "source_reference", "context": "proof" }, { "from": "old-eq:5.13b", "to": "lem:compensated-absolute-local-bounds", "type": "candidate_proof_dependency", "evidence": "fix $x_r=w_r=2$ and $z_r=z_\\infty>2$. For a selected prime off $u$, the four exponents in Equation~\\eqref{old-eq:5.13b} are $-2,-6z_\\infty,-6-6z_\\infty,-1-6z_\\infty$, so $G_p=O_{z_\\infty}(1)$. On $u$, the strict term after multiplication by $Q^{x_r", "anchor": { "source": "paper.tex", "line_start": 9186, "line_end": 9186, "char_start": 420134, "char_end": 420154 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:unselected-local-product", "to": "lem:compensated-absolute-local-bounds", "type": "candidate_proof_dependency", "evidence": "s well. A positive slot sum is consequently $O_{z_\\infty}(P_i^{z_\\infty})$. Together with Equation~\\eqref{eq:unselected-local-product}, this proves Equation~\\eqref{eq:absolute-local-tuple-bound} on the second lines. Equation~\\eqref{eq:compensated-poi", "anchor": { "source": "paper.tex", "line_start": 9194, "line_end": 9194, "char_start": 420640, "char_end": 420675 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:absolute-local-tuple-bound", "to": "lem:compensated-absolute-local-bounds", "type": "candidate_proof_dependency", "evidence": "(P_i^{z_\\infty})$. Together with Equation~\\eqref{eq:unselected-local-product}, this proves Equation~\\eqref{eq:absolute-local-tuple-bound} on the second lines. Equation~\\eqref{eq:compensated-poisson-identity} supplies the full high representation of the", "anchor": { "source": "paper.tex", "line_start": 9195, "line_end": 9195, "char_start": 420698, "char_end": 420735 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:compensated-poisson-identity", "to": "sec:local-compensation", "type": "internal_reference", "evidence": "s proves Equation~\\eqref{eq:absolute-local-tuple-bound} on the second lines. Equation~\\eqref{eq:compensated-poisson-identity} supplies the full high representation of the physical function just bounded on the low side. Proposition~\\ref{prop:probe-errors}", "anchor": { "source": "paper.tex", "line_start": 9199, "line_end": 9199, "char_start": 420780, "char_end": 420819 }, "origin": "source_reference", "context": "narrative" }, { "from": "prop:probe-errors", "to": "sec:local-compensation", "type": "internal_reference", "evidence": "ies the full high representation of the physical function just bounded on the low side. Proposition~\\ref{prop:probe-errors} controls the error factors jointly with the numerator, and Lemma~\\ref{lem:compensated-absolute-local-bounds} supplies the bounds", "anchor": { "source": "paper.tex", "line_start": 9201, "line_end": 9201, "char_start": 420925, "char_end": 420948 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:compensated-absolute-local-bounds", "to": "sec:local-compensation", "type": "internal_reference", "evidence": "roposition~\\ref{prop:probe-errors} controls the error factors jointly with the numerator, and Lemma~\\ref{lem:compensated-absolute-local-bounds} supplies the bounds on the remaining contour lines. The main factors are the prime polynomials displayed in Proposition~\\ref{prop", "anchor": { "source": "paper.tex", "line_start": 9202, "line_end": 9202, "char_start": 421014, "char_end": 421057 }, "origin": "source_reference", "context": "narrative" }, { "from": "prop:probe-errors", "to": "sec:local-compensation", "type": "internal_reference", "evidence": "on the remaining contour lines. The main factors are the prime polynomials displayed in Proposition~\\ref{prop:probe-errors}. To count rows on which those factors and a detector witness are large, we next prove the inverse and fourth-moment estimates use", "anchor": { "source": "paper.tex", "line_start": 9204, "line_end": 9204, "char_start": 421178, "char_end": 421201 }, "origin": "source_reference", "context": "narrative" }, { "from": "sec:refined-row-counts", "to": "sec:local-compensation", "type": "internal_reference", "evidence": "a detector witness are large, we next prove the inverse and fourth-moment estimates used in Section~\\ref{sec:refined-row-counts}. \\section{The inverse moment with prime factors} We now bound an inverse Dirichlet polynomial multiplied by", "anchor": { "source": "paper.tex", "line_start": 9207, "line_end": 9207, "char_start": 421344, "char_end": 421372 }, "origin": "source_reference", "context": "narrative" }, { "from": "sec:detector", "to": "sec:inverse", "type": "internal_reference", "evidence": "iplied by independently weighted prime sums. The estimate applies to the inverse witness in Section~\\ref{sec:detector} together with selected main factors from the local compensation. Its proof uses the completed row energy of Section~\\ref{sec:mark", "anchor": { "source": "paper.tex", "line_start": 9213, "line_end": 9213, "char_start": 421592, "char_end": 421610 }, "origin": "source_reference", "context": "narrative" }, { "from": "sec:marked-energy", "to": "sec:inverse", "type": "internal_reference", "evidence": "lected main factors from the local compensation. Its proof uses the completed row energy of Section~\\ref{sec:marked-energy}. A final amplification gives an unmarked estimate on sixth-power-free rows at longer column lengths. \\subsection{Statement of th", "anchor": { "source": "paper.tex", "line_start": 9215, "line_end": 9215, "char_start": 421727, "char_end": 421750 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:canonical-moment", "to": "sec:inverse", "type": "internal_reference", "evidence": "ext, in which a fourth-power residue symbol is averaged over an additional squarefree ideal. Lemma~\\ref{lem:canonical-moment} is proved by a terminal application of Lemma~\\ref{lem:reflected-energy} and a recursive step using two masked Poisson transformat", "anchor": { "source": "paper.tex", "line_start": 9273, "line_end": 9273, "char_start": 424273, "char_end": 424299 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:reflected-energy", "to": "sec:inverse", "type": "internal_reference", "evidence": "al squarefree ideal. Lemma~\\ref{lem:canonical-moment} is proved by a terminal application of Lemma~\\ref{lem:reflected-energy} and a recursive step using two masked Poisson transformations. After the principal contributions and tails have been handled, th", "anchor": { "source": "paper.tex", "line_start": 9274, "line_end": 9274, "char_start": 424345, "char_end": 424371 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:marked", "to": "sec:inverse", "type": "internal_reference", "evidence": "er row range. A separate initialization using one masked Poisson transformation then deduces Lemma~\\ref{lem:marked}, and the final subsection gives its longer unmarked consequence for sixth-power-free rows. \\subsection{The canonical estimate}", "anchor": { "source": "paper.tex", "line_start": 9278, "line_end": 9278, "char_start": 424687, "char_end": 424703 }, "origin": "source_reference", "context": "narrative" }, { "from": "sec:marked-energy", "to": "sec:inverse", "type": "internal_reference", "evidence": "the fixed puncture, squarefree coefficient, row character, and product-form mark defined in Section~\\ref{sec:marked-energy}. The additional squarefree label is now averaged, with a divisor-bounded weight depending on that label alone. The following is", "anchor": { "source": "paper.tex", "line_start": 9285, "line_end": 9285, "char_start": 424937, "char_end": 424960 }, "origin": "source_reference", "context": "narrative" }, { "from": "old-eq:4.2", "to": "lem:canonical-moment", "type": "internal_reference", "evidence": "puncture. Both are independent of \\(k,f\\). Let \\(\\mathfrak d\\) have the product form in Equation~\\eqref{old-eq:4.2}, with disjoint fixed prime lists of total length at most \\(z_0\\) and bounded coefficients independent of \\(k,f\\). No other row-de", "anchor": { "source": "paper.tex", "line_start": 9309, "line_end": 9309, "char_start": 426337, "char_end": 426355 }, "origin": "source_reference", "context": "statement" }, { "from": "sec:marked-energy", "to": "sec:inverse", "type": "internal_reference", "evidence": "original cap \\(z_0\\), including the empty collection. The reflected energy in Section~\\ref{sec:marked-energy} supplies the terminal estimate for this family. The remaining ranges will be reduced to the same family with shorter rows. Before", "anchor": { "source": "paper.tex", "line_start": 9342, "line_end": 9342, "char_start": 427723, "char_end": 427746 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:finite-seminorm-propagation", "to": "cor:indexed-seminorm-propagation", "type": "internal_reference", "evidence": "f} [Indexed finite propagation] In Lemma~\\ref{lem:finite-seminorm-propagation}, let $\\boldsymbol\\ell$ denote any admissible length and outer labels at a vertex $v$, and let $\\Phi_v(\\boldsymbol\\ell)$ be its de", "anchor": { "source": "paper.tex", "line_start": 9488, "line_end": 9488, "char_start": 434279, "char_end": 434316 }, "origin": "source_reference", "context": "statement" }, { "from": "lem:finite-seminorm-propagation", "to": "cor:indexed-seminorm-propagation", "type": "internal_reference", "evidence": "_e$ may vary with $Z$. Then $\\mathcal N_v/Z^{\\Phi_v(\\boldsymbol\\ell)}$ has the conclusion of Lemma~\\ref{lem:finite-seminorm-propagation}, uniformly in $\\boldsymbol\\ell$, with terminal exponents $\\delta_v$ and edge exponents $\\delta_e+\\delta_e^{\\mathrm{mass}}$. In p", "anchor": { "source": "paper.tex", "line_start": 9527, "line_end": 9527, "char_start": 436221, "char_end": 436258 }, "origin": "source_reference", "context": "statement" }, { "from": "eq:indexed-exponent-defect", "to": "cor:indexed-seminorm-propagation", "type": "candidate_proof_dependency", "evidence": "rline{\\mathcal N}_v=Z^{-\\Phi_v(\\boldsymbol\\ell)}\\mathcal N_v$. Substitution in an edge and Equation~\\eqref{eq:indexed-exponent-defect} bound its norm power by $Z^{\\delta_e}$ times the product of the normalized children. Insert their inductive bounds. As in the p", "anchor": { "source": "paper.tex", "line_start": 9535, "line_end": 9536, "char_start": 436628, "char_end": 436662 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:finite-seminorm-propagation", "to": "cor:indexed-seminorm-propagation", "type": "candidate_proof_dependency", "evidence": "oduct of the normalized children. Insert their inductive bounds. As in the product proof of Lemma~\\ref{lem:finite-seminorm-propagation}, the required coefficient moment is \\[ H_e=\\sum_r\\theta_{e,r} \\bigl(H_r+B_r c'_{e,r}(J_r)\\bigr), \\] which is finit", "anchor": { "source": "paper.tex", "line_start": 9538, "line_end": 9538, "char_start": 436814, "char_end": 436851 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:indexed-coefficient-measure", "to": "cor:indexed-seminorm-propagation", "type": "candidate_proof_dependency", "evidence": "\\bigl(H_r+B_r c'_{e,r}(J_r)\\bigr), \\] which is finite and independent of $\\gamma$. Equation~\\eqref{eq:indexed-coefficient-measure} at this order contributes $Z^{\\delta_e^{\\mathrm{mass}}}$ and only finite profile and height orders. Backward induction over the", "anchor": { "source": "paper.tex", "line_start": 9544, "line_end": 9545, "char_start": 437022, "char_end": 437060 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:indexed-coefficient-measure", "to": "sec:inverse", "type": "internal_reference", "evidence": "Gamma(\\gamma). Uniform weighted moments of $K_\\gamma$ now give Equation~\\eqref{eq:indexed-coefficient-measure} with only the remaining $Z^\\epsilon$ mass. Thus a displayed exponent that already includes the count $c$ must use this normalize", "anchor": { "source": "paper.tex", "line_start": 9576, "line_end": 9576, "char_start": 438572, "char_end": 438610 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:kernel-seminorms", "to": "sec:inverse", "type": "internal_reference", "evidence": "scale, or a radial $C_c^\\infty$ test constant near zero at that scale. After normalization, Lemma~\\ref{lem:kernel-seminorms} applies to its fixed transform, and moving scale ratios enter only as in Equation~\\eqref{eq:normalized-kernel-profile}. For exam", "anchor": { "source": "paper.tex", "line_start": 9649, "line_end": 9650, "char_start": 442608, "char_end": 442634 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:normalized-kernel-profile", "to": "sec:inverse", "type": "internal_reference", "evidence": "kernel-seminorms} applies to its fixed transform, and moving scale ratios enter only as in Equation~\\eqref{eq:normalized-kernel-profile}. For example $q_u^{-1/2}=U_0^{-1/2}y^{-1/2}$ on $q_u=U_0y$; the central power belongs to the norm exponent, and the annular fact", "anchor": { "source": "paper.tex", "line_start": 9651, "line_end": 9651, "char_start": 442717, "char_end": 442753 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:smooth-calculus", "to": "sec:inverse", "type": "internal_reference", "evidence": "endence on the individual labels, its Fourier density is common to those rows, as required by Lemma~\\ref{lem:smooth-calculus}; separate uniform bounds for row-dependent densities would not imply the same Hilbert-space inequality. Nonsmooth masks are reta", "anchor": { "source": "paper.tex", "line_start": 9671, "line_end": 9671, "char_start": 443973, "char_end": 443998 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:late-fourier-tail", "to": "sec:inverse", "type": "internal_reference", "evidence": "\\tau<\\delta/(1+H)$. Then $(1+T_1)^H\\le2^H Z^\\delta$ for $T_1=Z^\\tau$. After this choice, Equation~\\eqref{eq:late-fourier-tail} with weight $J_0$ and any fixed integer $N>(B_0+M)/\\tau$ makes an integrated external tail $O(Z^{-M})$ for any prescribed $M>0$.", "anchor": { "source": "paper.tex", "line_start": 9701, "line_end": 9701, "char_start": 445755, "char_end": 445783 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:pointwise-mellin-tail", "to": "sec:inverse", "type": "internal_reference", "evidence": "n integrated external tail $O(Z^{-M})$ for any prescribed $M>0$. For a horizontal join use Equation~\\eqref{eq:pointwise-mellin-tail} with the additional fixed height weight, or Equation~\\eqref{eq:joint-gaussian-slice} for a joint Gaussian slice. When other coor", "anchor": { "source": "paper.tex", "line_start": 9704, "line_end": 9704, "char_start": 445948, "char_end": 445980 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:joint-gaussian-slice", "to": "sec:inverse", "type": "internal_reference", "evidence": "use Equation~\\eqref{eq:pointwise-mellin-tail} with the additional fixed height weight, or Equation~\\eqref{eq:joint-gaussian-slice} for a joint Gaussian slice. When other coordinates on such a slice are extended to the whole real line, the accompanying arithme", "anchor": { "source": "paper.tex", "line_start": 9705, "line_end": 9706, "char_start": 446034, "char_end": 446065 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:marked", "to": "sec:inverse", "type": "internal_reference", "evidence": "sserted order of dependence. \\subsection{Finite Poisson summation with a mask} We now prove Lemma~\\ref{lem:marked}, using the arithmetic conventions of Section~\\ref{sec:arithmetic} and Lemma~\\ref{lem:reflected-energy} for the terminal case. Th", "anchor": { "source": "paper.tex", "line_start": 9716, "line_end": 9716, "char_start": 446628, "char_end": 446644 }, "origin": "source_reference", "context": "narrative" }, { "from": "sec:arithmetic", "to": "sec:inverse", "type": "internal_reference", "evidence": "tion with a mask} We now prove Lemma~\\ref{lem:marked}, using the arithmetic conventions of Section~\\ref{sec:arithmetic} and Lemma~\\ref{lem:reflected-energy} for the terminal case. The recursive reduction uses two applications of the following form", "anchor": { "source": "paper.tex", "line_start": 9717, "line_end": 9717, "char_start": 446690, "char_end": 446710 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:reflected-energy", "to": "sec:inverse", "type": "internal_reference", "evidence": "Lemma~\\ref{lem:marked}, using the arithmetic conventions of Section~\\ref{sec:arithmetic} and Lemma~\\ref{lem:reflected-energy} for the terminal case. The recursive reduction uses two applications of the following form of Poisson summation, and the initial", "anchor": { "source": "paper.tex", "line_start": 9717, "line_end": 9717, "char_start": 446721, "char_end": 446747 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:masked-poisson", "to": "lem:masked-poisson", "type": "candidate_proof_dependency", "evidence": "is zero by primitivity; for \\(m=1\\) the stated principal convention applies. This proves Equation~\\eqref{eq:masked-poisson}. For a nonprincipal primitive character, finite Parseval gives \\(\\sum_h|\\sum_x\\psi(x)e(hx/m)|^2=q_m\\sum_x|\\psi(x)|^2\\). Writing \\", "anchor": { "source": "paper.tex", "line_start": 9757, "line_end": 9757, "char_start": 448507, "char_end": 448532 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:canonical-moment", "to": "sec:inverse", "type": "internal_reference", "evidence": "ipal bound. \\subsection{The canonical reduction and its induction} We now prove Lemma~\\ref{lem:canonical-moment}. The reflected energy handles the short completion; the remaining blocks undergo two Poisson transformations to produce admissib", "anchor": { "source": "paper.tex", "line_start": 9785, "line_end": 9785, "char_start": 449827, "char_end": 449853 }, "origin": "source_reference", "context": "narrative" }, { "from": "cor:indexed-seminorm-propagation", "to": "lem:canonical-moment", "type": "candidate_proof_dependency", "evidence": "of of Lemma~\\ref{lem:canonical-moment}] The order of the analytic choices is organized by Corollary~\\ref{cor:indexed-seminorm-propagation} in Section~\\ref{sec:moment-propagation}. Its indexed conclusion will be used on each positive Cauchy-side sum separately; the pro", "anchor": { "source": "paper.tex", "line_start": 9791, "line_end": 9791, "char_start": 450138, "char_end": 450176 }, "origin": "source_reference", "context": "proof" }, { "from": "sec:moment-propagation", "to": "lem:canonical-moment", "type": "candidate_proof_dependency", "evidence": "of the analytic choices is organized by Corollary~\\ref{cor:indexed-seminorm-propagation} in Section~\\ref{sec:moment-propagation}. Its indexed conclusion will be used on each positive Cauchy-side sum separately; the proof below verifies the required common co", "anchor": { "source": "paper.tex", "line_start": 9792, "line_end": 9792, "char_start": 450188, "char_end": 450216 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:inverse-old-label-fibre", "to": "lem:canonical-moment", "type": "candidate_proof_dependency", "evidence": "measure \\(w(f)\\), and that factor is retained exactly once in every parent \\(f\\)-sum until Equation~\\eqref{eq:inverse-old-label-fibre} removes the old label. At each node keep the parameters \\(N,V,M\\) fixed, and write \\[ F=N+V,\\qquad Q=\\log_Zq_{\\mathfrak r_\\rho}", "anchor": { "source": "paper.tex", "line_start": 9798, "line_end": 9799, "char_start": 450577, "char_end": 450611 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:marked-completion", "to": "lem:canonical-moment", "type": "candidate_proof_dependency", "evidence": "l column norm. \\paragraph{The short completion.} M\\\"obius inversion of the cube factor in Equation~\\eqref{eq:marked-completion} gives the exact identity &Z^{-N/2}\\sum_{n\\ {\\rm sf}}a(n)\\c", "anchor": { "source": "paper.tex", "line_start": 9840, "line_end": 9840, "char_start": 452373, "char_end": 452401 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:reflected-energy", "to": "lem:canonical-moment", "type": "candidate_proof_dependency", "evidence": "hat h\\le\\ell_1+\\eta1\\), \\begin{equa", "anchor": { "source": "paper.tex", "line_start": 10123, "line_end": 10123, "char_start": 466869, "char_end": 466895 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:normalized-kernel-profile", "to": "lem:canonical-moment", "type": "candidate_proof_dependency", "evidence": "sson steps, these are the normalized-kernel bounds of Lemma~\\ref{lem:kernel-seminorms} and Equation~\\eqref{eq:normalized-kernel-profile}. The raw tail estimate we shall use is, for \\(Y\\ge1\\) and \\(A>1\\), \\sum_{h\\ne0:\\,a", "anchor": { "source": "paper.tex", "line_start": 10124, "line_end": 10124, "char_start": 466909, "char_end": 466945 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:canonical-block", "to": "lem:canonical-moment", "type": "candidate_proof_dependency", "evidence": "are fixed at the end. \\paragraph{The first Poisson transformation.} Expand the square in Equation~\\eqref{eq:canonical-block}. Put \\[ \\mathfrak B=\\operatorname{rad}(b_1b_2),\\qquad n_i=\\mathfrak A_i C u_i\\quad(i=1,2), \\] where \\(\\mathfrak A_i=(n_i,\\math", "anchor": { "source": "paper.tex", "line_start": 10146, "line_end": 10146, "char_start": 468100, "char_end": 468126 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:masked-poisson", "to": "lem:canonical-moment", "type": "candidate_proof_dependency", "evidence": "and the two factors \\(\\beta(b_1,f)\\overline{\\beta(b_2,f)}\\) remain in the expression. Apply Lemma~\\ref{lem:masked-poisson} with \\(K=Z^M\\), and denote its divisor by \\(d_k\\mid C\\mathfrak B/R_1\\), with center \\(\\delta\\) and \\(\\widehat\\delta=\\log_Zq_{d_k}", "anchor": { "source": "paper.tex", "line_start": 10180, "line_end": 10180, "char_start": 469550, "char_end": 469574 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:canonical-block", "to": "lem:canonical-moment", "type": "candidate_proof_dependency", "evidence": "equal columns, all \\(O(Z^{V+\\epsilon})\\) labels, and \\(O(Z^M)\\) rows, the normalization in Equation~\\eqref{eq:canonical-block} gives \\(O(Z^{M+\\epsilon})\\). If \\(r<0\\) but its fixed annular column window is nonempty, then \\(Z^{-r}\\) is bounded by its fixed", "anchor": { "source": "paper.tex", "line_start": 10198, "line_end": 10198, "char_start": 470527, "char_end": 470553 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:masked-poisson", "to": "lem:canonical-moment", "type": "candidate_proof_dependency", "evidence": "d below, they will carry the same outer ball mask as the nonprincipal terms; the last part of Lemma~\\ref{lem:masked-poisson} bounds them at this same cost. We next identify its column coefficients. For coprime squarefree primary \\(a,b\\), CRT and recipr", "anchor": { "source": "paper.tex", "line_start": 10205, "line_end": 10205, "char_start": 470984, "char_end": 471008 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:signal", "to": "lem:canonical-moment", "type": "candidate_proof_dependency", "evidence": "ro-extended symbols. Put \\(P_T=\\prod_{t_p\\ne0}p^{t_p}\\), regarded in the fixed ray group. Equation~\\eqref{eq:signal} and Equation~\\eqref{eq:G-quadratic-refinement} show that the remaining two-column ray factor is \\[ G(u_1u_2^{-1})\\mathcal R(u_1u", "anchor": { "source": "paper.tex", "line_start": 10228, "line_end": 10228, "char_start": 471885, "char_end": 471902 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:G-quadratic-refinement", "to": "lem:canonical-moment", "type": "candidate_proof_dependency", "evidence": "=\\prod_{t_p\\ne0}p^{t_p}\\), regarded in the fixed ray group. Equation~\\eqref{eq:signal} and Equation~\\eqref{eq:G-quadratic-refinement} show that the remaining two-column ray factor is \\[ G(u_1u_2^{-1})\\mathcal R(u_1u_2^{-1},P_T). \\] For example, before placing fa", "anchor": { "source": "paper.tex", "line_start": 10229, "line_end": 10229, "char_start": 471916, "char_end": 471949 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:first-poisson-column", "to": "lem:canonical-moment", "type": "candidate_proof_dependency", "evidence": "ter coefficients. The first transform has produced the inverse-type column coefficient in Equation~\\eqref{eq:first-poisson-column}. We next collect its fourth-power factors into one squarefree label before forming the positive row norm for the second transfor", "anchor": { "source": "paper.tex", "line_start": 10289, "line_end": 10289, "char_start": 474139, "char_end": 474170 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:first-poisson-lengths", "to": "lem:canonical-moment", "type": "candidate_proof_dependency", "evidence": "widehat j_2. \\] For \\(y=hf^2E\\), the implication \\(Z^Mq_h/(q_{d_k}q_{m_1})\\le Z^\\tau\\) and Equation~\\eqref{eq:first-poisson-lengths} give \\[ \\widehat y &\\le \\widehat n_1+\\widehat n_2-2\\widehat B-M +\\widehat s-2\\widehat j_2+\\widehat\\delta+", "anchor": { "source": "paper.tex", "line_start": 10338, "line_end": 10338, "char_start": 476244, "char_end": 476276 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:inverse-cube-actual", "to": "lem:canonical-moment", "type": "candidate_proof_dependency", "evidence": "idehat j +\\widehat\\delta+2\\widehat f+\\tau. \\] The second inequality is Equation~\\eqref{eq:inverse-cube-actual}. Define the formal center and the enclosing outer row scale by H_c=2r-2B-M+4", "anchor": { "source": "paper.tex", "line_start": 10349, "line_end": 10349, "char_start": 476606, "char_end": 476636 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:inverse-ratio-bounds", "to": "lem:canonical-moment", "type": "candidate_proof_dependency", "evidence": "-(\\widehat j-j)+(\\widehat\\delta-\\delta)+2(\\widehat f-V), \\] whose positive maximum under Equation~\\eqref{eq:inverse-ratio-bounds} is \\(12\\eta\\). Thus the actual-ratio inequality implies \\(q_y\\le Z^{H_{\\rm use}}\\) for every supported raw column pair. On that", "anchor": { "source": "paper.tex", "line_start": 10361, "line_end": 10361, "char_start": 477091, "char_end": 477122 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:inverse-raw-tail", "to": "lem:canonical-moment", "type": "candidate_proof_dependency", "evidence": "omplement is contained in the actual kernel tail \\(Z^Mq_h/(q_{d_k}q_{m_1})>Z^\\tau\\); apply Equation~\\eqref{eq:inverse-raw-tail} to discard that complement, with the raw counts and order fixed below. No indicator of the ratio inequality is inserted. Inside", "anchor": { "source": "paper.tex", "line_start": 10371, "line_end": 10371, "char_start": 477484, "char_end": 477511 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:inverse-ratio-bounds", "to": "lem:canonical-moment", "type": "candidate_proof_dependency", "evidence": "-V-\\widehat\\delta-\\widehat n_i +\\widehat A_i+\\widehat B-\\widehat R/2. \\] Equation~\\eqref{eq:inverse-ratio-bounds} gives on each side \\widehat\\kappa_i\\le\\kappa_i+\\tfrac92\\eta. \\end{equat", "anchor": { "source": "paper.tex", "line_start": 10426, "line_end": 10426, "char_start": 480211, "char_end": 480242 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:inverse-log-fourier", "to": "lem:canonical-moment", "type": "candidate_proof_dependency", "evidence": "ach real root once, also when the actual column norms differ. Its transform is defined by Equation~\\eqref{eq:inverse-log-fourier} on the ambient nine-dimensional box before any actual \\(f,h\\) or outer ideal is summed. The norms of \\(b_a,f,J,E,q_0\\) occur onl", "anchor": { "source": "paper.tex", "line_start": 10476, "line_end": 10476, "char_start": 482512, "char_end": 482542 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:first-poisson-column", "to": "lem:canonical-moment", "type": "candidate_proof_dependency", "evidence": ",J,E,q_0\\) occur only in outer coefficients, characters, masks or individual cutoffs after Equation~\\eqref{eq:first-poisson-column}; they require no additional coupled-profile coordinate. To expose the arithmetic extraction at \\(t'\\), put \\(y=hf^2E\\) and \\[ B", "anchor": { "source": "paper.tex", "line_start": 10479, "line_end": 10479, "char_start": 482752, "char_end": 482783 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:inverse-slot-priority", "to": "lem:canonical-moment", "type": "candidate_proof_dependency", "evidence": "ed at a zero. The one additional \\(\\mu(t')\\) from the mutual-gcd inversion is outer. Apply Equation~\\eqref{eq:inverse-slot-priority} with the ordered list \\((b_a,\\mathfrak A_a,C,t';x)\\). A surviving prime dividing \\(x\\) is already prime to \\(b_a,\\mathfrak A_a,C", "anchor": { "source": "paper.tex", "line_start": 10501, "line_end": 10501, "char_start": 483765, "char_end": 483797 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:inverse-first-polynomial", "to": "lem:canonical-moment", "type": "candidate_proof_dependency", "evidence": "). It is independent of \\(x_1,x_2\\); its definition is by that product, not by division by Equation~\\eqref{eq:inverse-first-polynomial}. Writing \\(\\mathcal O_1=\\{A_1,A_2,C,d,R,t,h\\}\\), set \\[ w_{1,\\sigma}(\\omega;\\boldsymbol t) =e_\\sigma(\\omega)\\mu(d_k)\\mu(t')B_{1", "anchor": { "source": "paper.tex", "line_start": 10527, "line_end": 10527, "char_start": 485300, "char_end": 485335 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:inverse-first-profile", "to": "lem:canonical-moment", "type": "candidate_proof_dependency", "evidence": "ials restores the two column modes; the seven outer modes restore the other coordinates of Equation~\\eqref{eq:inverse-first-profile}. Its larger cutoffs are one on the retained supports, and the fresh cutoffs are one wherever the old windows are nonzero. This", "anchor": { "source": "paper.tex", "line_start": 10549, "line_end": 10549, "char_start": 486250, "char_end": 486282 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:first-poisson-column", "to": "lem:canonical-moment", "type": "candidate_proof_dependency", "evidence": "d \\(J,q_0\\), the fixed dyadic length \\(A_i\\), and the fixed ray sector: this is precisely Equations~\\eqref{eq:first-poisson-column} and \\eqref{eq:retained-cube-label}, with the norm profiles separated. After the first weighted Cauchy inequality, \\(\\mathcal B_1\\", "anchor": { "source": "paper.tex", "line_start": 10586, "line_end": 10586, "char_start": 488110, "char_end": 488141 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:retained-cube-label", "to": "lem:canonical-moment", "type": "candidate_proof_dependency", "evidence": "\\(A_i\\), and the fixed ray sector: this is precisely Equations~\\eqref{eq:first-poisson-column} and \\eqref{eq:retained-cube-label}, with the norm profiles separated. After the first weighted Cauchy inequality, \\(\\mathcal B_1\\) is simply \\(1_{01+(B_{\\rm crude}+T)/\\tau. \\] Equation~\\eqref{eq:inverse-raw-tail} with \\(Y=Z^\\tau\\) then makes the discarded raw outer-ball complement \\(O(Z^{-T})\\), up to finitely many input seminorms and a fix", "anchor": { "source": "paper.tex", "line_start": 11549, "line_end": 11549, "char_start": 534902, "char_end": 534929 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:inverse-joint-fourier-moments", "to": "lem:canonical-moment", "type": "candidate_proof_dependency", "evidence": "and Euler differentiation of a full kernel at a norm monomial introduces no scale power. Equations~\\eqref{eq:inverse-joint-fourier-moments} and \\eqref{eq:inverse-normalized-triples} supply the common coefficient measure for each positive indexed Cauchy-side sum. Apply", "anchor": { "source": "paper.tex", "line_start": 11565, "line_end": 11566, "char_start": 535739, "char_end": 535779 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:inverse-normalized-triples", "to": "lem:canonical-moment", "type": "candidate_proof_dependency", "evidence": "a norm monomial introduces no scale power. Equations~\\eqref{eq:inverse-joint-fourier-moments} and \\eqref{eq:inverse-normalized-triples} supply the common coefficient measure for each positive indexed Cauchy-side sum. Apply Corollary~\\ref{cor:indexed-seminorm-propa", "anchor": { "source": "paper.tex", "line_start": 11566, "line_end": 11566, "char_start": 535784, "char_end": 535821 }, "origin": "source_reference", "context": "proof" }, { "from": "cor:indexed-seminorm-propagation", "to": "lem:canonical-moment", "type": "candidate_proof_dependency", "evidence": "} supply the common coefficient measure for each positive indexed Cauchy-side sum. Apply Corollary~\\ref{cor:indexed-seminorm-propagation} separately to those sums, using the child-profile bounds just proved from Lemma~\\ref{lem:smooth-calculus}, and then take the disp", "anchor": { "source": "paper.tex", "line_start": 11568, "line_end": 11568, "char_start": 535920, "char_end": 535958 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:smooth-calculus", "to": "lem:canonical-moment", "type": "candidate_proof_dependency", "evidence": "minorm-propagation} separately to those sums, using the child-profile bounds just proved from Lemma~\\ref{lem:smooth-calculus}, and then take the displayed geometric means. The two square roots retain one normalized discrete mass, as already shown, so gen", "anchor": { "source": "paper.tex", "line_start": 11570, "line_end": 11570, "char_start": 536039, "char_end": 536064 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:inverse-terminal-width", "to": "lem:canonical-moment", "type": "candidate_proof_dependency", "evidence": "gh for \\(\\log Z\\ge L_{\\rm win}/\\eta\\), the thresholds for \\(C_{\\rm res},C_{\\rm width}\\) in Equation~\\eqref{eq:inverse-terminal-width}, \\(\\log Z\\ge2\\log C_{\\rm ker}/\\tau_{\\rm ref}\\), \\(\\log Z\\ge4\\log q_\\lambda/\\eta\\), and the fixed \\(h\\)-annulus threshold used in", "anchor": { "source": "paper.tex", "line_start": 11578, "line_end": 11578, "char_start": 536489, "char_end": 536522 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:inverse-child-margins", "to": "lem:canonical-moment", "type": "candidate_proof_dependency", "evidence": "andled by increasing the final constant. Now \\(c_h=c_*-7h\\eta\\ge c_*/2\\) for \\(h\\le D\\). Equations~\\eqref{eq:inverse-child-margins} and \\eqref{eq:inverse-row-decrease} send every retained nonterminal child to the next depth and its row cap. At depth \\(D\\) that", "anchor": { "source": "paper.tex", "line_start": 11587, "line_end": 11587, "char_start": 536990, "char_end": 537022 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:inverse-row-decrease", "to": "lem:canonical-moment", "type": "candidate_proof_dependency", "evidence": "nt. Now \\(c_h=c_*-7h\\eta\\ge c_*/2\\) for \\(h\\le D\\). Equations~\\eqref{eq:inverse-child-margins} and \\eqref{eq:inverse-row-decrease} send every retained nonterminal child to the next depth and its row cap. At depth \\(D\\) that cap is negative, whereas every reta", "anchor": { "source": "paper.tex", "line_start": 11588, "line_end": 11588, "char_start": 537027, "char_end": 537058 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:marked", "to": "sec:inverse", "type": "internal_reference", "evidence": "} \\subsection{Initialization of the marked moment} We now convert the inverse polynomial in Lemma~\\ref{lem:marked} to the canonical family. Only one Poisson transformation is required. [Proof of Lemma~\\ref{lem:marked}] In the pro", "anchor": { "source": "paper.tex", "line_start": 11603, "line_end": 11603, "char_start": 537738, "char_end": 537754 }, "origin": "source_reference", "context": "narrative" }, { "from": "old-eq:4.2", "to": "lem:marked", "type": "candidate_proof_dependency", "evidence": "incorporated into their bounded coefficients. The coefficients thus remain of the form in Equation~\\eqref{old-eq:4.2}. For this fixed assigned subset define \\(G\\) from its nominal slot centers: \\[ G=\\sum_{i\\ {\\rm assigned}}z_i,\\qquad z_0=z-G,\\q", "anchor": { "source": "paper.tex", "line_start": 11621, "line_end": 11621, "char_start": 538459, "char_end": 538477 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:inverse-log-fourier", "to": "lem:marked", "type": "candidate_proof_dependency", "evidence": "t cutoff \\(\\omega_c\\) and the individual factors \\(W_i(y_i)\\) stay outside its transform. Equation~\\eqref{eq:inverse-log-fourier}, in dimension \\(1+|I_0|\\), therefore expresses Equation~\\eqref{eq:initial-overlap-polynomial} exactly as the same assigned scalar", "anchor": { "source": "paper.tex", "line_start": 11686, "line_end": 11686, "char_start": 541317, "char_end": 541347 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:initial-overlap-polynomial", "to": "lem:marked", "type": "candidate_proof_dependency", "evidence": "m. Equation~\\eqref{eq:inverse-log-fourier}, in dimension \\(1+|I_0|\\), therefore expresses Equation~\\eqref{eq:initial-overlap-polynomial} exactly as the same assigned scalar and \\(Z^{-G}\\psi_u(j)^2\\), times \\[ (2\\pi)^{-1-|I_0|}\\int_{\\mathbb R^{1+|I_0|}} \\widehat{\\m", "anchor": { "source": "paper.tex", "line_start": 11687, "line_end": 11688, "char_start": 541404, "char_end": 541441 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:inverse-fourier-moment", "to": "lem:marked", "type": "candidate_proof_dependency", "evidence": "sen for an actual surviving tuple. The measure may depend on the already fixed \\(j\\), but Equation~\\eqref{eq:inverse-fourier-moment} is uniform for its \\(y_j\\) in the fixed product interval. Minkowski's inequality is used in the whole row Hilbert space for this", "anchor": { "source": "paper.tex", "line_start": 11701, "line_end": 11702, "char_start": 542121, "char_end": 542154 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:masked-poisson", "to": "lem:marked", "type": "candidate_proof_dependency", "evidence": "\\). The row character has primitive modulus \\(z_1z_2\\) and remaining zero mask \\(C\\). Apply Lemma~\\ref{lem:masked-poisson}, with \\(d'\\mid C\\). Let \\(\\theta\\ge0\\) be its center and \\(\\widehat\\theta=\\log_Zq_{d'}\\). Set \\[ P_1=B-\\theta\\ge-2\\eta, \\] whe", "anchor": { "source": "paper.tex", "line_start": 11734, "line_end": 11734, "char_start": 543927, "char_end": 543951 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:inverse-raw-tail", "to": "lem:marked", "type": "candidate_proof_dependency", "evidence": ". \\] Its complement is contained in the actual ratio tail above for every supported pair. Equation~\\eqref{eq:inverse-raw-tail}, with the separate initial raw count and order specified below, discards precisely that complement. Retain the full smooth kerne", "anchor": { "source": "paper.tex", "line_start": 11765, "line_end": 11765, "char_start": 545198, "char_end": 545225 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:masked-poisson", "to": "lem:marked", "type": "candidate_proof_dependency", "evidence": "the same \\(\\mathcal B_{\\rm init}\\) mask and are bounded by the full principal restoration in Lemma~\\ref{lem:masked-poisson}, since \\(Z^m\\ge1\\). Insert frequency dyads of \\(q_{h'}\\) from one partition common to all \\(d'\\), not a label-recentered partiti", "anchor": { "source": "paper.tex", "line_start": 11793, "line_end": 11793, "char_start": 546584, "char_end": 546608 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:initial-residual-polynomial", "to": "lem:marked", "type": "candidate_proof_dependency", "evidence": "on the current \\(C,d',h'\\) dyads, evaluated at their normalized norms. The orientation in Equation~\\eqref{eq:initial-residual-polynomial} is now fixed. The calculation in the second canonical Poisson transformation replaces \\(\\mu(z)\\gamma_{-1}(z)\\) by \\(\\bar\\alpha(z", "anchor": { "source": "paper.tex", "line_start": 11803, "line_end": 11804, "char_start": 547106, "char_end": 547144 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:initial-formal-component", "to": "lem:marked", "type": "candidate_proof_dependency", "evidence": "plement removed before this equality were bounded separately above. Define the summand in Equation~\\eqref{eq:initial-formal-component} on all individually squarefree \\(z_1,z_2\\) prime to \\(Cj\\) by the displayed separate \\(A_*\\) factors, characters, formal product", "anchor": { "source": "paper.tex", "line_start": 11846, "line_end": 11847, "char_start": 549139, "char_end": 549174 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:initial-common-extraction", "to": "lem:marked", "type": "candidate_proof_dependency", "evidence": "d f=d's,\\qquad \\rho_{t,j}(n)=1_{(n,\\operatorname{rad}(tj))=1}. \\] The factors on \\(n\\) in Equation~\\eqref{eq:initial-common-extraction}, together with \\(\\rho_{t,j}\\), supply exactly the original exclusions at \\(C,s,j\\) and the numerator zeros at \\(d',h'\\). On the", "anchor": { "source": "paper.tex", "line_start": 11890, "line_end": 11890, "char_start": 551087, "char_end": 551123 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:inverse-slot-priority", "to": "lem:marked", "type": "candidate_proof_dependency", "evidence": "ator zeros at \\(d',h'\\). On the nonzero support \\(t,d',s,n\\) are pairwise coprime. Apply Equation~\\eqref{eq:inverse-slot-priority} to each mark with this ordered list. Assigned primes and their priority masks stay outer. A surviving prime is already prime to", "anchor": { "source": "paper.tex", "line_start": 11894, "line_end": 11894, "char_start": 551320, "char_end": 551352 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:initial-canonical-lengths", "to": "lem:marked", "type": "candidate_proof_dependency", "evidence": "_c^{\\rm init}+\\delta_{\\rm init}. \\] For a nonempty negative-center window, \\(q_n\\ge1\\) and Equation~\\eqref{eq:initial-canonical-lengths} give \\(\\delta_{\\rm init}\\le3\\eta\\), while its upper support endpoint gives \\(Z^{\\delta_{\\rm init}}\\le b\\). If \\(N_c^{\\rm init}\\g", "anchor": { "source": "paper.tex", "line_start": 11961, "line_end": 11961, "char_start": 554762, "char_end": 554798 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:initial-formal-component", "to": "lem:marked", "type": "candidate_proof_dependency", "evidence": "et \\(\\mathcal T_\\Sigma^{\\rm init}\\) be the fixed \\(s\\)-dyad and slot summand obtained from Equation~\\eqref{eq:initial-formal-component} by the complete M\\\"obius and priority expansions. Its exact separated identity is \\label{eq:initial-separated-id", "anchor": { "source": "paper.tex", "line_start": 12050, "line_end": 12051, "char_start": 558857, "char_end": 558892 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:initial-outer-weight", "to": "lem:marked", "type": "candidate_proof_dependency", "evidence": "ndeed \\(x_a=Z^{\\delta_{\\rm init}}z_{n_a}\\). The two column modes, with the last factor of Equation~\\eqref{eq:initial-outer-weight}, are exactly \\(x_1^{i\\xi_1}x_2^{i\\xi_2}\\), and the other four modes are outer. Fourier inversion restores the full profile; its", "anchor": { "source": "paper.tex", "line_start": 12068, "line_end": 12068, "char_start": 559561, "char_end": 559592 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:initial-root-kernel", "to": "lem:marked", "type": "candidate_proof_dependency", "evidence": "pports, and the fresh column cutoffs are one wherever the old \\(W_0\\) windows are nonzero. Equation~\\eqref{eq:initial-root-kernel} restores the full prefactor and kernel. Equation~\\eqref{eq:initial-common-extraction} and the priority expansion restore every a", "anchor": { "source": "paper.tex", "line_start": 12073, "line_end": 12073, "char_start": 559863, "char_end": 559893 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:initial-common-extraction", "to": "lem:marked", "type": "candidate_proof_dependency", "evidence": "nonzero. Equation~\\eqref{eq:initial-root-kernel} restores the full prefactor and kernel. Equation~\\eqref{eq:initial-common-extraction} and the priority expansion restore every arithmetic factor term by term. Extra combinations in the full fresh windows cancel thro", "anchor": { "source": "paper.tex", "line_start": 12074, "line_end": 12074, "char_start": 559944, "char_end": 559980 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:initial-separated-identity", "to": "lem:marked", "type": "candidate_proof_dependency", "evidence": "s cancel through the old windows in this complex identity before Cauchy. The transform in Equation~\\eqref{eq:initial-separated-identity} is chosen from the fixed ambient profile before any actual \\(C,d',s,h'\\), and hence before any \\(t,k,f\\), is evaluated. It depen", "anchor": { "source": "paper.tex", "line_start": 12079, "line_end": 12080, "char_start": 560197, "char_end": 560234 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:inverse-fourier-moment", "to": "lem:marked", "type": "candidate_proof_dependency", "evidence": "in the displayed characters. The relation \\(C=td'\\) only restricts evaluation points. By Equation~\\eqref{eq:inverse-fourier-moment}, for each fixed \\(J\\) there are fixed \\(L,b_0\\) such that \\[ \\int_{\\mathbb R^6} |\\widehat{\\mathfrak H}_{{\\rm init},\\Sigma,\\bold", "anchor": { "source": "paper.tex", "line_start": 12088, "line_end": 12088, "char_start": 560751, "char_end": 560784 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:initial-separated-identity", "to": "lem:marked", "type": "candidate_proof_dependency", "evidence": "earlier row-space Minkowski inequality. No derivative order introduces a power of \\(Z\\). Equation~\\eqref{eq:initial-separated-identity} now expresses the component in polynomials of canonical shape with one common Fourier density chosen before the current outer ide", "anchor": { "source": "paper.tex", "line_start": 12107, "line_end": 12107, "char_start": 561616, "char_end": 561653 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:weighted-cauchy", "to": "lem:marked", "type": "candidate_proof_dependency", "evidence": "ity from source witnesses, and restore the normalization in Lemma~\\ref{lem:marked}. Apply Equation~\\eqref{eq:weighted-cauchy} on the whole \\(\\Omega_\\Sigma\\), including all quotients \\(t\\), before fixing one for the child. Only afterward take absolute wei", "anchor": { "source": "paper.tex", "line_start": 12113, "line_end": 12113, "char_start": 561986, "char_end": 562012 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:canonical-energy", "to": "lem:marked", "type": "candidate_proof_dependency", "evidence": "ldsymbol\\xi}(k,f)|^2 =Z^{F_{\\rm init}}\\mathcal E_{a,t}^{\\rm init}. This is Equation~\\eqref{eq:canonical-energy} with the same row ball; with a larger fixed canonical ball the equality is replaced by the corresponding upper bound. The fibre", "anchor": { "source": "paper.tex", "line_start": 12176, "line_end": 12176, "char_start": 564920, "char_end": 564947 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:inverse-initial-row", "to": "lem:marked", "type": "candidate_proof_dependency", "evidence": "m-2r-8z+10G+2P_1. \\] Use the marked premises, \\(G\\ge0\\), \\(P_1\\ge-2\\eta\\), Equations~\\eqref{eq:inverse-initial-row} and \\eqref{old-eq:4.13}, and the favorable clipping signs. The actual canonical margins are \\label{eq:inverse-in", "anchor": { "source": "paper.tex", "line_start": 12213, "line_end": 12213, "char_start": 566595, "char_end": 566625 }, "origin": "source_reference", "context": "proof" }, { "from": "old-eq:4.13", "to": "lem:marked", "type": "candidate_proof_dependency", "evidence": "Use the marked premises, \\(G\\ge0\\), \\(P_1\\ge-2\\eta\\), Equations~\\eqref{eq:inverse-initial-row} and \\eqref{old-eq:4.13}, and the favorable clipping signs. The actual canonical margins are \\begin{a", "anchor": { "source": "paper.tex", "line_start": 12213, "line_end": 12213, "char_start": 566630, "char_end": 566649 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:initial-positive-child", "to": "lem:marked", "type": "candidate_proof_dependency", "evidence": "nly for \\(t\\in\\mathfrak T_\\Sigma\\), where the preceding puncture and margin checks hold. Equations~\\eqref{eq:initial-positive-child} and \\eqref{eq:initial-weighted-count} then bound the unscaled signed outer sum by \\[ C(\\boldsymbol\\upsilon,\\boldsymbol\\xi) Z^{P", "anchor": { "source": "paper.tex", "line_start": 12230, "line_end": 12231, "char_start": 567372, "char_end": 567405 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:initial-weighted-count", "to": "lem:marked", "type": "candidate_proof_dependency", "evidence": "ere the preceding puncture and margin checks hold. Equations~\\eqref{eq:initial-positive-child} and \\eqref{eq:initial-weighted-count} then bound the unscaled signed outer sum by \\[ C(\\boldsymbol\\upsilon,\\boldsymbol\\xi) Z^{P_1+2\\eta+2F_{\\rm init}+\\epsilon_c} \\|", "anchor": { "source": "paper.tex", "line_start": 12231, "line_end": 12231, "char_start": 567410, "char_end": 567443 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:initial-normalized-quotients", "to": "lem:marked", "type": "candidate_proof_dependency", "evidence": "P_1+2\\eta+2F_{\\rm init}+\\epsilon_c} \\|\\nu_\\Sigma^{\\rm init}\\|. \\] The one mass is that of Equation~\\eqref{eq:initial-normalized-quotients}; the label \\(d's\\) remains inside the child average. The common Fourier density is integrated separately, once, with the uniform", "anchor": { "source": "paper.tex", "line_start": 12238, "line_end": 12238, "char_start": 567634, "char_end": 567673 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:inverse-parameter-order", "to": "lem:marked", "type": "candidate_proof_dependency", "evidence": "m init}\\le D_{\\max}+5\\eta\\) bounds both nonnegative child parameters. Make the choices in Equation~\\eqref{eq:inverse-parameter-order} with \\(\\epsilon_c=\\epsilon/2\\), imposing in addition \\[ \\eta,\\tau_{\\rm init}\\le\\bar c/100,\\qquad \\eta\\le\\epsilon/88,\\qquad \\pi_", "anchor": { "source": "paper.tex", "line_start": 12279, "line_end": 12280, "char_start": 569610, "char_end": 569644 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:inverse-initial-margins", "to": "lem:marked", "type": "candidate_proof_dependency", "evidence": "le\\epsilon/8. \\] One may take \\(\\tau_{\\rm init}=\\tau\\) after imposing both sets of bounds. Equation~\\eqref{eq:inverse-initial-margins} then gives both margins at least \\(3\\bar c/4>c_*\\), so Lemma~\\ref{lem:canonical-moment} applies. Equation~\\eqref{eq:inverse-initi", "anchor": { "source": "paper.tex", "line_start": 12286, "line_end": 12286, "char_start": 569885, "char_end": 569919 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:canonical-moment", "to": "lem:marked", "type": "candidate_proof_dependency", "evidence": "ion~\\eqref{eq:inverse-initial-margins} then gives both margins at least \\(3\\bar c/4>c_*\\), so Lemma~\\ref{lem:canonical-moment} applies. Equation~\\eqref{eq:inverse-initial-energy} adds at most \\(\\epsilon/4\\) to its \\(\\epsilon_c=\\epsilon/2\\) loss. With thes", "anchor": { "source": "paper.tex", "line_start": 12287, "line_end": 12287, "char_start": 569981, "char_end": 570007 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:inverse-initial-energy", "to": "lem:marked", "type": "candidate_proof_dependency", "evidence": "ives both margins at least \\(3\\bar c/4>c_*\\), so Lemma~\\ref{lem:canonical-moment} applies. Equation~\\eqref{eq:inverse-initial-energy} adds at most \\(\\epsilon/4\\) to its \\(\\epsilon_c=\\epsilon/2\\) loss. With these bounded initial ranges fixed, choose the subsidiar", "anchor": { "source": "paper.tex", "line_start": 12288, "line_end": 12288, "char_start": 570026, "char_end": 570059 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:inverse-raw-tail", "to": "lem:marked", "type": "candidate_proof_dependency", "evidence": ",init}\\) and then \\[ A_{\\rm init}>1+(B_{\\rm crude,init}+T_{\\rm init})/\\tau_{\\rm init}. \\] Equation~\\eqref{eq:inverse-raw-tail} makes the raw \\(\\mathcal B_{\\rm init}=0\\) complement \\(O(Z^{-T_{\\rm init}})\\), with fixed seminorm and polynomial-height factors.", "anchor": { "source": "paper.tex", "line_start": 12323, "line_end": 12323, "char_start": 571538, "char_end": 571565 }, "origin": "source_reference", "context": "proof" }, { "from": "old-eq:4.1", "to": "lem:marked", "type": "candidate_proof_dependency", "evidence": "d measures give the stated finite- seminorm and polynomial-height uniformity. This proves Equation~\\eqref{old-eq:4.1}. \\subsection{Amplification on sixth-power-free rows} We first extract an unmarked consequence on all element rows.", "anchor": { "source": "paper.tex", "line_start": 12340, "line_end": 12340, "char_start": 572601, "char_end": 572619 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:marked", "to": "sec:inverse", "type": "internal_reference", "evidence": "on sixth-power-free rows} We first extract an unmarked consequence on all element rows. In Lemma~\\ref{lem:marked}, take \\(Z=H,\\ m=1\\), no slots, and \\(r=\\log_H D\\). If \\(H\\ge D^{1+c}\\) for a fixed \\(c>0\\), then \\(1-r\\ge c/(1+c)\\); the second", "anchor": { "source": "paper.tex", "line_start": 12346, "line_end": 12346, "char_start": 572759, "char_end": 572775 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:marked", "to": "lem:inverse-amplification", "type": "internal_reference", "evidence": "el{lem:inverse-amplification} Let \\(\\nu,W,\\varepsilon_\\chi\\) and the zero extensions be as in Lemma~\\ref{lem:marked}, with no prime slots. Let \\(U,D\\ge1\\), and sum over elements \\(u\\) with \\(q_u\\asymp U\\) such that every prime valuation of the i", "anchor": { "source": "paper.tex", "line_start": 12364, "line_end": 12364, "char_start": 573666, "char_end": 573682 }, "origin": "source_reference", "context": "statement" }, { "from": "eq:raw-moment", "to": "lem:inverse-amplification", "type": "candidate_proof_dependency", "evidence": "_u|\\le T_1\\), at a cost \\((1+T_1)^A\\) for a fixed \\(A\\). First, Equation~\\eqref{eq:raw-moment} and Lemma~\\ref{lem:smooth-calculus} imply the rowwise scale bound \\sum_{\\subst", "anchor": { "source": "paper.tex", "line_start": 12392, "line_end": 12392, "char_start": 574683, "char_end": 574704 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:smooth-calculus", "to": "lem:inverse-amplification", "type": "candidate_proof_dependency", "evidence": ")^A\\) for a fixed \\(A\\). First, Equation~\\eqref{eq:raw-moment} and Lemma~\\ref{lem:smooth-calculus} imply the rowwise scale bound \\sum_{\\substack{v\\in\\mathcal O\\\\00", "anchor": { "source": "paper.tex", "line_start": 12408, "line_end": 12408, "char_start": 575507, "char_end": 575540 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:inverse-scale-supremum", "to": "lem:inverse-amplification", "type": "candidate_proof_dependency", "evidence": "unit. This argument does not require \\((u,a)=1\\). Also \\(q_{ua^6}\\ll UP^6\\ll H\\). Thus Equation~\\eqref{eq:inverse-scale-supremum} gives \\[ \\sum_u|M_u(D;W)|^2 \\ll \\frac HP(UD)^\\epsilon =U^{1/6}H^{5/6}(UD)^\\epsilon, \\] after choosing the preliminary \\(\\delta", "anchor": { "source": "paper.tex", "line_start": 12445, "line_end": 12445, "char_start": 577059, "char_end": 577092 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:amplified-note", "to": "lem:inverse-amplification", "type": "candidate_proof_dependency", "evidence": "eliminary \\(\\delta\\) and power losses small enough. Since \\(H=\\max(2U,D^{1+c})\\), this is Equation~\\eqref{eq:amplified-note}. For bounded \\(r\\), choose \\(c>0\\) sufficiently small in terms of \\(r\\)'s bound and the desired power loss; the stated formula f", "anchor": { "source": "paper.tex", "line_start": 12453, "line_end": 12453, "char_start": 577304, "char_end": 577329 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:smooth-calculus", "to": "lem:inverse-amplification", "type": "candidate_proof_dependency", "evidence": "_{\\sigma,t}\\) insert bounded powers of \\(\\log y\\). Apply the parameter Sobolev inequality in Lemma~\\ref{lem:smooth-calculus} on a bounded cover in \\(\\sigma\\) and \\(O(1+T_1)\\) unit intervals in \\(t\\), summing the row moments before integrating the derivat", "anchor": { "source": "paper.tex", "line_start": 12460, "line_end": 12460, "char_start": 577654, "char_end": 577679 }, "origin": "source_reference", "context": "proof" }, { "from": "sec:arithmetic", "to": "sec:plain", "type": "internal_reference", "evidence": "el. The estimate itself concerns the original, uncentered product. We use the notation of Section~\\ref{sec:arithmetic} and the coefficient conventions of Section~\\ref{sec:additional-arithmetic}. In particular, the rows are elements \\(k\\in\\mathcal", "anchor": { "source": "paper.tex", "line_start": 12492, "line_end": 12492, "char_start": 579400, "char_end": 579420 }, "origin": "source_reference", "context": "narrative" }, { "from": "sec:additional-arithmetic", "to": "sec:plain", "type": "internal_reference", "evidence": "ct. We use the notation of Section~\\ref{sec:arithmetic} and the coefficient conventions of Section~\\ref{sec:additional-arithmetic}. In particular, the rows are elements \\(k\\in\\mathcal O\\) with \\(00\\), the left side of Equation~\\eqref{old-eq:3.9} has decreased; if \\(z'=0\\), the zero-slot assertion has no length restriction and \\(\\mathcal R_z\\subseteq\\mathcal R_0\\). For eve", "anchor": { "source": "paper.tex", "line_start": 12654, "line_end": 12654, "char_start": 587207, "char_end": 587225 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:deleted-euler-factors", "to": "sec:plain", "type": "internal_reference", "evidence": "-1} \\ll_{\\epsilon_1,B} Z^{\\epsilon_1}. This follows from Lemma~\\ref{lem:deleted-euler-factors}. The corresponding mass for a frozen slot is also \\(Z^{\\epsilon_1}\\): on its annular support, \\(P_i^{-1/2}\\ll q_p^{-1/2}\\), and", "anchor": { "source": "paper.tex", "line_start": 12665, "line_end": 12665, "char_start": 587579, "char_end": 587610 }, "origin": "source_reference", "context": "narrative" }, { "from": "old-eq:2.1", "to": "sec:plain", "type": "internal_reference", "evidence": "mid\\mathfrak R\\). The number of slots is fixed. Minkowski's inequality therefore reduces Equation~\\eqref{old-eq:2.1} to the natural assertion at the same width, with an arbitrarily small power loss. A nonempty annular scale below one is bounded", "anchor": { "source": "paper.tex", "line_start": 12669, "line_end": 12669, "char_start": 587881, "char_end": 587899 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:fixed-numerator-ray", "to": "sec:plain", "type": "internal_reference", "evidence": "ion of the row radical and the declared moving radical, whose norm is at most \\(q_kZ^q\\). By Lemma~\\ref{lem:fixed-numerator-ray}, the unit and \\(\\mathcal S\\)-supported parts of a row, with valuations reduced modulo six when evaluated on primary elements prim", "anchor": { "source": "paper.tex", "line_start": 12693, "line_end": 12693, "char_start": 589070, "char_end": 589099 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:hecke-strip-growth", "to": "sec:plain", "type": "internal_reference", "evidence": "only a fixed factor. Apply the primitive Hecke functional equation recorded in the proof of Lemma~\\ref{lem:hecke-strip-growth} to \\(\\psi_k^*\\), using the entireness of \\(L(s,\\psi_k^*)\\) stated there. Let \\(\\varepsilon_k\\) be its root number, so \\(|\\vareps", "anchor": { "source": "paper.tex", "line_start": 12701, "line_end": 12701, "char_start": 589433, "char_end": 589461 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:smooth-calculus", "to": "sec:plain", "type": "internal_reference", "evidence": "thcal M W(1-s)\\frac{\\Gamma(s)}{\\Gamma(1-s)}. \\] The Mellin transform convention is the one in Lemma~\\ref{lem:smooth-calculus}. The quotient of gamma functions has poles only at the nonpositive integers and zeros at the positive integers. Shifting the in", "anchor": { "source": "paper.tex", "line_start": 12712, "line_end": 12712, "char_start": 589954, "char_end": 589979 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:smooth-calculus", "to": "sec:plain", "type": "internal_reference", "evidence": "ivative orders. The pointwise Mellin integration-by-parts estimate on a compact real strip in Lemma~\\ref{lem:smooth-calculus} justifies the horizontal joins in these shifts; an integrated Fourier tail is not used to bound a fixed horizontal trace. This p", "anchor": { "source": "paper.tex", "line_start": 12728, "line_end": 12728, "char_start": 590995, "char_end": 591020 }, "origin": "source_reference", "context": "narrative" }, { "from": "old-eq:2.1c", "to": "sec:plain", "type": "internal_reference", "evidence": "quad \\operatorname{rad}(h_0)\\mid\\mathfrak R_{0,k}. By Equation~\\eqref{old-eq:2.1c}, there is a fixed \\(C_{\\mathcal A}\\ge1\\) such that \\[ C_kq_{\\mathfrak R_{0,k}}\\le C_{\\mathcal A}Z^M,\\qquad Y\\le C_{\\mathcal A}Z", "anchor": { "source": "paper.tex", "line_start": 12756, "line_end": 12756, "char_start": 592062, "char_end": 592081 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:smooth-calculus", "to": "sec:plain", "type": "internal_reference", "evidence": "are handled by their coefficient mass followed by a rowwise supremum in the resulting scales. Lemma~\\ref{lem:smooth-calculus} bounds a supremum over two polynomial-range scales using \\(O((\\log Z)^2)\\) unit boxes and derivatives of the profiles. The row c", "anchor": { "source": "paper.tex", "line_start": 12782, "line_end": 12782, "char_start": 593227, "char_end": 593252 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:centered-reflection-length", "to": "sec:plain", "type": "internal_reference", "evidence": "original factor whose length exceeds \\(M/2\\) once, and leave the other factor unchanged. Equation~\\eqref{eq:centered-reflection-length} bounds every retained reflected factor; the fixed box multiplier for an unreflected factor is included in \\(C_{\\rm ref}\\). After", "anchor": { "source": "paper.tex", "line_start": 12811, "line_end": 12812, "char_start": 594778, "char_end": 594815 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:logarithmic-control", "to": "sec:plain", "type": "internal_reference", "evidence": "t \\(s_\\kappa=(1+\\kappa)/2\\). Under the hypothesis \\(s_\\kappa\\ge\\beta_*\\), the global part of Lemma~\\ref{lem:logarithmic-control} gives, for every fixed \\(e>0\\), \\frac{L'}{L}(s_\\kappa+e+it,\\psi) \\ll_e \\log\\!\\bigl(2Q_\\psi(3+|t|)^2\\bigr) \\la", "anchor": { "source": "paper.tex", "line_start": 12835, "line_end": 12835, "char_start": 595890, "char_end": 595919 }, "origin": "source_reference", "context": "narrative" }, { "from": "old-eq:3.4", "to": "sec:plain", "type": "internal_reference", "evidence": "are no poles in the region of the shift, and the Mellin transform has arbitrary decay, so Equation~\\eqref{old-eq:3.4} bounds the new integral by \\(P^{s_\\kappa+e}\\) times a logarithm of the conductor and a fixed polynomial in the height. Prime pow", "anchor": { "source": "paper.tex", "line_start": 12846, "line_end": 12846, "char_start": 596386, "char_end": 596404 }, "origin": "source_reference", "context": "narrative" }, { "from": "old-eq:3.5", "to": "sec:plain", "type": "internal_reference", "evidence": "he stated power loss because \\(P_i\\ge1\\). For \\(\\kappa=1\\), absolute prime counting gives Equation~\\eqref{old-eq:3.5} without a zero-free assumption and with \\(h=0\\) for undifferentiated pure twists, whose modulus is one. For \\(\\kappa<1\\) the hyp", "anchor": { "source": "paper.tex", "line_start": 12894, "line_end": 12894, "char_start": 599053, "char_end": 599071 }, "origin": "source_reference", "context": "narrative" }, { "from": "old-eq:3.5", "to": "sec:plain", "type": "internal_reference", "evidence": "iform in \\(\\kappa\\in[3/4,1]\\), since the length ranges are bounded. In every later use of Equation~\\eqref{old-eq:3.5}, its fixed seminorm and height factor is retained in the weighted Fourier estimates; it is not included in the exponent of \\(Z\\).", "anchor": { "source": "paper.tex", "line_start": 12899, "line_end": 12899, "char_start": 599372, "char_end": 599390 }, "origin": "source_reference", "context": "narrative" }, { "from": "old-eq:2.1h", "to": "sec:plain", "type": "internal_reference", "evidence": "t band, first prove the uncentered assertion for \\(A\\le5M/6\\), and then the padded core in Equation~\\eqref{old-eq:2.1h}. Reflection then supplies all zero-slot lengths in that band. Within a positive-slot band, first prove the uncentered assertion", "anchor": { "source": "paper.tex", "line_start": 12912, "line_end": 12912, "char_start": 600127, "char_end": 600146 }, "origin": "source_reference", "context": "narrative" }, { "from": "old-eq:3.9", "to": "sec:plain", "type": "internal_reference", "evidence": "hin a positive-slot band, first prove the uncentered assertion for \\(A\\le5M/6\\) subject to Equation~\\eqref{old-eq:3.9}, and then the remaining part of that region by centering. A completed earlier band therefore includes the unrestricted zero-slot", "anchor": { "source": "paper.tex", "line_start": 12915, "line_end": 12915, "char_start": 600314, "char_end": 600332 }, "origin": "source_reference", "context": "narrative" }, { "from": "old-eq:3.5", "to": "sec:plain", "type": "internal_reference", "evidence": "\\(A-q\\le\\rho+\\delta\\). For positive slots, combine the completed zero-slot estimate with Equation~\\eqref{old-eq:3.5}. In the region of Equation~\\eqref{old-eq:3.9}, \\(A\\ge z\\), whence z\\le\\frac{M}{6\\kappa}\\le\\frac{2M}{9}, \\qqua", "anchor": { "source": "paper.tex", "line_start": 12929, "line_end": 12929, "char_start": 600984, "char_end": 601002 }, "origin": "source_reference", "context": "narrative" }, { "from": "old-eq:3.9", "to": "sec:plain", "type": "internal_reference", "evidence": "mbine the completed zero-slot estimate with Equation~\\eqref{old-eq:3.5}. In the region of Equation~\\eqref{old-eq:3.9}, \\(A\\ge z\\), whence z\\le\\frac{M}{6\\kappa}\\le\\frac{2M}{9}, \\qquad \\kappa z\\le\\frac M6. \\end", "anchor": { "source": "paper.tex", "line_start": 12930, "line_end": 12930, "char_start": 601031, "char_end": 601049 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:centered-reflection-length", "to": "sec:plain", "type": "internal_reference", "evidence": "reflection is at most \\[ b+\\{M-(A-z-b)\\}+z+\\xi \\le 3M/2-A+2z+\\xi, \\] where \\(b\\le L\\). Equation~\\eqref{eq:centered-reflection-length} includes the clipped reflected scale and the paired unreflected box multiplier in this same \\(\\xi\\); the discarded tails have the", "anchor": { "source": "paper.tex", "line_start": 12961, "line_end": 12962, "char_start": 602241, "char_end": 602278 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:centered-reflection-length", "to": "sec:plain", "type": "internal_reference", "evidence": "least \\(M/15\\) before the \\(\\xi\\) term. Choose \\(\\xi\\le\\rho/30\\). After the threshold in Equation~\\eqref{eq:centered-reflection-length}, the unit-box multipliers are already included in \\(\\xi\\), so every retained reflected comparison calls the previously proved unc", "anchor": { "source": "paper.tex", "line_start": 12982, "line_end": 12982, "char_start": 603020, "char_end": 603057 }, "origin": "source_reference", "context": "narrative" }, { "from": "old-eq:2.18a", "to": "sec:plain", "type": "internal_reference", "evidence": "angle, that rectangle's profile is zero on the corresponding sum; the formal difference in Equation~\\eqref{old-eq:2.18a} is nevertheless retained. The reason for retaining these common data is already visible in the exceptional case. For a fixed \\(", "anchor": { "source": "paper.tex", "line_start": 13029, "line_end": 13029, "char_start": 604894, "char_end": 604914 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:centered-lattice-cancellation", "to": "sec:plain", "type": "internal_reference", "evidence": "1^{1+it}T_2^{1+it}I_1(t)I_2(t)\\), which agrees for the two rectangles when \\(T_1T_2\\) agrees. Lemma~\\ref{lem:centered-lattice-cancellation} will quantify the remaining error. This cancellation is used only on the transformed rows inducing characters in \\(\\Theta\\); the", "anchor": { "source": "paper.tex", "line_start": 13046, "line_end": 13046, "char_start": 605640, "char_end": 605679 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:centered-localization-threshold", "to": "sec:plain", "type": "internal_reference", "evidence": "2H_N}T_{\\rm sec})\\). On every discarded weight it is therefore at least \\(Z^{\\xi/4}\\), by Equation~\\eqref{eq:centered-localization-threshold}. Lemma~\\ref{lem:kernel-seminorms} gives arbitrary decay \\((1+\\text{argument})^{-B}\\). For the first formula use \\(|G(a,h)|\\le q_", "anchor": { "source": "paper.tex", "line_start": 13155, "line_end": 13155, "char_start": 610868, "char_end": 610910 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:kernel-seminorms", "to": "sec:plain", "type": "internal_reference", "evidence": "is therefore at least \\(Z^{\\xi/4}\\), by Equation~\\eqref{eq:centered-localization-threshold}. Lemma~\\ref{lem:kernel-seminorms} gives arbitrary decay \\((1+\\text{argument})^{-B}\\). For the first formula use \\(|G(a,h)|\\le q_a^{1/2}\\), and for the second use", "anchor": { "source": "paper.tex", "line_start": 13156, "line_end": 13156, "char_start": 610918, "char_end": 610944 }, "origin": "source_reference", "context": "narrative" }, { "from": "sec:arithmetic", "to": "sec:plain", "type": "internal_reference", "evidence": "ne{\\xi_{\\mathfrak r}(b)}. \\] Residue-class Poisson, with the self-dual lattice measure from Section~\\ref{sec:arithmetic}, gives for this allocation the following nonzero-frequency expression, up to a scalar of bounded modulus in the frozen labels: \\b", "anchor": { "source": "paper.tex", "line_start": 13234, "line_end": 13234, "char_start": 614962, "char_end": 614982 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:first-poisson-bridge", "to": "sec:plain", "type": "internal_reference", "evidence": "b)}\\); the remaining factors are precisely \\(\\tau_C(a)\\overline{\\tau_D(b)}\\). This proves Equation~\\eqref{eq:first-poisson-bridge}. Each new character acts on its whole residual product. Its new moving primes belong to the extracted support and puncture ever", "anchor": { "source": "paper.tex", "line_start": 13272, "line_end": 13272, "char_start": 616459, "char_end": 616490 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:first-poisson-bridge", "to": "sec:plain", "type": "internal_reference", "evidence": "upport convention gives \\(q_a/(X/q_C),q_b/(X/q_D)\\in\\exp([-H_N,H_N])\\), so the argument in Equation~\\eqref{eq:first-poisson-bridge} is at least \\(q_h/(e^{2H_N}T_{\\rm sec})\\) for the supremum of \\(T_1\\) in the outer sector. Apply the preceding whole-dyad locali", "anchor": { "source": "paper.tex", "line_start": 13288, "line_end": 13288, "char_start": 617166, "char_end": 617197 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:centered-aggregate-support", "to": "sec:plain", "type": "internal_reference", "evidence": "outer ratio as another coordinate. The cutoffs are chosen on the common product annulus of Equation~\\eqref{eq:centered-aggregate-support}, before fixing live slot labels. Thus one Fourier coefficient measure is common to the rows and all live labels. It supplies a", "anchor": { "source": "paper.tex", "line_start": 13326, "line_end": 13326, "char_start": 619092, "char_end": 619129 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:centered-coefficient-form", "to": "sec:plain", "type": "internal_reference", "evidence": "main present. Complete extraction and multiplicativity therefore leave the coefficient in Equation~\\eqref{eq:centered-coefficient-form}, now multiplied by \\(G(a,h)\\). In particular the two rectangles retain the same allocated plain powers, character, puncture, and", "anchor": { "source": "paper.tex", "line_start": 13346, "line_end": 13346, "char_start": 620144, "char_end": 620180 }, "origin": "source_reference", "context": "narrative" }, { "from": "old-eq:2.6", "to": "sec:plain", "type": "internal_reference", "evidence": "m of positive parts. Multiplication by each prime's logarithmic norm and summation proves Equation~\\eqref{old-eq:2.6}. Thus it remains to establish Equation~\\eqref{old-eq:2.5}. \\subsection{Enlarging the Gauss-row norm}\\label{sec:plain-gauss-enla", "anchor": { "source": "paper.tex", "line_start": 13408, "line_end": 13408, "char_start": 622773, "char_end": 622791 }, "origin": "source_reference", "context": "narrative" }, { "from": "old-eq:2.5", "to": "sec:plain", "type": "internal_reference", "evidence": "thmic norm and summation proves Equation~\\eqref{old-eq:2.6}. Thus it remains to establish Equation~\\eqref{old-eq:2.5}. \\subsection{Enlarging the Gauss-row norm} This subsection describes three possible positive", "anchor": { "source": "paper.tex", "line_start": 13409, "line_end": 13409, "char_start": 622832, "char_end": 622850 }, "origin": "source_reference", "context": "narrative" }, { "from": "old-eq:2.5", "to": "sec:plain", "type": "internal_reference", "evidence": "a_0=A-c-w\\). When a squared extraction coefficient of size \\(Z^{-w_o}\\) has been removed, Equation~\\eqref{old-eq:2.5} allows the unweighted remaining norm the exponent \\Lambda_c= a_0+\\widetilde q+w_o+w+\\mathcal B_c-s_0+\\epsilon_", "anchor": { "source": "paper.tex", "line_start": 13418, "line_end": 13419, "char_start": 623285, "char_end": 623303 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:gauss-local", "to": "sec:plain", "type": "internal_reference", "evidence": "p^6)=G(u,h)\\), since \\(p\\) is a unit modulo \\(u\\) and \\(\\chi_u(p^6)=1\\). For \\(p\\nmid h\\), Equation~\\eqref{eq:gauss-local} shows that replacing \\(h\\) by \\(hp^6\\) changes only \\(i=1,6,7\\). The factors \\(G(p^i,h)\\) that occur are a scalar in the row and", "anchor": { "source": "paper.tex", "line_start": 13485, "line_end": 13485, "char_start": 625828, "char_end": 625850 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:centered-coefficient-form", "to": "sec:plain", "type": "internal_reference", "evidence": "cation just described are applied to the whole centered coefficient, so each error retains Equation~\\eqref{eq:centered-coefficient-form}. On the support of the original row weight, the new row \\(hp^6\\) has norm \\(O(Z^{K+6\\ell_*})\\). A fixed output row has only bou", "anchor": { "source": "paper.tex", "line_start": 13513, "line_end": 13513, "char_start": 627228, "char_end": 627264 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:gauss-local", "to": "sec:plain", "type": "internal_reference", "evidence": "g\\) contains all the new rows. For the extracted terms put \\(P=q_p\\). When \\(p\\nmid h\\), Equation~\\eqref{eq:gauss-local} gives the following central coefficients. At valuation one, the old Gauss sum has modulus one and the new one is zero, so extrac", "anchor": { "source": "paper.tex", "line_start": 13530, "line_end": 13530, "char_start": 627967, "char_end": 627989 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:gauss-local", "to": "sec:plain", "type": "internal_reference", "evidence": "A nonempty column shell satisfies \\(Y_{\\rm col}\\ge1\\), and hence \\(a_0\\ge-\\theta_N\\). By Equation~\\eqref{eq:gauss-local}, \\(G(u,0)=0\\) unless \\(u\\) is a sixth power, and \\(|G(u,0)|\\le q_u^{1/2}\\le Y_{\\rm col}^{1/2}\\). There are \\(O(Y_{\\rm col}^{1/6})", "anchor": { "source": "paper.tex", "line_start": 13575, "line_end": 13575, "char_start": 629947, "char_end": 629969 }, "origin": "source_reference", "context": "narrative" }, { "from": "old-eq:2.7", "to": "sec:plain", "type": "internal_reference", "evidence": "ast inequality follows from \\(a_0\\ge-\\theta_N\\) and the displayed bound for \\(s_0\\). Thus Equation~\\eqref{old-eq:2.7} bounds this added row with the numerical \\(2\\theta_N\\) correction included in the \\(C_*\\xi\\) stage allowance. Here \\(C_*\\ge1\\) d", "anchor": { "source": "paper.tex", "line_start": 13588, "line_end": 13588, "char_start": 630603, "char_end": 630621 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:full-correlation", "to": "sec:plain", "type": "internal_reference", "evidence": "+g}) \\left|Z^{-a_0/2}\\sum_u B(u)\\tau_1(u)q_u^{it}G(u,h)\\right|^2. \\] The Fourier identity in Lemma~\\ref{lem:full-correlation} gives its exact expansion \\[ Z^{K+g-\\ell-a_0}\\sum_{u,v} \\frac{B(u)\\overline{B(v)}\\tau_1(u)\\overline{\\tau_1(v)}", "anchor": { "source": "paper.tex", "line_start": 13610, "line_end": 13610, "char_start": 631732, "char_end": 631758 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:full-correlation", "to": "sec:plain", "type": "internal_reference", "evidence": "cular the formula includes every live prime and every shared-prime multiplicity. At \\(j=0\\), Lemma~\\ref{lem:full-correlation} leaves only \\(u=v\\), with \\(F(u,u;0)=\\varphi(u)\\le q_u\\). The number of supported moduli divisible by \\(s\\), on a nonempty shell", "anchor": { "source": "paper.tex", "line_start": 13630, "line_end": 13630, "char_start": 632407, "char_end": 632433 }, "origin": "source_reference", "context": "narrative" }, { "from": "old-eq:2.7", "to": "sec:plain", "type": "internal_reference", "evidence": "\\theta_N\\), including when \\(a_0-s_0\\) is slightly negative. Subtracting the allowance in Equation~\\eqref{old-eq:2.7} from its nominal part, without \\(\\epsilon_G\\), gives &(K+g-\\ell-s_0) -(a_0+\\widetilde q+w_o+w+\\", "anchor": { "source": "paper.tex", "line_start": 13640, "line_end": 13641, "char_start": 632922, "char_end": 632940 }, "origin": "source_reference", "context": "narrative" }, { "from": "old-eq:2.18b", "to": "sec:plain", "type": "internal_reference", "evidence": "bda). \\] This common row weight satisfies \\(0\\le\\Omega_{\\rm ret}\\le1\\). The full kernel in Equation~\\eqref{old-eq:2.18b} remains on every retained term. All subsequent signed Fourier separations are performed one retained \\(\\lambda\\) at a time, so t", "anchor": { "source": "paper.tex", "line_start": 13677, "line_end": 13677, "char_start": 634524, "char_end": 634544 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:complete-support-correlation", "to": "sec:plain", "type": "internal_reference", "evidence": "). Allocate all extracted powers to the plain variables and slots as at the first transform. Lemma~\\ref{lem:complete-support-correlation} applies on this genuine locus and gives \\[ F(D_2a,E_2b;j)=F(D_2,E_2;j)\\mathcal R(a,E_2) \\overline{\\mathcal R(b,D_2)}\\mathcal", "anchor": { "source": "paper.tex", "line_start": 13689, "line_end": 13689, "char_start": 635275, "char_end": 635313 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:correlation-local", "to": "sec:plain", "type": "internal_reference", "evidence": "both remaining plain variables at \\(p\\). Equal multiplicities have the scalar factors in Equation~\\eqref{eq:correlation-local} and the same puncture conclusion. Complete rather than gcd-only extraction is what makes these local factors independent of the", "anchor": { "source": "paper.tex", "line_start": 13720, "line_end": 13720, "char_start": 636778, "char_end": 636806 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:correlation-local", "to": "sec:plain", "type": "internal_reference", "evidence": "\\(V_{\\rm id}\\) be the product of these nonunit primes and put \\(V=\\log_Zq_{V_{\\rm id}}\\). Equation~\\eqref{eq:correlation-local} gives the following absolute local bounds: \\[ {c|c} \\text{common multiplicities and divided frequency}& \\te", "anchor": { "source": "paper.tex", "line_start": 13740, "line_end": 13740, "char_start": 637541, "char_end": 637569 }, "origin": "source_reference", "context": "narrative" }, { "from": "old-eq:2.3", "to": "sec:plain", "type": "internal_reference", "evidence": "he identities follow by substituting \\(a_0=A-c-w\\) and the definition of \\(K_0\\) preceding Equation~\\eqref{old-eq:2.3}. Put \\[ 0\\le\\delta_{{\\rm fr},2}:=\\frac\\xi2+ \\frac{\\log(C_dC_{\\rm sec})}{\\log Z}<\\xi. \\] The sector constants here are th", "anchor": { "source": "paper.tex", "line_start": 13804, "line_end": 13804, "char_start": 640115, "char_end": 640133 }, "origin": "source_reference", "context": "narrative" }, { "from": "old-eq:2.7", "to": "sec:plain", "type": "internal_reference", "evidence": "ir sum is \\[ K+g-\\ell-a_0+p_2-s_0+g_2-t_2-b_2. \\] Subtracting this sum from the allowance Equation~\\eqref{old-eq:2.7} leaves the exponent permitted for the inner plain products. This is the bound to be supplied either by smaller-width moments or", "anchor": { "source": "paper.tex", "line_start": 13864, "line_end": 13864, "char_start": 642759, "char_end": 642777 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:centered-coefficient-form", "to": "lem:centered-coefficient-invariant", "type": "internal_reference", "evidence": "rve the following coefficient data. Each Gauss polynomial and each amplifier error retains Equation~\\eqref{eq:centered-coefficient-form}, multiplied by \\(G(u,h)\\), apart from bounded frozen scalars and row scalars of modulus at most one. For an uncentered input, omi", "anchor": { "source": "paper.tex", "line_start": 13958, "line_end": 13958, "char_start": 647238, "char_end": 647274 }, "origin": "source_reference", "context": "statement" }, { "from": "old-eq:2.14", "to": "sec:plain", "type": "internal_reference", "evidence": "nly count below includes them. Both sides use the same eligibility class and compare with Equation~\\eqref{old-eq:2.14}. Consider first rows whose child inducing character is outside \\(\\Theta\\). The two sides have the same eligibility condition be", "anchor": { "source": "paper.tex", "line_start": 14117, "line_end": 14117, "char_start": 655163, "char_end": 655182 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:centered-divisor-boundary", "to": "sec:plain", "type": "internal_reference", "evidence": "varrho})/2}. \\] For a fixed pair of retained rectangles, suppress their rectangle indices. Equation~\\eqref{eq:centered-divisor-boundary} then bounds the paired divisor count and these coefficients by t_- -\\frac{r_{{\\rm clip},1}+r_{{\\", "anchor": { "source": "paper.tex", "line_start": 14167, "line_end": 14167, "char_start": 657643, "char_end": 657679 }, "origin": "source_reference", "context": "narrative" }, { "from": "old-eq:2.9", "to": "sec:plain", "type": "internal_reference", "evidence": "{\\rm fr},1}\\) and the \\(1\\)-Lipschitz property of the positive part. The three choices in Equation~\\eqref{old-eq:2.9}, together with \\(\\ell_p\\ge\\sigma/6\\), give \\(w-w_o+\\sigma\\le6w\\). This proves Equation~\\eqref{old-eq:3.14}. For each retained c", "anchor": { "source": "paper.tex", "line_start": 14203, "line_end": 14203, "char_start": 658923, "char_end": 658941 }, "origin": "source_reference", "context": "narrative" }, { "from": "old-eq:3.14", "to": "sec:plain", "type": "internal_reference", "evidence": "old-eq:2.9}, together with \\(\\ell_p\\ge\\sigma/6\\), give \\(w-w_o+\\sigma\\le6w\\). This proves Equation~\\eqref{old-eq:3.14}. For each retained clipped rectangle, if no slot survives, apply the completed unrestricted zero-slot assertion at the declared", "anchor": { "source": "paper.tex", "line_start": 14206, "line_end": 14206, "char_start": 659030, "char_end": 659049 }, "origin": "source_reference", "context": "narrative" }, { "from": "old-eq:2.1a", "to": "sec:plain", "type": "internal_reference", "evidence": "red width \\(M'_{\\rm act}\\). If slots survive, first apply the algebraic mask deletion in Equations~\\eqref{old-eq:2.1a}--\\eqref{old-eq:2.1b}. It expresses this standard clipped product as natural products and only decreases its nonnegative affine ex", "anchor": { "source": "paper.tex", "line_start": 14211, "line_end": 14211, "char_start": 659275, "char_end": 659294 }, "origin": "source_reference", "context": "narrative" }, { "from": "old-eq:2.1b", "to": "sec:plain", "type": "internal_reference", "evidence": "ct}\\). If slots survive, first apply the algebraic mask deletion in Equations~\\eqref{old-eq:2.1a}--\\eqref{old-eq:2.1b}. It expresses this standard clipped product as natural products and only decreases its nonnegative affine expression. For each a", "anchor": { "source": "paper.tex", "line_start": 14211, "line_end": 14211, "char_start": 659296, "char_end": 659315 }, "origin": "source_reference", "context": "narrative" }, { "from": "old-eq:3.9", "to": "sec:plain", "type": "internal_reference", "evidence": "} =\\bigl(A_{\\rm act}-M'_{\\rm act}+(6\\kappa-1)z_{\\rm act}\\bigr)_+. \\] The parent satisfies Equation~\\eqref{old-eq:3.9}. Before this mask deletion, let \\(z'\\le z\\) be the surviving slot length of the clipped rectangle. Equations~\\eqref{old-eq:3.14", "anchor": { "source": "paper.tex", "line_start": 14221, "line_end": 14221, "char_start": 659737, "char_end": 659755 }, "origin": "source_reference", "context": "narrative" }, { "from": "old-eq:3.14", "to": "sec:plain", "type": "internal_reference", "evidence": "is mask deletion, let \\(z'\\le z\\) be the surviving slot length of the clipped rectangle. Equations~\\eqref{old-eq:3.14} and \\eqref{eq:centered-clipped-rectangle}, together with \\(M'_{\\rm act}\\ge M'\\), give \\[ A_{{\\rm clip},j,\\varrho}", "anchor": { "source": "paper.tex", "line_start": 14223, "line_end": 14223, "char_start": 659867, "char_end": 659886 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:centered-clipped-rectangle", "to": "sec:plain", "type": "internal_reference", "evidence": "z'\\le z\\) be the surviving slot length of the clipped rectangle. Equations~\\eqref{old-eq:3.14} and \\eqref{eq:centered-clipped-rectangle}, together with \\(M'_{\\rm act}\\ge M'\\), give \\[ A_{{\\rm clip},j,\\varrho}-M'_{\\rm act}+(6\\kappa-1)z' &\\le \\alpha_j", "anchor": { "source": "paper.tex", "line_start": 14224, "line_end": 14224, "char_start": 659891, "char_end": 659928 }, "origin": "source_reference", "context": "narrative" }, { "from": "old-eq:3.9", "to": "sec:plain", "type": "internal_reference", "evidence": "_{\\rm act}>0\\), remove whole slots from this product until the remaining product satisfies Equation~\\eqref{old-eq:3.9} or until no slot remains. Removal here means applying the pointwise bound Equation~\\eqref{old-eq:3.5} to the entire selected pri", "anchor": { "source": "paper.tex", "line_start": 14255, "line_end": 14255, "char_start": 661298, "char_end": 661316 }, "origin": "source_reference", "context": "narrative" }, { "from": "old-eq:3.5", "to": "sec:plain", "type": "internal_reference", "evidence": "ref{old-eq:3.9} or until no slot remains. Removal here means applying the pointwise bound Equation~\\eqref{old-eq:3.5} to the entire selected prime polynomial; no prime label is frozen. A removed slot of length \\(d\\) decreases \\(A_{\\rm act}+(6\\kap", "anchor": { "source": "paper.tex", "line_start": 14257, "line_end": 14257, "char_start": 661401, "char_end": 661419 }, "origin": "source_reference", "context": "narrative" }, { "from": "old-eq:3.16", "to": "sec:plain", "type": "internal_reference", "evidence": "en, so their possibly different rectangles and slot sets do not double \\(\\eta\\). Combining Equation~\\eqref{old-eq:3.16} with the paired coefficient bound in Equation~\\eqref{eq:centered-clipped-shell} and \\(M'_{\\rm act}-M'\\le\\delta_{{\\rm fr},2}\\), th", "anchor": { "source": "paper.tex", "line_start": 14286, "line_end": 14286, "char_start": 662991, "char_end": 663010 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:centered-clipped-shell", "to": "sec:plain", "type": "internal_reference", "evidence": "uble \\(\\eta\\). Combining Equation~\\eqref{old-eq:3.16} with the paired coefficient bound in Equation~\\eqref{eq:centered-clipped-shell} and \\(M'_{\\rm act}-M'\\le\\delta_{{\\rm fr},2}\\), the strict-edge scale cost beyond the child envelope is at most", "anchor": { "source": "paper.tex", "line_start": 14287, "line_end": 14287, "char_start": 663057, "char_end": 663090 }, "origin": "source_reference", "context": "narrative" }, { "from": "old-eq:2.7", "to": "sec:plain", "type": "internal_reference", "evidence": "ons give the single \\(O(\\xi+\\eta)\\) edge loss in the \\(\\epsilon_G\\) reserve inherited from Equation~\\eqref{old-eq:2.7} when using Equation~\\eqref{old-eq:2.14}; the existing local small-power and fixed analytic factors remain separate. The pointwis", "anchor": { "source": "paper.tex", "line_start": 14298, "line_end": 14298, "char_start": 663591, "char_end": 663609 }, "origin": "source_reference", "context": "narrative" }, { "from": "old-eq:2.14", "to": "sec:plain", "type": "internal_reference", "evidence": "e loss in the \\(\\epsilon_G\\) reserve inherited from Equation~\\eqref{old-eq:2.7} when using Equation~\\eqref{old-eq:2.14}; the existing local small-power and fixed analytic factors remain separate. The pointwise estimate is requested once for the ent", "anchor": { "source": "paper.tex", "line_start": 14299, "line_end": 14299, "char_start": 663630, "char_end": 663649 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:row-fixed-ray-reduction", "to": "sec:plain", "type": "internal_reference", "evidence": "on. We make explicit the finite-ray reduction for row factors at the fixed primes. As in Equation~\\eqref{eq:row-fixed-ray-reduction}, write a nonzero row as \\(h'=\\varepsilon h_{\\mathcal S}h_{\\rm good}\\), with \\(\\varepsilon\\) a unit and the two other factors supp", "anchor": { "source": "paper.tex", "line_start": 14329, "line_end": 14329, "char_start": 665334, "char_end": 665368 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:fixed-numerator-ray", "to": "sec:plain", "type": "internal_reference", "evidence": "cal S}) \\mathcal R(n,h_{\\rm good}) \\prod_{p\\mid h_{\\rm good}}\\chi_p(n)^{v_p(h')}. \\] By Lemma~\\ref{lem:fixed-numerator-ray}, the first factor belongs to a finite family of ray characters supported on \\(\\mathcal S\\) after the unit and the \\(\\mathcal S\\)-", "anchor": { "source": "paper.tex", "line_start": 14339, "line_end": 14339, "char_start": 665783, "char_end": 665812 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:exceptional-row-count", "to": "sec:plain", "type": "internal_reference", "evidence": "rows take the absolute product for each already allocated pair of children and multiply by Equation~\\eqref{eq:exceptional-row-count}. Let \\(c_{\\mathfrak t,\\mathbf J}(h')\\) denote its central extraction coefficient and bounded extracted scalars. For an allocati", "anchor": { "source": "paper.tex", "line_start": 14394, "line_end": 14394, "char_start": 668525, "char_end": 668557 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:centered-raw-normalization", "to": "sec:plain", "type": "internal_reference", "evidence": "ormalized live slots contribute \\(O_N(\\prod_{i\\in\\mathcal I_j\\setminus\\mathcal J_j}P_i)\\). Equation~\\eqref{eq:centered-raw-normalization} therefore gives the raw normalized volume \\[ |P_{j,\\mathfrak t,\\mathbf J}(h')| \\ll_N N_{{\\rm post},j}^{-1/2}T'_j \\", "anchor": { "source": "paper.tex", "line_start": 14404, "line_end": 14404, "char_start": 669136, "char_end": 669173 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:centered-divisor-boundary", "to": "sec:plain", "type": "internal_reference", "evidence": "detilde r_1-\\widetilde r_2 \\le a_0-b_2+e_1+(e_2+\\omega_2) \\le a_0-b_2+2\\theta_N. \\] Here Equation~\\eqref{eq:centered-divisor-boundary} gives \\(t_- -\\widetilde r_2\\le\\omega_2\\), and \\(\\widetilde r_1\\ge0\\). Consequently \\[ \\sum_{\\mathfrak t,\\mathbf J} |c_{\\mathfra", "anchor": { "source": "paper.tex", "line_start": 14420, "line_end": 14420, "char_start": 669712, "char_end": 669748 }, "origin": "source_reference", "context": "narrative" }, { "from": "old-eq:2.15", "to": "sec:plain", "type": "internal_reference", "evidence": "abel{old-eq:2.16} The actual excess is bounded by the nominal expression in Equation~\\eqref{old-eq:2.15} plus \\(\\delta_{{\\rm fr},2}/6+2\\theta_N+\\epsilon_1\\). For an explicit check, the left side of Equation~\\eqref{old-eq:2.15} first e", "anchor": { "source": "paper.tex", "line_start": 14451, "line_end": 14451, "char_start": 670743, "char_end": 670762 }, "origin": "source_reference", "context": "narrative" }, { "from": "old-eq:2.15", "to": "sec:plain", "type": "internal_reference", "evidence": "us \\(\\delta_{{\\rm fr},2}/6+2\\theta_N+\\epsilon_1\\). For an explicit check, the left side of Equation~\\eqref{old-eq:2.15} first equals \\[ a_0-2b_2+p_2-w-\\mathcal B_c-\\ell -\\tfrac56m'-q'-f/6. \\] Substitution of Equation~\\eqref{old-eq:2.13} and", "anchor": { "source": "paper.tex", "line_start": 14454, "line_end": 14454, "char_start": 670865, "char_end": 670884 }, "origin": "source_reference", "context": "narrative" }, { "from": "old-eq:2.13", "to": "sec:plain", "type": "internal_reference", "evidence": "equals \\[ a_0-2b_2+p_2-w-\\mathcal B_c-\\ell -\\tfrac56m'-q'-f/6. \\] Substitution of Equation~\\eqref{old-eq:2.13} and \\(K=2A-c-d+R+E-m-(K_0-K)\\) gives Equations~\\eqref{old-eq:2.15}--\\eqref{old-eq:2.16}. The following lower bounds are the reas", "anchor": { "source": "paper.tex", "line_start": 14459, "line_end": 14459, "char_start": 670990, "char_end": 671009 }, "origin": "source_reference", "context": "narrative" }, { "from": "old-eq:2.15", "to": "sec:plain", "type": "internal_reference", "evidence": "f/6. \\] Substitution of Equation~\\eqref{old-eq:2.13} and \\(K=2A-c-d+R+E-m-(K_0-K)\\) gives Equations~\\eqref{old-eq:2.15}--\\eqref{old-eq:2.16}. The following lower bounds are the reason complete common supports do not consume the available exponent:", "anchor": { "source": "paper.tex", "line_start": 14461, "line_end": 14461, "char_start": 671057, "char_end": 671076 }, "origin": "source_reference", "context": "narrative" }, { "from": "old-eq:2.16", "to": "sec:plain", "type": "internal_reference", "evidence": "of Equation~\\eqref{old-eq:2.13} and \\(K=2A-c-d+R+E-m-(K_0-K)\\) gives Equations~\\eqref{old-eq:2.15}--\\eqref{old-eq:2.16}. The following lower bounds are the reason complete common supports do not consume the available exponent: F_1", "anchor": { "source": "paper.tex", "line_start": 14461, "line_end": 14461, "char_start": 671078, "char_end": 671097 }, "origin": "source_reference", "context": "narrative" }, { "from": "old-eq:2.10", "to": "sec:plain", "type": "internal_reference", "evidence": "c56\\delta_{{\\rm fr},1}. \\] The actual choices have \\(g\\le J_++2\\sigma\\) and \\(\\ell\\ge0\\). Equation~\\eqref{old-eq:2.10} therefore bounds the additional loss from \\(2(c+w)/3\\) by \\[ w/3+5\\sigma/3+\\tfrac56\\delta_{{\\rm fr},1} \\le22\\sigma/9+\\tfrac56\\d", "anchor": { "source": "paper.tex", "line_start": 14496, "line_end": 14496, "char_start": 672131, "char_end": 672150 }, "origin": "source_reference", "context": "narrative" }, { "from": "old-eq:2.17", "to": "sec:plain", "type": "internal_reference", "evidence": "delta_{{\\rm fr},1} <3\\sigma+\\tfrac56\\delta_{{\\rm fr},1}. \\] For the second inequality in Equation~\\eqref{old-eq:2.17}, compute prime by prime from the definition of \\(F_2\\). In units of the prime's logarithmic norm, the contributions are \\[ \\beg", "anchor": { "source": "paper.tex", "line_start": 14504, "line_end": 14504, "char_start": 672379, "char_end": 672398 }, "origin": "source_reference", "context": "narrative" }, { "from": "old-eq:2.15", "to": "sec:plain", "type": "internal_reference", "evidence": "\\) and \\(i\\ge j_0+1\\). Summing proves the claim. For the uncentered range \\(A\\le5M/6\\), Equations~\\eqref{old-eq:2.15}--\\eqref{old-eq:2.17} bound the nominal exceptional excess by \\(3\\sigma+5\\delta_{{\\rm fr},1}/6\\). The actual-count and aggregate", "anchor": { "source": "paper.tex", "line_start": 14527, "line_end": 14527, "char_start": 673235, "char_end": 673254 }, "origin": "source_reference", "context": "narrative" }, { "from": "old-eq:2.17", "to": "sec:plain", "type": "internal_reference", "evidence": ". Summing proves the claim. For the uncentered range \\(A\\le5M/6\\), Equations~\\eqref{old-eq:2.15}--\\eqref{old-eq:2.17} bound the nominal exceptional excess by \\(3\\sigma+5\\delta_{{\\rm fr},1}/6\\). The actual-count and aggregate support corrections r", "anchor": { "source": "paper.tex", "line_start": 14527, "line_end": 14528, "char_start": 673256, "char_end": 673275 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:centered-coefficient-invariant", "to": "sec:plain", "type": "internal_reference", "evidence": "en \\(A>5M/6\\). For one separated side of a fixed exceptional row, fix the live slot labels. Lemma~\\ref{lem:centered-coefficient-invariant} then gives the same character, puncture, and norm power in the two plain variables and in both rectangle terms. The next lemma s", "anchor": { "source": "paper.tex", "line_start": 14539, "line_end": 14539, "char_start": 673862, "char_end": 673902 }, "origin": "source_reference", "context": "narrative" }, { "from": "old-eq:2.18a", "to": "lem:centered-lattice-cancellation", "type": "internal_reference", "evidence": "), for the stated masks, and for the finite character set. Let \\(D_{\\mathbf b}\\) be as in Equation~\\eqref{old-eq:2.18a}, and put \\[ U_i=X_i/q_{\\mathfrak b_i},\\qquad V_i=Y_i/q_{\\mathfrak b_i},\\qquad T=U_1U_2=V_1V_2. \\] If \\(U_i,V_i\\ge Z^r\\) for \\(", "anchor": { "source": "paper.tex", "line_start": 14574, "line_end": 14574, "char_start": 675243, "char_end": 675263 }, "origin": "source_reference", "context": "statement" }, { "from": "old-eq:2.18d", "to": "lem:centered-lattice-cancellation", "type": "candidate_proof_dependency", "evidence": "\\) is the fixed weighted mean of the residue classes. This proves the unmasked version of Equation~\\eqref{old-eq:2.18d}. Now insert the mask by inclusion--exclusion. First remove from \\(\\mathfrak R_*\\) its primes in \\(\\mathcal S\\), since every sum", "anchor": { "source": "paper.tex", "line_start": 14628, "line_end": 14628, "char_start": 676930, "char_end": 676950 }, "origin": "source_reference", "context": "proof" }, { "from": "old-eq:2.18d", "to": "lem:centered-lattice-cancellation", "type": "candidate_proof_dependency", "evidence": "lso \\(Z^{\\epsilon_1}\\)-bounded by the polynomial-size Euler-product estimate. This proves Equation~\\eqref{old-eq:2.18d}. For Equation~\\eqref{old-eq:2.18e}, take absolute values in the original sum. It is empty below a fixed positive scale; wheneve", "anchor": { "source": "paper.tex", "line_start": 14653, "line_end": 14653, "char_start": 677923, "char_end": 677943 }, "origin": "source_reference", "context": "proof" }, { "from": "old-eq:2.18e", "to": "lem:centered-lattice-cancellation", "type": "candidate_proof_dependency", "evidence": "e polynomial-size Euler-product estimate. This proves Equation~\\eqref{old-eq:2.18d}. For Equation~\\eqref{old-eq:2.18e}, take absolute values in the original sum. It is empty below a fixed positive scale; whenever it is nonempty, ideal counting on", "anchor": { "source": "paper.tex", "line_start": 14655, "line_end": 14655, "char_start": 677959, "char_end": 677979 }, "origin": "source_reference", "context": "proof" }, { "from": "old-eq:2.18f", "to": "lem:centered-lattice-cancellation", "type": "candidate_proof_dependency", "evidence": "nulus gives \\(O(X)\\) points. This is uniform in \\(t\\) and in the mask. The masked sum in Equation~\\eqref{old-eq:2.18f} is exactly \\[ \\mathcal L_1(U_1,t)\\mathcal L_2(U_2,t) -\\mathcal L_1(V_1,t)\\mathcal L_2(V_2,t). \\] The two product main terms in", "anchor": { "source": "paper.tex", "line_start": 14660, "line_end": 14660, "char_start": 678221, "char_end": 678241 }, "origin": "source_reference", "context": "proof" }, { "from": "old-eq:2.18d", "to": "lem:centered-lattice-cancellation", "type": "candidate_proof_dependency", "evidence": "hcal L_2(U_2,t) -\\mathcal L_1(V_1,t)\\mathcal L_2(V_2,t). \\] The two product main terms in Equation~\\eqref{old-eq:2.18d} are both \\(c_{\\vartheta,\\mathfrak R_*}^2T^{1+it}I_1(t)I_2(t)\\). They cancel. Each cross term with one error is bounded by \\(Z^{\\", "anchor": { "source": "paper.tex", "line_start": 14666, "line_end": 14666, "char_start": 678380, "char_end": 678400 }, "origin": "source_reference", "context": "proof" }, { "from": "old-eq:2.18f", "to": "lem:centered-lattice-cancellation", "type": "candidate_proof_dependency", "evidence": "ence is therefore \\(O(Z^{\\epsilon_1}(1+|t|)^{2J}TZ^{-r})\\). Division by \\(T^{1/2}\\) proves Equation~\\eqref{old-eq:2.18f}. When no nonnegative lower length is available, Equation~\\eqref{old-eq:2.18e} bounds each product by \\(O(T)\\), giving the last a", "anchor": { "source": "paper.tex", "line_start": 14677, "line_end": 14677, "char_start": 678903, "char_end": 678923 }, "origin": "source_reference", "context": "proof" }, { "from": "old-eq:2.18e", "to": "lem:centered-lattice-cancellation", "type": "candidate_proof_dependency", "evidence": "2}\\) proves Equation~\\eqref{old-eq:2.18f}. When no nonnegative lower length is available, Equation~\\eqref{old-eq:2.18e} bounds each product by \\(O(T)\\), giving the last assertion. Apply this lemma to each already allocated exceptional c", "anchor": { "source": "paper.tex", "line_start": 14678, "line_end": 14678, "char_start": 678982, "char_end": 679002 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:centered-coefficient-invariant", "to": "sec:plain", "type": "internal_reference", "evidence": "y allocated exceptional child. The hypotheses on its coefficient and common mask follow from Lemma~\\ref{lem:centered-coefficient-invariant}; in particular the main terms cancel for the same row, mask, and whole-product norm power before any absolute value. Initially a", "anchor": { "source": "paper.tex", "line_start": 14684, "line_end": 14684, "char_start": 679207, "char_end": 679247 }, "origin": "source_reference", "context": "narrative" }, { "from": "old-eq:2.18f", "to": "sec:plain", "type": "internal_reference", "evidence": "plain scales have logarithmic length at least \\(r-a_{1,i}\\ge r-\\widetilde r_1>0\\). Apply Equation~\\eqref{old-eq:2.18f} on these exact equal-product formal scales. If \\(\\widetilde r_1\\ge r\\), including equality, or if \\(r=0\\), use the all-scale abso", "anchor": { "source": "paper.tex", "line_start": 14701, "line_end": 14701, "char_start": 680118, "char_end": 680138 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:centered-divisor-boundary", "to": "sec:plain", "type": "internal_reference", "evidence": "and reduction \\(-\\widetilde r_1-\\widetilde r_2\\), while the divisor count gives \\(t_-\\). Equation~\\eqref{eq:centered-divisor-boundary} gives the one-line inequality t_- -\\widetilde r_1-\\widetilde r_2-(r-\\widetilde r_1)_+ &=(t_- -\\", "anchor": { "source": "paper.tex", "line_start": 14723, "line_end": 14723, "char_start": 681141, "char_end": 681177 }, "origin": "source_reference", "context": "narrative" }, { "from": "old-eq:2.18", "to": "sec:plain", "type": "internal_reference", "evidence": "r\\le2\\theta_N-r\\). The divisor-bounded allocations contribute only \\(\\epsilon_1\\), proving Equation~\\eqref{old-eq:2.18}. This applies to the artificial terms even when quotients retain \\(\\mathfrak t\\)-primes, because the coefficient lemma preserves", "anchor": { "source": "paper.tex", "line_start": 14736, "line_end": 14736, "char_start": 681569, "char_end": 681588 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:centered-raw-normalization", "to": "sec:plain", "type": "internal_reference", "evidence": "). Combining this with the nominal raw normalizer gives the full \\(e_j\\) volume factor in Equation~\\eqref{eq:centered-raw-normalization}, as included above; it is not replaced by a factor-product normalizer. The Fourier measure was fixed before their labels, so con", "anchor": { "source": "paper.tex", "line_start": 14746, "line_end": 14746, "char_start": 682084, "char_end": 682121 }, "origin": "source_reference", "context": "narrative" }, { "from": "old-eq:2.17", "to": "sec:plain", "type": "internal_reference", "evidence": "scalar is discarded only in the present absolute bound. Set \\(v=c+w+\\min(c_2,d_2)\\). By Equation~\\eqref{old-eq:2.17}, \\(F_1+F_2\\) is at least \\(2v/3-3\\sigma-5\\delta_{{\\rm fr},1}/6\\), since \\(b_2\\ge\\min(c_2,d_2)\\). The centered saving in Equation~", "anchor": { "source": "paper.tex", "line_start": 14753, "line_end": 14753, "char_start": 682458, "char_end": 682477 }, "origin": "source_reference", "context": "narrative" }, { "from": "old-eq:2.18", "to": "sec:plain", "type": "internal_reference", "evidence": "/3-3\\sigma-5\\delta_{{\\rm fr},1}/6\\), since \\(b_2\\ge\\min(c_2,d_2)\\). The centered saving in Equation~\\eqref{old-eq:2.18} now bounds the exceptional deficit, apart from \\(3\\sigma\\) and the recorded frequency and aggregate support perturbations, by \\be", "anchor": { "source": "paper.tex", "line_start": 14756, "line_end": 14756, "char_start": 682607, "char_end": 682626 }, "origin": "source_reference", "context": "narrative" }, { "from": "old-eq:3.9", "to": "sec:plain", "type": "internal_reference", "evidence": "decreases. Its maximum is attained at \\(v=L=M/4\\) and equals \\(A-M\\). For positive slots, Equation~\\eqref{old-eq:3.9} implies \\(A\\le M\\). For the zero-slot core, Equation~\\eqref{old-eq:2.1h} gives \\(A-M\\le\\delta\\). Thus the centered exceptional t", "anchor": { "source": "paper.tex", "line_start": 14768, "line_end": 14768, "char_start": 683085, "char_end": 683103 }, "origin": "source_reference", "context": "narrative" }, { "from": "old-eq:2.1h", "to": "sec:plain", "type": "internal_reference", "evidence": "r positive slots, Equation~\\eqref{old-eq:3.9} implies \\(A\\le M\\). For the zero-slot core, Equation~\\eqref{old-eq:2.1h} gives \\(A-M\\le\\delta\\). Thus the centered exceptional terms require only the terminal loss \\(3\\sigma\\), or \\(\\delta+3\\sigma\\) in", "anchor": { "source": "paper.tex", "line_start": 14770, "line_end": 14770, "char_start": 683158, "char_end": 683177 }, "origin": "source_reference", "context": "narrative" }, { "from": "old-eq:2.12", "to": "sec:plain", "type": "internal_reference", "evidence": "and \\(\\theta_N\\) terms are included in the same \\(C_*\\xi\\) stage allowance. Together with Equation~\\eqref{old-eq:2.12}, this completes the centered stage at the current width. \\subsection{Completion of the finite induction} We finish by verifying", "anchor": { "source": "paper.tex", "line_start": 14776, "line_end": 14776, "char_start": 683488, "char_end": 683507 }, "origin": "source_reference", "context": "narrative" }, { "from": "old-eq:2.13", "to": "sec:plain", "type": "internal_reference", "evidence": "nd the finite induction. Every nonterminal call is to a child with nominal width \\(M'\\) in Equation~\\eqref{old-eq:2.13}, at least \\(\\sigma\\) below its parent. The declared row width is \\(M'_{\\rm act}\\), and all nonexceptional moment-input lengths ar", "anchor": { "source": "paper.tex", "line_start": 14783, "line_end": 14783, "char_start": 683760, "char_end": 683779 }, "origin": "source_reference", "context": "narrative" }, { "from": "old-eq:2.1a", "to": "sec:plain", "type": "internal_reference", "evidence": "already been included by its enclosing length and clipping error. At each smaller width, Equations \\eqref{old-eq:2.1a}--\\eqref{old-eq:2.1b} delete the fixed extra masks before the natural positive-slot estimate. The unrestricted zero-slot assertio", "anchor": { "source": "paper.tex", "line_start": 14796, "line_end": 14796, "char_start": 684482, "char_end": 684501 }, "origin": "source_reference", "context": "narrative" }, { "from": "old-eq:2.1b", "to": "sec:plain", "type": "internal_reference", "evidence": "by its enclosing length and clipping error. At each smaller width, Equations \\eqref{old-eq:2.1a}--\\eqref{old-eq:2.1b} delete the fixed extra masks before the natural positive-slot estimate. The unrestricted zero-slot assertion follows once from t", "anchor": { "source": "paper.tex", "line_start": 14796, "line_end": 14796, "char_start": 684503, "char_end": 684522 }, "origin": "source_reference", "context": "narrative" }, { "from": "old-eq:3.16", "to": "sec:plain", "type": "internal_reference", "evidence": "uss norm has at most one amplification, whose errors go directly to the second transform. Equation~\\eqref{old-eq:3.16} is invoked once for each actual natural child, after all its boundary defects have been included in \\(F_{\\rm act}\\). Thus there", "anchor": { "source": "paper.tex", "line_start": 14802, "line_end": 14802, "char_start": 684914, "char_end": 684933 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:centered-aggregate-support", "to": "sec:plain", "type": "internal_reference", "evidence": "). Here is why \\(C_*\\) and the mesh can be chosen independently of the fixed slot count. Equation~\\eqref{eq:centered-aggregate-support} bounds the entire logarithmic error of any slot subset by \\(\\theta_N=H_N/\\log Z\\). Each extracted plain power is measured exactl", "anchor": { "source": "paper.tex", "line_start": 14810, "line_end": 14811, "char_start": 685349, "char_end": 685386 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:kernel-seminorms", "to": "sec:plain", "type": "internal_reference", "evidence": "g norms. Every derivative is a fixed combination of Euler derivatives of \\(\\widehat\\Phi_i\\). Lemma~\\ref{lem:kernel-seminorms} bounds these uniformly for all \\(R_{\\rm sc}>0\\), with any prescribed radial decay and derivative orders. One does not use the wea", "anchor": { "source": "paper.tex", "line_start": 14839, "line_end": 14839, "char_start": 687021, "char_end": 687047 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:smooth-calculus", "to": "sec:plain", "type": "internal_reference", "evidence": "he fixed box; its derivatives cost only fixed constants depending on \\(H_N\\) and their order. Lemma~\\ref{lem:smooth-calculus} uses the full weighted Fourier integral, not a supremum over heights below \\(Z^\\xi\\). A required height degree \\(J\\), which may", "anchor": { "source": "paper.tex", "line_start": 14846, "line_end": 14846, "char_start": 687435, "char_end": 687460 }, "origin": "source_reference", "context": "narrative" }, { "from": "old-eq:3.5", "to": "sec:plain", "type": "internal_reference", "evidence": "the upper-annulus subshare \\(\\xi/2\\) remains fixed. The fixed polynomial height factor in Equation~\\eqref{old-eq:3.5} is integrated by these weighted Fourier measures at its required finite order, without changing the previously chosen exponent of", "anchor": { "source": "paper.tex", "line_start": 14853, "line_end": 14854, "char_start": 687932, "char_end": 687950 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:centered-reflection-length", "to": "sec:plain", "type": "internal_reference", "evidence": "constants. After imposing an eventual threshold \\(\\log Z\\ge C H_N/\\xi\\), the threshold in Equation~\\eqref{eq:centered-reflection-length}, and Equation~\\eqref{eq:centered-localization-threshold}, all occurring column, radical, and mask norms have bounded logarithmic", "anchor": { "source": "paper.tex", "line_start": 14897, "line_end": 14897, "char_start": 689984, "char_end": 690021 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:centered-localization-threshold", "to": "sec:plain", "type": "internal_reference", "evidence": "log Z\\ge C H_N/\\xi\\), the threshold in Equation~\\eqref{eq:centered-reflection-length}, and Equation~\\eqref{eq:centered-localization-threshold}, all occurring column, radical, and mask norms have bounded logarithmic ranges independent of moving labels. In those ranges cho", "anchor": { "source": "paper.tex", "line_start": 14898, "line_end": 14898, "char_start": 690036, "char_end": 690078 }, "origin": "source_reference", "context": "narrative" }, { "from": "old-eq:2.5", "to": "sec:plain", "type": "internal_reference", "evidence": "C_*(\\eta+\\xi+\\epsilon_0)\\), giving \\(\\mathcal E_d\\). The Gauss allowance \\(\\epsilon_G\\) in Equation~\\eqref{old-eq:2.5} denotes this appropriate envelope and the current local shares; it is not a request to reapply the moment lemma with an \\(N\\)-dep", "anchor": { "source": "paper.tex", "line_start": 14932, "line_end": 14932, "char_start": 691846, "char_end": 691864 }, "origin": "source_reference", "context": "narrative" }, { "from": "old-eq:2.1i", "to": "sec:plain", "type": "internal_reference", "evidence": "The largest path error is at most \\(T_{\\rm term}+C_*D(\\eta+\\xi+\\epsilon_0)<\\epsilon\\) by Equation~\\eqref{old-eq:2.1i}. Finally choose the required finite kernel, reflection, prime, and terminal seminorm and polynomial-height orders backwards thro", "anchor": { "source": "paper.tex", "line_start": 14940, "line_end": 14940, "char_start": 692337, "char_end": 692356 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:smooth-calculus", "to": "sec:plain", "type": "internal_reference", "evidence": "ding the reflection-kernel orders, are fixed first. The later external Fourier-tail order in Lemma~\\ref{lem:smooth-calculus} lies outside this internal propagation: it raises only the external input seminorm and does not change a previously fixed interna", "anchor": { "source": "paper.tex", "line_start": 14968, "line_end": 14968, "char_start": 693871, "char_end": 693896 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:plain", "to": "sec:plain", "type": "internal_reference", "evidence": "ity. The finite induction proves the natural assertion, and the initial mask deletion proves Lemma~\\ref{lem:plain} in its full stated form. For the later physical application, \\(q=0\\) and the row \\(u\\) is sixth-power-free. If a prime outside", "anchor": { "source": "paper.tex", "line_start": 14973, "line_end": 14973, "char_start": 694188, "char_end": 694203 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:exceptional-row-count", "to": "sec:plain", "type": "internal_reference", "evidence": "oving twists, additional exceptional rows can arise by cancellation. They were included in Equation~\\eqref{eq:exceptional-row-count} and the fixed-character volume argument. \\section{Prime amplitudes and refined row counts} We use", "anchor": { "source": "paper.tex", "line_start": 14983, "line_end": 14984, "char_start": 694761, "char_end": 694793 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:compensated-probe-definition", "to": "sec:refined-row-counts", "type": "internal_reference", "evidence": ":refined-row-counts} We use the current Part~II arithmetic data and the physical slots of Equation~\\eqref{eq:compensated-probe-definition}. Instantiate the shared zero detector of Section~\\ref{sec:detector} with these same fixed data. The main and error factors below", "anchor": { "source": "paper.tex", "line_start": 14988, "line_end": 14988, "char_start": 694994, "char_end": 695033 }, "origin": "source_reference", "context": "narrative" }, { "from": "sec:detector", "to": "sec:refined-row-counts", "type": "internal_reference", "evidence": "Equation~\\eqref{eq:compensated-probe-definition}. Instantiate the shared zero detector of Section~\\ref{sec:detector} with these same fixed data. The main and error factors below belong to the retained dynamic decomposition in Equation~\\eqref{eq:d", "anchor": { "source": "paper.tex", "line_start": 14989, "line_end": 14989, "char_start": 695084, "char_end": 695102 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:dynamic-compensated-decomposition", "to": "sec:refined-row-counts", "type": "internal_reference", "evidence": "xed data. The main and error factors below belong to the retained dynamic decomposition in Equation~\\eqref{eq:dynamic-compensated-decomposition}; no individual local quotient is asserted outside that region. Every retained external coordinate obeys the single height alloca", "anchor": { "source": "paper.tex", "line_start": 14991, "line_end": 14991, "char_start": 695221, "char_end": 695265 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:stage-height-allocation", "to": "sec:refined-row-counts", "type": "internal_reference", "evidence": "ide that region. Every retained external coordinate obeys the single height allocation in Equation~\\eqref{eq:stage-height-allocation}. \\subsection{Prime amplitudes} For a main physical slot of scale \\(P_i=Z^{\\ell_i}\\), write its central fa", "anchor": { "source": "paper.tex", "line_start": 14993, "line_end": 14993, "char_start": 695412, "char_end": 695446 }, "origin": "source_reference", "context": "narrative" }, { "from": "prop:probe-errors", "to": "sec:refined-row-counts", "type": "internal_reference", "evidence": "\\overline{\\chi_p(u)}W_i(q_p/P_i)(q_p/P_i)^{z-1}. \\] The main part \\(\\mathcal Q_i\\) in Proposition~\\ref{prop:probe-errors} is exactly \\[ \\mathcal Q_i(u;z)=-\\sum_{p\\in1_T}\\overline{\\chi_p(u)}q_p^{z-1}W_i(q_p/P_i) =-P_i^{z-1/2}Q_i(u;z). \\] As in Sectio", "anchor": { "source": "paper.tex", "line_start": 15003, "line_end": 15003, "char_start": 695723, "char_end": 695746 }, "origin": "source_reference", "context": "narrative" }, { "from": "sec:compensation", "to": "sec:refined-row-counts", "type": "internal_reference", "evidence": ")=-\\sum_{p\\in1_T}\\overline{\\chi_p(u)}q_p^{z-1}W_i(q_p/P_i) =-P_i^{z-1/2}Q_i(u;z). \\] As in Section~\\ref{sec:compensation}, each sum over \\(p\\in1_T\\) retains the same allowed set \\(\\mathcal P_i(Z)\\), including its exclusion of \\(S\\). The original zero", "anchor": { "source": "paper.tex", "line_start": 15008, "line_end": 15008, "char_start": 695878, "char_end": 695900 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:detector-dyads", "to": "lem:bin-prime-bound", "type": "internal_reference", "evidence": "ance. [Prime bound in a bin] Under the hypotheses of Lemma~\\ref{lem:detector-dyads}, with the prime-annulus Mellin frequency included in its cumulative allowance, for every \\(\\epsilon_1>0\\) \\[ |Q_i(u;z)|\\ll_{\\mat", "anchor": { "source": "paper.tex", "line_start": 15016, "line_end": 15016, "char_start": 696365, "char_end": 696389 }, "origin": "source_reference", "context": "statement" }, { "from": "lem:logarithmic-control", "to": "lem:bin-prime-bound", "type": "candidate_proof_dependency", "evidence": "\\theta\\in\\widehat T}\\theta(p)\\). Every resulting prime character belongs to \\(\\mathcal X_u\\). Lemma~\\ref{lem:logarithmic-control}, applied to the buffered disks, bounds its logarithmic derivative on \\(\\Re s=a+8e\\) by \\(O_{\\mathcal A,e}(\\log U)\\); the logarith", "anchor": { "source": "paper.tex", "line_start": 15030, "line_end": 15030, "char_start": 696863, "char_end": 696892 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:pointwise-mellin-tail", "to": "lem:bin-prime-bound", "type": "candidate_proof_dependency", "evidence": "n the starting line \\(\\Re s=2\\) it is absolutely bounded. Apply the pointwise estimate in Equation~\\eqref{eq:pointwise-mellin-tail} to the untwisted transform for the joins, and Equation~\\eqref{eq:late-fourier-tail} for the absolute-line tails. Choosing the ex", "anchor": { "source": "paper.tex", "line_start": 15043, "line_end": 15043, "char_start": 697636, "char_end": 697668 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:late-fourier-tail", "to": "lem:bin-prime-bound", "type": "candidate_proof_dependency", "evidence": "in Equation~\\eqref{eq:pointwise-mellin-tail} to the untwisted transform for the joins, and Equation~\\eqref{eq:late-fourier-tail} for the absolute-line tails. Choosing the external order after their fixed polynomial scale and height bounds gives the asserted", "anchor": { "source": "paper.tex", "line_start": 15044, "line_end": 15044, "char_start": 697724, "char_end": 697752 }, "origin": "source_reference", "context": "proof" }, { "from": "old-eq:5.13e", "to": "sec:refined-row-counts", "type": "internal_reference", "evidence": "ght)\\right\\}. \\] Put \\(g_i=0\\) if \\(Q_i=0\\), and also put \\(g_i=0\\) for an error slot from Equation~\\eqref{old-eq:5.13e}. Then, for a main slot, \\[ |Q_i|\\le P_i^{g_i+\\vartheta},\\qquad g_i>0\\ \\Longrightarrow\\ |Q_i|\\ge P_i^{g_i}. \\] No lower bound i", "anchor": { "source": "paper.tex", "line_start": 15078, "line_end": 15078, "char_start": 699223, "char_end": 699243 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:part-II-bootstrap", "to": "sec:refined-row-counts", "type": "internal_reference", "evidence": "d prime slots and the row counts} Use \\(\\Delta\\) and \\(\\kappa\\) from Equation~\\eqref{eq:part-II-bootstrap}: \\[ 0<\\Delta\\le\\frac1{24},\\qquad \\kappa=2\\beta_*-1=\\frac34+2\\Delta\\in(3/4,5/6],\\qquad \\alpha:=\\frac56. \\] The parameter \\(\\kap", "anchor": { "source": "paper.tex", "line_start": 15097, "line_end": 15097, "char_start": 699954, "char_end": 699982 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:plain", "to": "sec:refined-row-counts", "type": "internal_reference", "evidence": "lification. Since \\(\\kappa<1\\) and \\(\\beta_*=(1+\\kappa)/2\\), the positive-slot hypothesis of Lemma~\\ref{lem:plain} is satisfied by equality. Equation~\\eqref{eq:part-II-bin-ceiling} gives \\(\\delta\\le\\kappa\\le\\alpha\\) for the current Part~II bin", "anchor": { "source": "paper.tex", "line_start": 15106, "line_end": 15106, "char_start": 700284, "char_end": 700299 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:part-II-bin-ceiling", "to": "sec:refined-row-counts", "type": "internal_reference", "evidence": "ppa)/2\\), the positive-slot hypothesis of Lemma~\\ref{lem:plain} is satisfied by equality. Equation~\\eqref{eq:part-II-bin-ceiling} gives \\(\\delta\\le\\kappa\\le\\alpha\\) for the current Part~II bins, including the possible endpoint \\(\\delta=\\alpha\\). Fix a dynami", "anchor": { "source": "paper.tex", "line_start": 15106, "line_end": 15107, "char_start": 700336, "char_end": 700366 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:plain", "to": "sec:refined-row-counts", "type": "internal_reference", "evidence": "sjoint before masks. The marked moment permits this negative common row orientation. To apply Lemma~\\ref{lem:plain}, whose row character has positive orientation, conjugate the whole product of both plain witness factors and all selected prime f", "anchor": { "source": "paper.tex", "line_start": 15152, "line_end": 15152, "char_start": 702606, "char_end": 702621 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:smooth-calculus", "to": "sec:refined-row-counts", "type": "internal_reference", "evidence": "ta\\), since \\(\\Theta\\) is a group. The witness and physical heights need not be equal. Apply Lemma~\\ref{lem:smooth-calculus} to the rowwise witness parameters after fixing the physical parameters and selected indices. Derivatives in those parameters inse", "anchor": { "source": "paper.tex", "line_start": 15164, "line_end": 15164, "char_start": 703316, "char_end": 703341 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:plain", "to": "sec:refined-row-counts", "type": "internal_reference", "evidence": "(\\Theta\\), because it ramifies at a prime outside \\(S\\). Thus the \\(z>0\\) family condition of Lemma~\\ref{lem:plain} holds. For an actual plain length \\(m\\le1/2\\), its largest zero-loss capacity from Lemma~\\ref{lem:plain}, at effective row width", "anchor": { "source": "paper.tex", "line_start": 15170, "line_end": 15170, "char_start": 703735, "char_end": 703750 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:plain", "to": "sec:refined-row-counts", "type": "internal_reference", "evidence": "em:plain} holds. For an actual plain length \\(m\\le1/2\\), its largest zero-loss capacity from Lemma~\\ref{lem:plain}, at effective row width one, is z_P(m)=\\frac{1-2m}{6\\kappa} =\\frac{1-2m}{9/2+12\\Delta}. \\label{eq:plain", "anchor": { "source": "paper.tex", "line_start": 15173, "line_end": 15173, "char_start": 703841, "char_end": 703856 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:marked", "to": "sec:refined-row-counts", "type": "internal_reference", "evidence": "1\\), put \\(z_M(r)=(1-r)/2\\). We use this capacity only where the second strict inequality in Lemma~\\ref{lem:marked} also has a fixed margin. [Row counts from the witnesses] Let \\(\\mathcal B\\) be a", "anchor": { "source": "paper.tex", "line_start": 15183, "line_end": 15183, "char_start": 704266, "char_end": 704282 }, "origin": "source_reference", "context": "narrative" }, { "from": "prop:detector-witness", "to": "prop:detector-counts", "type": "internal_reference", "evidence": "t)=1-\\delta+(\\alpha-\\delta)(t-1). \\] The estimate is valid for the rowwise witnesses of Proposition~\\ref{prop:detector-witness}, with its height condition. If no prime slots are selected, then \\(t=1\\) instead gives \\[ \\#\\mathcal B\\ll_{\\mathcal A,\\epsilon}", "anchor": { "source": "paper.tex", "line_start": 15211, "line_end": 15211, "char_start": 705340, "char_end": 705367 }, "origin": "source_reference", "context": "statement" }, { "from": "lem:inverse-amplification", "to": "prop:detector-counts", "type": "internal_reference", "evidence": "(1+T_1)^{A_{\\mathcal A}}, \\] without a prime-supply hypothesis. At zero inverse capacity use Lemma~\\ref{lem:inverse-amplification}; at zero plain capacity use the zero-slot case of Lemma~\\ref{lem:plain}. Subdivide by the witnes", "anchor": { "source": "paper.tex", "line_start": 15218, "line_end": 15218, "char_start": 705622, "char_end": 705653 }, "origin": "source_reference", "context": "statement" }, { "from": "lem:plain", "to": "prop:detector-counts", "type": "internal_reference", "evidence": "y use Lemma~\\ref{lem:inverse-amplification}; at zero plain capacity use the zero-slot case of Lemma~\\ref{lem:plain}. Subdivide by the witness presentation and dyadic pair. The number of choices is \\(O_{\\mathcal", "anchor": { "source": "paper.tex", "line_start": 15219, "line_end": 15219, "char_start": 705710, "char_end": 705725 }, "origin": "source_reference", "context": "statement" }, { "from": "prop:detector-witness", "to": "prop:detector-counts", "type": "candidate_proof_dependency", "evidence": "vision, denoting it again by \\(\\mathcal B\\), and let \\(r,m\\) be the actual lengths from Proposition~\\ref{prop:detector-witness}. Whenever the selected slots satisfy the corresponding moment hypotheses and have total requested capacity \\(z\\), their spike an", "anchor": { "source": "paper.tex", "line_start": 15228, "line_end": 15228, "char_start": 706078, "char_end": 706105 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:marked", "to": "prop:detector-counts", "type": "candidate_proof_dependency", "evidence": "ray}{ll} \\#\\mathcal B \\ll U^{1-\\delta r-2qz+\\epsilon}(1+T_1)^{A_{\\mathcal A}}, &\\text{from Lemma~\\ref{lem:marked}},\\\\[2mm] \\#\\mathcal B \\ll U^{1-2\\delta m-2qz+\\epsilon}(1+T_1)^{A_{\\mathcal A}}, &\\text{from Lemma~\\ref{lem:plain}}. \\end{arra", "anchor": { "source": "paper.tex", "line_start": 15236, "line_end": 15236, "char_start": 706422, "char_end": 706438 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:plain", "to": "prop:detector-counts", "type": "candidate_proof_dependency", "evidence": "\\\\[2mm] \\#\\mathcal B \\ll U^{1-2\\delta m-2qz+\\epsilon}(1+T_1)^{A_{\\mathcal A}}, &\\text{from Lemma~\\ref{lem:plain}}. \\] In the second line use two copies of \\(S_m\\), so the denominator is \\(|S_m|^4\\), not \\(|S_m|^2\\). The effectiv", "anchor": { "source": "paper.tex", "line_start": 15239, "line_end": 15239, "char_start": 706540, "char_end": 706555 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:inverse-amplification", "to": "prop:detector-counts", "type": "candidate_proof_dependency", "evidence": "d primes.} Whenever no inverse slot is selected, use instead the sixth-power amplification of Lemma~\\ref{lem:inverse-amplification}. It applies to these physical rows because their ideal valuations are at most five and their original zero extensions are unchan", "anchor": { "source": "paper.tex", "line_start": 15250, "line_end": 15250, "char_start": 707029, "char_end": 707060 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:inverse-amplification", "to": "prop:detector-counts", "type": "candidate_proof_dependency", "evidence": "re twist \\(-(\\gamma-\\nu)\\), of absolute value at most \\((3I+1)T_1\\). The rowwise assertion of Lemma~\\ref{lem:inverse-amplification} therefore applies with that height range. Replacing its factor \\((1+(3I+1)T_1)^A\\) by \\((3I+2)^A(1+T_1)^A\\) changes only a fixed", "anchor": { "source": "paper.tex", "line_start": 15263, "line_end": 15264, "char_start": 707680, "char_end": 707711 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:amplified-note", "to": "prop:detector-counts", "type": "candidate_proof_dependency", "evidence": "ity, this use is uniform even when \\(r\\) approaches one with \\(U\\). If \\(0\\le r\\le R_0\\), Equation~\\eqref{eq:amplified-note} with preliminary loss \\(\\epsilon_0\\) has exponent \\[ \\max\\left\\{1,\\frac{1+5(1+c)r}{6}\\right\\} +(1+r)\\epsilon_0 \\le e(r)+", "anchor": { "source": "paper.tex", "line_start": 15269, "line_end": 15270, "char_start": 707961, "char_end": 707986 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:no-slot-inverse-count", "to": "prop:detector-counts", "type": "candidate_proof_dependency", "evidence": "which a witness already gives the required count without selecting primes. For \\(r\\ge1\\), Equation~\\eqref{eq:no-slot-inverse-count} uses effective moment exponent \\((1+5r)/6=1-\\alpha+\\alpha r\\). Division by the inverse spike gives \\(1-\\alpha+(\\alpha-\\delta)r\\).", "anchor": { "source": "paper.tex", "line_start": 15299, "line_end": 15299, "char_start": 709037, "char_end": 709069 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:no-slot-inverse-count", "to": "prop:detector-counts", "type": "candidate_proof_dependency", "evidence": "t} up to \\(O(\\epsilon)\\). This includes \\(r=1\\), where the two branches in Equation~\\eqref{eq:no-slot-inverse-count} agree. If \\(m\\ge1/2\\), the zero-slot case of Lemma~\\ref{lem:plain} gives count exponent \\(1-2\\delta m\\le1-\\delta\\). Since \\(R_{\\r", "anchor": { "source": "paper.tex", "line_start": 15311, "line_end": 15311, "char_start": 709477, "char_end": 709509 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:plain", "to": "prop:detector-counts", "type": "candidate_proof_dependency", "evidence": "hes in Equation~\\eqref{eq:no-slot-inverse-count} agree. If \\(m\\ge1/2\\), the zero-slot case of Lemma~\\ref{lem:plain} gives count exponent \\(1-2\\delta m\\le1-\\delta\\). Since \\(R_{\\rm short}(t)\\ge1-\\delta\\), this also satisfies the stated bound. We", "anchor": { "source": "paper.tex", "line_start": 15312, "line_end": 15313, "char_start": 709561, "char_end": 709576 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:selected-counts", "to": "prop:detector-counts", "type": "candidate_proof_dependency", "evidence": "O(\\epsilon)\\) replacement of \\(m\\) is therefore legitimate. The two affine expressions in Equation~\\eqref{eq:selected-counts} cross at \\[ r_*(t)= \\frac{(2-8x/9)t-5x/9}{D_x}. \\] Here \\(D_x\\ge37/18>0\\). Directly, \\[ r_*(3/2)=1,\\qquad r_*(1)=\\frac{2-13x", "anchor": { "source": "paper.tex", "line_start": 15346, "line_end": 15347, "char_start": 710830, "char_end": 710856 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:marked", "to": "prop:detector-counts", "type": "candidate_proof_dependency", "evidence": "/37\\) before its positive first term. These are precisely the two strict width conditions of Lemma~\\ref{lem:marked} at row width one. The inverse capacity is at most \\((1-23/37)/2=7/37\\). On the plain side, \\(m\\ge1/3-O(\\epsilon)\\), so its capac", "anchor": { "source": "paper.tex", "line_start": 15380, "line_end": 15380, "char_start": 711742, "char_end": 711758 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:plain-zero-loss-capacity", "to": "prop:detector-counts", "type": "candidate_proof_dependency", "evidence": "n just made. On the plain side, replacing the baseline capacity by the actual capacity in Equation~\\eqref{eq:plain-zero-loss-capacity} raises the count exponent by 2q(1-2m) \\left(\\frac29-\\frac1{9/2+12\\Delta}\\right) &=\\frac{24q(1-", "anchor": { "source": "paper.tex", "line_start": 15393, "line_end": 15393, "char_start": 712238, "char_end": 712273 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:no-slot-inverse-count", "to": "prop:detector-counts", "type": "candidate_proof_dependency", "evidence": "any subdivisions. Finally, use no prime slots and take \\(t=1\\). For the inverse side use Equation~\\eqref{eq:no-slot-inverse-count}, including \\(e(r)=1\\) for every \\(r\\le1\\); for the plain side use the zero-slot case of Lemma~\\ref{lem:plain}. The same short ca", "anchor": { "source": "paper.tex", "line_start": 15439, "line_end": 15439, "char_start": 713979, "char_end": 714011 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:plain", "to": "prop:detector-counts", "type": "candidate_proof_dependency", "evidence": "ount}, including \\(e(r)=1\\) for every \\(r\\le1\\); for the plain side use the zero-slot case of Lemma~\\ref{lem:plain}. The same short calculation with selection gain zero, that is, with \\(x=0\\) in the two comparison lines, gives \\(R_{\\rm short}(1", "anchor": { "source": "paper.tex", "line_start": 15441, "line_end": 15441, "char_start": 714106, "char_end": 714121 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:no-slot-inverse-count", "to": "remark:line:15448", "type": "internal_reference", "evidence": "gin{remark}[Unselected inverse witnesses beyond the Part II bin ceiling] The derivation of Equation~\\eqref{eq:no-slot-inverse-count} uses only the inverse spike from Proposition~\\ref{prop:detector-witness} and Lemma~\\ref{lem:inverse-amplification}. It can there", "anchor": { "source": "paper.tex", "line_start": 15449, "line_end": 15449, "char_start": 714496, "char_end": 714528 }, "origin": "source_reference", "context": "statement" }, { "from": "prop:detector-witness", "to": "remark:line:15448", "type": "internal_reference", "evidence": "erivation of Equation~\\eqref{eq:no-slot-inverse-count} uses only the inverse spike from Proposition~\\ref{prop:detector-witness} and Lemma~\\ref{lem:inverse-amplification}. It can therefore be repeated for any instance of the shared detector satisfying those", "anchor": { "source": "paper.tex", "line_start": 15450, "line_end": 15450, "char_start": 714574, "char_end": 714601 }, "origin": "source_reference", "context": "statement" }, { "from": "lem:inverse-amplification", "to": "remark:line:15448", "type": "internal_reference", "evidence": "t-inverse-count} uses only the inverse spike from Proposition~\\ref{prop:detector-witness} and Lemma~\\ref{lem:inverse-amplification}. It can therefore be repeated for any instance of the shared detector satisfying those hypotheses, independently of the current", "anchor": { "source": "paper.tex", "line_start": 15451, "line_end": 15451, "char_start": 714612, "char_end": 714643 }, "origin": "source_reference", "context": "statement" }, { "from": "eq:no-slot-inverse-count", "to": "remark:line:15448", "type": "internal_reference", "evidence": "e current Part~II bin ceiling. Consequently, for each fixed \\(t\\in[1,3/2]\\), the bound in Equation~\\eqref{eq:no-slot-inverse-count} holds on each fixed presentation/dyadic subdivision of a fixed dynamic bin of retained rows with actual witnesses from that propo", "anchor": { "source": "paper.tex", "line_start": 15454, "line_end": 15455, "char_start": 714863, "char_end": 714895 }, "origin": "source_reference", "context": "statement" }, { "from": "prop:detector-counts", "to": "sec:refined-row-counts", "type": "internal_reference", "evidence": "bin \\(a=51/100\\), which need not have an actual witness. The formulas in Proposition~\\ref{prop:detector-counts} describe the only row exponents needed below. The cardinality factor \\((1+T_1)^{A_{\\mathcal A}}\\) is kept explicit here. We ver", "anchor": { "source": "paper.tex", "line_start": 15469, "line_end": 15469, "char_start": 715521, "char_end": 715547 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:late-height-closure", "to": "sec:refined-row-counts", "type": "internal_reference", "evidence": "is kept explicit here. We verify the additional target-dependent height ceiling required by Lemma~\\ref{lem:late-height-closure}. Let \\(\\epsilon_{\\rm ht}>0\\) be the minimum of the finitely many detector height allowances already chosen with the real losses.", "anchor": { "source": "paper.tex", "line_start": 15473, "line_end": 15473, "char_start": 715746, "char_end": 715775 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:late-height-closure", "to": "sec:refined-row-counts", "type": "internal_reference", "evidence": "ncreasing that order changes only external test seminorms, not these internal height orders. Lemma~\\ref{lem:late-height-closure} then makes the final height choice without changing the positive real margins or the slot mesh. \\section{The seven-eighths bound", "anchor": { "source": "paper.tex", "line_start": 15492, "line_end": 15492, "char_start": 716612, "char_end": 716641 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:part-II-bootstrap", "to": "sec:conclusion", "type": "internal_reference", "evidence": "nd} We complete the contradiction assumed in Part~II. Recall from Equations~\\eqref{eq:part-II-bootstrap} and~\\eqref{eq:part-II-bin-ceiling} that \\[ 0<\\Delta=\\beta_*-\\frac78\\le\\frac1{24},\\qquad \\kappa=\\frac34+2\\Delta\\le\\frac56,\\qquad", "anchor": { "source": "paper.tex", "line_start": 15498, "line_end": 15498, "char_start": 716869, "char_end": 716897 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:part-II-bin-ceiling", "to": "sec:conclusion", "type": "internal_reference", "evidence": "plete the contradiction assumed in Part~II. Recall from Equations~\\eqref{eq:part-II-bootstrap} and~\\eqref{eq:part-II-bin-ceiling} that \\[ 0<\\Delta=\\beta_*-\\frac78\\le\\frac1{24},\\qquad \\kappa=\\frac34+2\\Delta\\le\\frac56,\\qquad \\delta=2a-1\\le\\kappa \\] for every", "anchor": { "source": "paper.tex", "line_start": 15498, "line_end": 15499, "char_start": 716902, "char_end": 716932 }, "origin": "source_reference", "context": "narrative" }, { "from": "prop:probe-low", "to": "sec:conclusion", "type": "internal_reference", "evidence": "qquad \\delta=2a-1\\le\\kappa \\] for every retained bin. The compensated low estimate is Proposition~\\ref{prop:probe-low}. For the same physical probe, we first normalize the principal term of its high expansion, then bound the remaining rows and veri", "anchor": { "source": "paper.tex", "line_start": 15506, "line_end": 15506, "char_start": 717122, "char_end": 717142 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:continuation-criterion", "to": "sec:conclusion", "type": "internal_reference", "evidence": "on, then bound the remaining rows and verify the target-independent margins required by Proposition~\\ref{lem:continuation-criterion}. The geometry and Mellin exponent are \\[ h=\\frac{13}{16},\\qquad \\ell=\\frac16,\\qquad l_x=\\frac{17}{48},\\qquad l_y=\\frac{23}{48}", "anchor": { "source": "paper.tex", "line_start": 15509, "line_end": 15509, "char_start": 717330, "char_end": 717362 }, "origin": "source_reference", "context": "narrative" }, { "from": "old-eq:5.1", "to": "sec:conclusion", "type": "internal_reference", "evidence": "C_{\\mathrm{II}}(s)=s-\\frac{11}{16}, \\qquad C(7/8)=\\frac3{16}. \\] These are the values in Equations~\\eqref{old-eq:5.1} and~\\eqref{eq:part-II-mellin-target}. Write \\(s\\) for the local variable called \\(x\\) in Section~\\ref{sec:local-compensation}; t", "anchor": { "source": "paper.tex", "line_start": 15521, "line_end": 15522, "char_start": 717627, "char_end": 717645 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:part-II-mellin-target", "to": "sec:conclusion", "type": "internal_reference", "evidence": "rac{11}{16}, \\qquad C(7/8)=\\frac3{16}. \\] These are the values in Equations~\\eqref{old-eq:5.1} and~\\eqref{eq:part-II-mellin-target}. Write \\(s\\) for the local variable called \\(x\\) in Section~\\ref{sec:local-compensation}; the amplitude ratio \\(x=q/\\delta\\) bel", "anchor": { "source": "paper.tex", "line_start": 15522, "line_end": 15522, "char_start": 717650, "char_end": 717682 }, "origin": "source_reference", "context": "narrative" }, { "from": "sec:local-compensation", "to": "sec:conclusion", "type": "internal_reference", "evidence": "} and~\\eqref{eq:part-II-mellin-target}. Write \\(s\\) for the local variable called \\(x\\) in Section~\\ref{sec:local-compensation}; the amplitude ratio \\(x=q/\\delta\\) below is a different real number. Use Definition~\\ref{def:stage-high-data} with \\[ \\mathscr", "anchor": { "source": "paper.tex", "line_start": 15523, "line_end": 15523, "char_start": 717744, "char_end": 717772 }, "origin": "source_reference", "context": "narrative" }, { "from": "def:stage-high-data", "to": "sec:conclusion", "type": "internal_reference", "evidence": "compensation}; the amplitude ratio \\(x=q/\\delta\\) below is a different real number. Use Definition~\\ref{def:stage-high-data} with \\[ \\mathscr I_\\eta(Z)=I_{\\eta,\\mathrm{modified}}(Z),\\qquad \\mathfrak H_{\\eta,u,Z}(s,w,z) \\ \\text{as in Equation~\\eqre", "anchor": { "source": "paper.tex", "line_start": 15526, "line_end": 15526, "char_start": 717859, "char_end": 717884 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:holomorphic-selected-tuple", "to": "sec:conclusion", "type": "internal_reference", "evidence": "(Z)=I_{\\eta,\\mathrm{modified}}(Z),\\qquad \\mathfrak H_{\\eta,u,Z}(s,w,z) \\ \\text{as in Equation~\\eqref{eq:holomorphic-selected-tuple}}. \\] In the shared analytic lemmas take \\(\\sigma_0=7/8\\). Section~\\ref{sec:local-compensation} verifies every part of these data:", "anchor": { "source": "paper.tex", "line_start": 15530, "line_end": 15530, "char_start": 718009, "char_end": 718046 }, "origin": "source_reference", "context": "narrative" }, { "from": "sec:local-compensation", "to": "sec:conclusion", "type": "internal_reference", "evidence": "ef{eq:holomorphic-selected-tuple}}. \\] In the shared analytic lemmas take \\(\\sigma_0=7/8\\). Section~\\ref{sec:local-compensation} verifies every part of these data: Equation~\\eqref{eq:compensated-poisson-identity} is the absolutely convergent high identity fo", "anchor": { "source": "paper.tex", "line_start": 15533, "line_end": 15533, "char_start": 718113, "char_end": 718141 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:compensated-poisson-identity", "to": "sec:conclusion", "type": "internal_reference", "evidence": "\\(\\sigma_0=7/8\\). Section~\\ref{sec:local-compensation} verifies every part of these data: Equation~\\eqref{eq:compensated-poisson-identity} is the absolutely convergent high identity for the independently defined finite expression in Equation~\\eqref{eq:compensated-prob", "anchor": { "source": "paper.tex", "line_start": 15534, "line_end": 15534, "char_start": 718186, "char_end": 718225 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:compensated-probe-definition", "to": "sec:conclusion", "type": "internal_reference", "evidence": "the absolutely convergent high identity for the independently defined finite expression in Equation~\\eqref{eq:compensated-probe-definition}, and the full correction is holomorphic in the two required Euler regions and satisfies Equation~\\eqref{eq:stage-all-height-major", "anchor": { "source": "paper.tex", "line_start": 15536, "line_end": 15536, "char_start": 718329, "char_end": 718368 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:stage-all-height-majorant", "to": "sec:conclusion", "type": "internal_reference", "evidence": "n}, and the full correction is holomorphic in the two required Euler regions and satisfies Equation~\\eqref{eq:stage-all-height-majorant}. The excluded set, physical row masks, and calibration are unchanged. In particular there is no additional Euler factor outside", "anchor": { "source": "paper.tex", "line_start": 15538, "line_end": 15538, "char_start": 718466, "char_end": 718502 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:principal-residue-interface", "to": "sec:conclusion", "type": "internal_reference", "evidence": "principal normalizer} We verify the two correction hypotheses of Lemma~\\ref{lem:principal-residue-interface}. First, Equation~\\eqref{eq:principal-local-tuple-bound} gives \\[ |\\mathfrak H_{\\eta,1,Z}(s,w,z)|\\ll Z^{\\ell\\Re z} \\] uniformly", "anchor": { "source": "paper.tex", "line_start": 15551, "line_end": 15551, "char_start": 719058, "char_end": 719095 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:principal-local-tuple-bound", "to": "sec:conclusion", "type": "internal_reference", "evidence": "rify the two correction hypotheses of Lemma~\\ref{lem:principal-residue-interface}. First, Equation~\\eqref{eq:principal-local-tuple-bound} gives \\[ |\\mathfrak H_{\\eta,1,Z}(s,w,z)|\\ll Z^{\\ell\\Re z} \\] uniformly in all imaginary parts on every fixed real box in the sec", "anchor": { "source": "paper.tex", "line_start": 15552, "line_end": 15552, "char_start": 719114, "char_end": 719152 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:principal-correction-bound", "to": "sec:conclusion", "type": "internal_reference", "evidence": "real box in the second Euler region. In particular this holds on the entire rectangle in Equation~\\eqref{eq:principal-correction-bound}, with height degree zero. This is a bound for the full tuple correction. For the residue value, define \\[ S_i(Z)=\\sum_{p\\in\\mat", "anchor": { "source": "paper.tex", "line_start": 15558, "line_end": 15558, "char_start": 719362, "char_end": 719399 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:ray-prime-normalizer", "to": "sec:conclusion", "type": "internal_reference", "evidence": "efine \\[ S_i(Z)=\\sum_{p\\in\\mathcal P_i(Z)}W_i(q_p/P_i)q_p^{-5/6}, \\qquad P_i=Z^{\\ell_i}. \\] Lemma~\\ref{lem:ray-prime-normalizer}, including its assertion about deletion of a fixed finite set, gives \\[ S_i(Z)\\sim\\frac{P_i^{1/6}}{|T|\\log P_i} \\int_0^\\i", "anchor": { "source": "paper.tex", "line_start": 15566, "line_end": 15566, "char_start": 719598, "char_end": 719628 }, "origin": "source_reference", "context": "narrative" }, { "from": "sec:local-compensation", "to": "sec:conclusion", "type": "internal_reference", "evidence": "asymptotic in a moving ray conductor is being used. On the principal second Euler region, Section~\\ref{sec:local-compensation} defines \\(\\mathcal B_p=G_p/H_p\\) with \\(H_p\\ne0\\), and Equation~\\eqref{eq:principal-local-approximation} gives \\(\\mathcal B_p=-1+", "anchor": { "source": "paper.tex", "line_start": 15594, "line_end": 15595, "char_start": 720657, "char_end": 720685 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:principal-local-approximation", "to": "sec:conclusion", "type": "internal_reference", "evidence": "ection~\\ref{sec:local-compensation} defines \\(\\mathcal B_p=G_p/H_p\\) with \\(H_p\\ne0\\), and Equation~\\eqref{eq:principal-local-approximation} gives \\(\\mathcal B_p=-1+O(q_p^{-7/8})\\), uniformly in all imaginary parts and target unit phases. Apply the principal factorizat", "anchor": { "source": "paper.tex", "line_start": 15596, "line_end": 15596, "char_start": 720750, "char_end": 720790 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:principal-slot-factorization", "to": "sec:conclusion", "type": "internal_reference", "evidence": "ormly in all imaginary parts and target unit phases. Apply the principal factorization in Equation~\\eqref{eq:principal-slot-factorization} at \\(w=1,z=1/6\\), where it remains valid even if the original quotient by \\(P_p\\) was undefined. It gives \\[ \\mathfrak H_{\\eta,1", "anchor": { "source": "paper.tex", "line_start": 15599, "line_end": 15599, "char_start": 720936, "char_end": 720975 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:shared-principal-data", "to": "sec:conclusion", "type": "internal_reference", "evidence": "W_i(q_p/P_i)q_p^{-5/6}\\mathcal B_p\\right), \\] where \\(H_\\eta\\) is the function in Equation~\\eqref{eq:shared-principal-data} for this same excluded set \\(S\\). By nonnegativity of \\(W_i\\), its \\(i\\)th slot equals \\[ -S_i(Z)\\{1+\\rho_i(s)\\},\\qquad |\\rho_", "anchor": { "source": "paper.tex", "line_start": 15608, "line_end": 15609, "char_start": 721285, "char_end": 721317 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:principal-correction-residue", "to": "sec:conclusion", "type": "internal_reference", "evidence": "H_{\\eta,1,Z}(s,1,1/6) =H_\\eta(s)Z^{\\ell/6}A_T(Z)\\{1+\\mathcal R_{\\eta,Z}(s)\\}, \\] which is Equation~\\eqref{eq:principal-correction-residue} with \\(A_\\eta=A_T\\) and \\(\\mu=\\kappa_P\\). This factorization is asserted only at the principal residue; it is not the correction", "anchor": { "source": "paper.tex", "line_start": 15628, "line_end": 15628, "char_start": 721981, "char_end": 722020 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:shared-principal-data", "to": "sec:conclusion", "type": "internal_reference", "evidence": "; it is not the correction used for general contour moves. Let \\(c_S>0\\) be the scalar in Equation~\\eqref{eq:shared-principal-data}, whose positivity is part of Lemma~\\ref{lem:principal-residue-interface}. For sufficiently large \\(Z\\), define the normalized phy", "anchor": { "source": "paper.tex", "line_start": 15633, "line_end": 15633, "char_start": 722224, "char_end": 722256 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:principal-residue-interface", "to": "sec:conclusion", "type": "internal_reference", "evidence": "S>0\\) be the scalar in Equation~\\eqref{eq:shared-principal-data}, whose positivity is part of Lemma~\\ref{lem:principal-residue-interface}. For sufficiently large \\(Z\\), define the normalized physical probe \\[ J_{\\mathrm{II},\\eta}(Z) =\\frac{I_{\\eta,\\mathrm{modifie", "anchor": { "source": "paper.tex", "line_start": 15634, "line_end": 15634, "char_start": 722292, "char_end": 722329 }, "origin": "source_reference", "context": "narrative" }, { "from": "prop:probe-low", "to": "sec:conclusion", "type": "internal_reference", "evidence": ", and both its numerator and normalizer are independent of the analysis height \\(T_1\\). Proposition~\\ref{prop:probe-low} and the subpower bound for \\(A_T^{-1}\\) give, for every \\(\\epsilon>0\\), |J_{\\m", "anchor": { "source": "paper.tex", "line_start": 15642, "line_end": 15642, "char_start": 722627, "char_end": 722647 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:shared-principal-nonvanishing", "to": "sec:conclusion", "type": "internal_reference", "evidence": "The same scalar is used for the high comparison. Choose the fixed cutoff as required for Equation~\\eqref{eq:shared-principal-nonvanishing}. The associated \\(H_\\eta\\) is holomorphic on \\(\\Re s>7/8\\), satisfies the contraction there, and uses this same \\(S\\). Let \\(\\m", "anchor": { "source": "paper.tex", "line_start": 15651, "line_end": 15651, "char_start": 722953, "char_end": 722993 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:compensated-poisson-identity", "to": "sec:conclusion", "type": "internal_reference", "evidence": "nd uses this same \\(S\\). Let \\(\\mathscr P_{\\mathrm{II},\\eta}\\) denote the \\(u=1\\) term of Equation~\\eqref{eq:compensated-poisson-identity}, and set \\[ f_{\\mathrm{II},\\eta}(Z)=\\frac1{2\\pi i}\\int_{(2)} Z^{C(s)}e^{(s-5/6)^2}\\frac{H_\\eta(s)}{L_F^S(s,\\eta)}\\,ds. \\] All h", "anchor": { "source": "paper.tex", "line_start": 15654, "line_end": 15654, "char_start": 723189, "char_end": 723228 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:principal-residue-interface", "to": "sec:conclusion", "type": "internal_reference", "evidence": "i}\\int_{(2)} Z^{C(s)}e^{(s-5/6)^2}\\frac{H_\\eta(s)}{L_F^S(s,\\eta)}\\,ds. \\] All hypotheses of Lemma~\\ref{lem:principal-residue-interface} have now been verified with \\(\\sigma_0=7/8\\). Its remainder estimate is \\[ \\left|\\frac{\\mathscr P_{\\mathrm{II},\\", "anchor": { "source": "paper.tex", "line_start": 15660, "line_end": 15660, "char_start": 723377, "char_end": 723414 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:principal-residue-interface", "to": "sec:conclusion", "type": "internal_reference", "evidence": "m_w:=\\frac{l_y}{20}=\\frac{23}{960},\\qquad m_z:=\\frac h{600}=\\frac{13}{9600}. Lemma~\\ref{lem:principal-residue-interface} includes the residue paths, the scalar Jacobian, and the rightward shift of the main integral. In particular the last term is th", "anchor": { "source": "paper.tex", "line_start": 15677, "line_end": 15677, "char_start": 723909, "char_end": 723946 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:buffered-bins", "to": "lem:high-bin", "type": "internal_reference", "evidence": "fixed dynamic bin \\((i,a)\\). Put \\(\\delta=2a-1\\). Assume the bin and height hypotheses of Lemmas~\\ref{lem:buffered-bins}, \\ref{lem:detector-dyads}, and \\ref{lem:bin-prime-bound}, with \\(02\\), \\(0<\\tau\\le d_{\\min}/100\\), and", "anchor": { "source": "paper.tex", "line_start": 15705, "line_end": 15705, "char_start": 725198, "char_end": 725221 }, "origin": "source_reference", "context": "statement" }, { "from": "lem:detector-dyads", "to": "lem:high-bin", "type": "internal_reference", "evidence": ")\\). Put \\(\\delta=2a-1\\). Assume the bin and height hypotheses of Lemmas~\\ref{lem:buffered-bins}, \\ref{lem:detector-dyads}, and \\ref{lem:bin-prime-bound}, with \\(02\\), \\(0<\\tau\\le d_{\\min}/100\\), and the cumulative allocation", "anchor": { "source": "paper.tex", "line_start": 15705, "line_end": 15705, "char_start": 725223, "char_end": 725247 }, "origin": "source_reference", "context": "statement" }, { "from": "lem:bin-prime-bound", "to": "lem:high-bin", "type": "internal_reference", "evidence": "sume the bin and height hypotheses of Lemmas~\\ref{lem:buffered-bins}, \\ref{lem:detector-dyads}, and \\ref{lem:bin-prime-bound}, with \\(02\\), \\(0<\\tau\\le d_{\\min}/100\\), and the cumulative allocation in Equation~\\eqref{eq:stage-he", "anchor": { "source": "paper.tex", "line_start": 15706, "line_end": 15706, "char_start": 725253, "char_end": 725278 }, "origin": "source_reference", "context": "statement" }, { "from": "eq:stage-height-allocation", "to": "lem:high-bin", "type": "internal_reference", "evidence": "<10^{-3}\\), \\(T_1=Z^\\tau>2\\), \\(0<\\tau\\le d_{\\min}/100\\), and the cumulative allocation in Equation~\\eqref{eq:stage-height-allocation}. The bounded real ranges, the positive losses \\(e,\\vartheta,\\epsilon\\), and the slot system are fixed before the target. For eac", "anchor": { "source": "paper.tex", "line_start": 15708, "line_end": 15708, "char_start": 725390, "char_end": 725424 }, "origin": "source_reference", "context": "statement" }, { "from": "sec:amplitudes", "to": "lem:high-bin", "type": "internal_reference", "evidence": "d each subset of error slots, partition \\(\\mathcal B\\) pointwise into the amplitude sets of Section~\\ref{sec:amplitudes}, and into the witness subdivisions of Section~\\ref{sec:selection} when a witness count is used. Suppose each such set \\(\\mathcal", "anchor": { "source": "paper.tex", "line_start": 15714, "line_end": 15714, "char_start": 725696, "char_end": 725716 }, "origin": "source_reference", "context": "statement" }, { "from": "sec:selection", "to": "lem:high-bin", "type": "internal_reference", "evidence": "to the amplitude sets of Section~\\ref{sec:amplitudes}, and into the witness subdivisions of Section~\\ref{sec:selection} when a witness count is used. Suppose each such set \\(\\mathcal C\\) has main-slot mean \\(q\\in[0,\\delta/2]\\), as in Equation~\\eqre", "anchor": { "source": "paper.tex", "line_start": 15715, "line_end": 15715, "char_start": 725763, "char_end": 725782 }, "origin": "source_reference", "context": "statement" }, { "from": "old-eq:6.6", "to": "lem:high-bin", "type": "internal_reference", "evidence": "used. Suppose each such set \\(\\mathcal C\\) has main-slot mean \\(q\\in[0,\\delta/2]\\), as in Equation~\\eqref{old-eq:6.6}, and \\[ \\#\\mathcal C\\ll_{\\mathcal A,\\epsilon} U^{R+\\epsilon}(1+T_1)^{A_{\\mathcal A}}, \\] uniformly in the retained tuple and mo", "anchor": { "source": "paper.tex", "line_start": 15717, "line_end": 15717, "char_start": 725907, "char_end": 725925 }, "origin": "source_reference", "context": "statement" }, { "from": "lem:bin-contour-accounting", "to": "lem:high-bin", "type": "candidate_proof_dependency", "evidence": "d \\(B\\) and every fixed \\(N\\). We verify the central hypothesis of Lemma~\\ref{lem:bin-contour-accounting}, which supplies the contour move and its tails for the full correction just specified. Its bin ceiling holds by Equation~\\eqref{", "anchor": { "source": "paper.tex", "line_start": 15761, "line_end": 15761, "char_start": 727555, "char_end": 727587 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:part-II-bin-ceiling", "to": "lem:high-bin", "type": "candidate_proof_dependency", "evidence": "ntour move and its tails for the full correction just specified. Its bin ceiling holds by Equation~\\eqref{eq:part-II-bin-ceiling}. The physical \\(\\Im z\\) coordinate is included in its height allocation because it enters the prime profiles. The other witness", "anchor": { "source": "paper.tex", "line_start": 15763, "line_end": 15763, "char_start": 727710, "char_end": 727740 }, "origin": "source_reference", "context": "proof" }, { "from": "sec:selection", "to": "lem:high-bin", "type": "candidate_proof_dependency", "evidence": "s, dyadic, and prime-annulus coordinates are the fixed finite list used in the estimates of Section~\\ref{sec:selection}. Their complete pointwise bounds include the discarded auxiliary integrals on global or absolute lines. Thus the single cumulat", "anchor": { "source": "paper.tex", "line_start": 15767, "line_end": 15767, "char_start": 727969, "char_end": 727988 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:stage-height-allocation", "to": "lem:high-bin", "type": "candidate_proof_dependency", "evidence": "auxiliary integrals on global or absolute lines. Thus the single cumulative allocation in Equation~\\eqref{eq:stage-height-allocation} applies, without renewing an allowance at a later estimate. Only on the retained contours, Proposition~\\ref{prop:probe-errors} g", "anchor": { "source": "paper.tex", "line_start": 15769, "line_end": 15770, "char_start": 728145, "char_end": 728179 }, "origin": "source_reference", "context": "proof" }, { "from": "prop:probe-errors", "to": "lem:high-bin", "type": "candidate_proof_dependency", "evidence": "ies, without renewing an allowance at a later estimate. Only on the retained contours, Proposition~\\ref{prop:probe-errors} gives the decomposition in Equation~\\eqref{eq:dynamic-compensated-decomposition}. For \\(I\\subseteq\\{1,\\ldots,K\\}\\), its summand i", "anchor": { "source": "paper.tex", "line_start": 15772, "line_end": 15772, "char_start": 728284, "char_end": 728307 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:dynamic-compensated-decomposition", "to": "lem:high-bin", "type": "candidate_proof_dependency", "evidence": "y on the retained contours, Proposition~\\ref{prop:probe-errors} gives the decomposition in Equation~\\eqref{eq:dynamic-compensated-decomposition}. For \\(I\\subseteq\\{1,\\ldots,K\\}\\), its summand is \\[ \\mathfrak H_{\\eta,u,Z}^{(I)} =\\mathcal H_{\\eta,u}(s,w,z) \\prod_{i\\i", "anchor": { "source": "paper.tex", "line_start": 15773, "line_end": 15773, "char_start": 728344, "char_end": 728388 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:deleted-euler-factors", "to": "lem:high-bin", "type": "candidate_proof_dependency", "evidence": "ws from the buffered estimate for the numerator presentation and its conjugate, together with Lemma~\\ref{lem:deleted-euler-factors}, as verified before Proposition~\\ref{prop:probe-errors}. The retained \\(w\\) heights lie in its allowed set. Every retained nume", "anchor": { "source": "paper.tex", "line_start": 15783, "line_end": 15783, "char_start": 728738, "char_end": 728769 }, "origin": "source_reference", "context": "proof" }, { "from": "prop:probe-errors", "to": "lem:high-bin", "type": "candidate_proof_dependency", "evidence": "its conjugate, together with Lemma~\\ref{lem:deleted-euler-factors}, as verified before Proposition~\\ref{prop:probe-errors}. The retained \\(w\\) heights lie in its allowed set. Every retained numerator is nonprincipal. Fix one amplitude set at one ret", "anchor": { "source": "paper.tex", "line_start": 15784, "line_end": 15784, "char_start": 728802, "char_end": 728825 }, "origin": "source_reference", "context": "proof" }, { "from": "old-eq:5.13e", "to": "lem:high-bin", "type": "candidate_proof_dependency", "evidence": "_i\\) and \\(\\lvert Q_i\\rvert\\le P_i^{g_i+\\vartheta}\\). Assign \\(g_i=0\\) to an error slot. Equation~\\eqref{old-eq:5.13e} bounds all error slots in \\(I\\) jointly with the numerator; its proof uses the actual conductor deficit in Equation~\\eqref{old-eq", "anchor": { "source": "paper.tex", "line_start": 15790, "line_end": 15790, "char_start": 729114, "char_end": 729134 }, "origin": "source_reference", "context": "proof" }, { "from": "old-eq:5.13d", "to": "lem:high-bin", "type": "candidate_proof_dependency", "evidence": "slots in \\(I\\) jointly with the numerator; its proof uses the actual conductor deficit in Equation~\\eqref{old-eq:5.13d} simultaneously for the distinct strict ramified labels. It is not a separate numerator allowance for each slot. Since \\(\\mathca", "anchor": { "source": "paper.tex", "line_start": 15792, "line_end": 15792, "char_start": 729251, "char_end": 729271 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:stage-central-bound", "to": "lem:high-bin", "type": "candidate_proof_dependency", "evidence": "lower bound on an error slot has been used. Multiplying by the assumed cardinality proves Equation~\\eqref{eq:stage-central-bound} with \\(g=q\\ell\\), after taking \\(\\varepsilon_c\\) to be a fixed sufficiently large multiple of \\(e+\\vartheta+\\epsilon\\). This mul", "anchor": { "source": "paper.tex", "line_start": 15811, "line_end": 15811, "char_start": 730000, "char_end": 730030 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:bin-contour-accounting", "to": "lem:high-bin", "type": "candidate_proof_dependency", "evidence": "Z\\). Thus the required pointwise multiplicity bound holds. These sets are formed only after Lemma~\\ref{lem:bin-contour-accounting} has moved the original dynamic-bin sum using \\(\\mathfrak H\\). They are not used for continuation and are not integrated separate", "anchor": { "source": "paper.tex", "line_start": 15818, "line_end": 15818, "char_start": 730482, "char_end": 730514 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:bin-contour-accounting", "to": "lem:high-bin", "type": "candidate_proof_dependency", "evidence": "(\\mathfrak H\\). They are not used for continuation and are not integrated separately. Apply Lemma~\\ref{lem:bin-contour-accounting} with \\(\\sigma_0=7/8\\) and \\(g=q\\ell\\). Its exponent in Equation~\\eqref{eq:stage-high-exponent} is \\[ a-\\frac78+h\\left(\\frac{17}", "anchor": { "source": "paper.tex", "line_start": 15823, "line_end": 15823, "char_start": 730661, "char_end": 730693 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:stage-high-exponent", "to": "lem:high-bin", "type": "candidate_proof_dependency", "evidence": "a~\\ref{lem:bin-contour-accounting} with \\(\\sigma_0=7/8\\) and \\(g=q\\ell\\). Its exponent in Equation~\\eqref{eq:stage-high-exponent} is \\[ a-\\frac78+h\\left(\\frac{17}{50}-\\frac16\\right)-a l_y-\\frac{\\ell}{2} +q\\ell+d\\left(R+\\frac\\delta2-\\frac{17}{50}\\right", "anchor": { "source": "paper.tex", "line_start": 15825, "line_end": 15825, "char_start": 730759, "char_end": 730789 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:common-high-exponent", "to": "lem:high-bin", "type": "candidate_proof_dependency", "evidence": "ight). \\] Because \\(-(1-a)\\ell-(\\delta/2-q)\\ell=-\\ell/2+q\\ell\\), this is the first line of Equation~\\eqref{eq:common-high-exponent}. Substituting the geometry and \\(a=(1+\\delta)/2\\) gives its second line. In particular \\(q\\ell=q/6\\) is a base-\\(Z\\) exponent, no", "anchor": { "source": "paper.tex", "line_start": 15832, "line_end": 15832, "char_start": 731015, "char_end": 731046 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:stage-central-contribution", "to": "lem:high-bin", "type": "candidate_proof_dependency", "evidence": "In particular \\(q\\ell=q/6\\) is a base-\\(Z\\) exponent, not \\(dq/6\\). The explicit errors in Equation~\\eqref{eq:stage-central-contribution} are \\(O(e+\\vartheta+\\epsilon)\\) on the bounded \\(d\\)-range after the remaining preliminary powers are chosen. Its supremum is ex", "anchor": { "source": "paper.tex", "line_start": 15835, "line_end": 15836, "char_start": 731220, "char_end": 731257 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:bin-contour-accounting", "to": "lem:high-bin", "type": "candidate_proof_dependency", "evidence": "remaining preliminary powers are chosen. Its supremum is exactly the one in the statement. Lemma~\\ref{lem:bin-contour-accounting} also gives the stated external tails with a scale degree fixed before \\(N\\). The floor bin \\(a=51/100\\) has \\(\\delta", "anchor": { "source": "paper.tex", "line_start": 15838, "line_end": 15838, "char_start": 731426, "char_end": 731458 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:compensated-absolute-local-bounds", "to": "sec:conclusion", "type": "internal_reference", "evidence": "led below. \\subsection{Small and large row norms} Set \\(d_{\\min}=1/100\\). Lemma~\\ref{lem:compensated-absolute-local-bounds} bounds the positive sum of the full selected tuples, with every \\(G_p\\) retained before taking absolute values. On the small-row", "anchor": { "source": "paper.tex", "line_start": 15855, "line_end": 15856, "char_start": 732029, "char_end": 732072 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:absolute-local-tuple-bound", "to": "sec:conclusion", "type": "internal_reference", "evidence": "values. On the small-row lines \\[ \\Re s=\\beta_*+e,\\qquad \\Re w=1/2,\\qquad \\Re z=17/50 \\] Equation~\\eqref{eq:absolute-local-tuple-bound} gives \\[ |\\mathfrak H_{\\eta,u,Z}(s,w,z)| \\ll_\\epsilon U^\\epsilon\\prod_iP_i^{17/50} =U^\\epsilon Z^{17\\ell/50}, \\] uniformly in", "anchor": { "source": "paper.tex", "line_start": 15861, "line_end": 15861, "char_start": 732277, "char_end": 732314 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:small-correction-bound", "to": "sec:conclusion", "type": "internal_reference", "evidence": "rod_iP_i^{17/50} =U^\\epsilon Z^{17\\ell/50}, \\] uniformly in all imaginary parts. This is Equation~\\eqref{eq:small-correction-bound}, with height degree zero. It remains valid at zeros of individual local factors. The data for Lemma~\\ref{lem:outer-row-tails} we", "anchor": { "source": "paper.tex", "line_start": 15868, "line_end": 15868, "char_start": 732483, "char_end": 732516 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:outer-row-tails", "to": "sec:conclusion", "type": "internal_reference", "evidence": "with height degree zero. It remains valid at zeros of individual local factors. The data for Lemma~\\ref{lem:outer-row-tails} were already verified above, so it applies to every physical row \\(u\\ne1\\) in these dyads. Its classification includes nontrivia", "anchor": { "source": "paper.tex", "line_start": 15870, "line_end": 15870, "char_start": 732618, "char_end": 732643 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:small-row-bound", "to": "sec:conclusion", "type": "internal_reference", "evidence": "17}{50}-\\frac16\\right)-\\frac{l_y}{2} =-\\frac{79}{800}. The row exponent in Equation~\\eqref{eq:small-row-bound} is \\(63/50<2\\). Thus the small dyadic sum in Equation~\\eqref{eq:small-row-sum}, measured relative to \\(C(\\beta_*)\\), has exponen", "anchor": { "source": "paper.tex", "line_start": 15880, "line_end": 15880, "char_start": 733101, "char_end": 733127 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:small-row-sum", "to": "sec:conclusion", "type": "internal_reference", "evidence": "onent in Equation~\\eqref{eq:small-row-bound} is \\(63/50<2\\). Thus the small dyadic sum in Equation~\\eqref{eq:small-row-sum}, measured relative to \\(C(\\beta_*)\\), has exponent at most \\[ -\\frac{79}{800}+2d_{\\min}+O(e+\\epsilon) =-\\frac{63}{800}+O(e+\\eps", "anchor": { "source": "paper.tex", "line_start": 15882, "line_end": 15882, "char_start": 733183, "char_end": 733207 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:absolute-local-tuple-bound", "to": "sec:conclusion", "type": "internal_reference", "evidence": "r the large rows, fix \\(z_\\infty>2\\). On \\((\\Re s,\\Re w,\\Re z)=(2,2,z_\\infty)\\), the same Equation~\\eqref{eq:absolute-local-tuple-bound} gives \\[ |\\mathfrak H_{\\eta,u,Z}(s,w,z)| \\ll_\\epsilon U^\\epsilon Z^{\\ell z_\\infty} \\] for every physical row and all imaginary", "anchor": { "source": "paper.tex", "line_start": 15893, "line_end": 15893, "char_start": 733594, "char_end": 733631 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:large-correction-bound", "to": "sec:conclusion", "type": "internal_reference", "evidence": "psilon Z^{\\ell z_\\infty} \\] for every physical row and all imaginary parts. This verifies Equation~\\eqref{eq:large-correction-bound}. Lemma~\\ref{lem:outer-row-tails} therefore gives, with \\(B_0=l_x/2+1+l_y=53/32\\), \\[ |\\mathscr R_\\eta(U;Z)| \\ll Z^{B_0+hz_\\inf", "anchor": { "source": "paper.tex", "line_start": 15899, "line_end": 15899, "char_start": 733792, "char_end": 733825 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:outer-row-tails", "to": "sec:conclusion", "type": "internal_reference", "evidence": "ical row and all imaginary parts. This verifies Equation~\\eqref{eq:large-correction-bound}. Lemma~\\ref{lem:outer-row-tails} therefore gives, with \\(B_0=l_x/2+1+l_y=53/32\\), \\[ |\\mathscr R_\\eta(U;Z)| \\ll Z^{B_0+hz_\\infty+\\epsilon}U^{1+\\epsilon-z_\\infty", "anchor": { "source": "paper.tex", "line_start": 15899, "line_end": 15900, "char_start": 733834, "char_end": 733859 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:outer-row-tails", "to": "sec:conclusion", "type": "internal_reference", "evidence": "(1+\\epsilon)-\\zeta z_\\infty+\\epsilon}. \\] Here \\(\\mathscr R_\\eta\\) is the row contribution of Lemma~\\ref{lem:outer-row-tails} for the present physical expression. A fixed sufficiently large \\(z_\\infty\\) makes the last exponent as negative as required. I", "anchor": { "source": "paper.tex", "line_start": 15913, "line_end": 15913, "char_start": 734239, "char_end": 734264 }, "origin": "source_reference", "context": "narrative" }, { "from": "sec:local-compensation", "to": "sec:conclusion", "type": "internal_reference", "evidence": "r \\(\\zeta\\) but before the target. These invocations use the quotient-free tuple bounds in Section~\\ref{sec:local-compensation}, not the central main/error factorization. \\subsection{The endpoint inequality} We next bound \\(E(", "anchor": { "source": "paper.tex", "line_start": 15917, "line_end": 15917, "char_start": 734510, "char_end": 734538 }, "origin": "source_reference", "context": "narrative" }, { "from": "prop:detector-witness", "to": "sec:conclusion", "type": "internal_reference", "evidence": "quad \\alpha=\\frac56. \\] At the live upper endpoint \\(\\delta=\\alpha\\), take \\(t=3/2\\) in Proposition~\\ref{prop:detector-witness}. Then \\(r\\ge1-O(\\epsilon)\\). Apply the no-slot amplified estimate in Equation~\\eqref{eq:no-slot-inverse-count} on both sides of", "anchor": { "source": "paper.tex", "line_start": 15929, "line_end": 15929, "char_start": 734920, "char_end": 734947 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:no-slot-inverse-count", "to": "sec:conclusion", "type": "internal_reference", "evidence": "p:detector-witness}. Then \\(r\\ge1-O(\\epsilon)\\). Apply the no-slot amplified estimate in Equation~\\eqref{eq:no-slot-inverse-count} on both sides of \\(r=1\\). For \\(r\\ge1\\), its exponent is \\[ 1-\\alpha+(\\alpha-\\delta)r=1-\\delta. \\] For \\(r<1\\) in the stated \\(", "anchor": { "source": "paper.tex", "line_start": 15931, "line_end": 15931, "char_start": 735028, "char_end": 735060 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:common-high-exponent", "to": "sec:conclusion", "type": "internal_reference", "evidence": "primes are needed and the ideal row exponent is \\(R=1-\\delta\\). Using \\(q\\le\\delta/2\\) in Equation~\\eqref{eq:common-high-exponent} gives E(h)\\le C_0+\\left(\\frac34-h\\right)\\delta+O(\\epsilon) =-\\frac1{48}-\\frac{\\delta}{16}+O(\\epsilon)<0.", "anchor": { "source": "paper.tex", "line_start": 15942, "line_end": 15942, "char_start": 735497, "char_end": 735528 }, "origin": "source_reference", "context": "narrative" }, { "from": "prop:detector-counts", "to": "sec:conclusion", "type": "internal_reference", "evidence": "delta<\\alpha\\). Put \\(x=q/\\delta\\in[0,1/2]\\), and use \\(D_x,P_x,R_{\\rm short},L\\) from Proposition~\\ref{prop:detector-counts}. The parameter \\(\\kappa\\) remains the dynamic value \\(3/4+2\\Delta\\), not the upper bound \\(5/6\\). Thus the actual plain capacit", "anchor": { "source": "paper.tex", "line_start": 15951, "line_end": 15951, "char_start": 735827, "char_end": 735853 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:plain-zero-loss-capacity", "to": "sec:conclusion", "type": "internal_reference", "evidence": "Proposition is \\[ z_P(m)=\\frac{1-2m}{6\\kappa} =\\frac{1-2m}{9/2+12\\Delta}, \\] as in Equation~\\eqref{eq:plain-zero-loss-capacity}. The comparison in Equation~\\eqref{eq:capacity-comparison} is for this exact capacity and raises the row exponent by at most \\(\\", "anchor": { "source": "paper.tex", "line_start": 15958, "line_end": 15958, "char_start": 736098, "char_end": 736133 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:capacity-comparison", "to": "sec:conclusion", "type": "internal_reference", "evidence": "}{9/2+12\\Delta}, \\] as in Equation~\\eqref{eq:plain-zero-loss-capacity}. The comparison in Equation~\\eqref{eq:capacity-comparison} is for this exact capacity and raises the row exponent by at most \\(\\Delta/4\\), apart from requested small losses. Define \\begin", "anchor": { "source": "paper.tex", "line_start": 15959, "line_end": 15959, "char_start": 736163, "char_end": 736193 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:no-slot-inverse-count", "to": "sec:conclusion", "type": "internal_reference", "evidence": "has \\(r_*(3/2)=1\\), so its inverse capacity is zero. The preceding equality argument uses Equation~\\eqref{eq:no-slot-inverse-count} at that boundary, not a marked estimate with a vanishing margin. [Compensated endpoint certificate]\\label{lem:balan", "anchor": { "source": "paper.tex", "line_start": 15991, "line_end": 15991, "char_start": 737268, "char_end": 737300 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:target-cutoff", "to": "lem:balanced-endpoint", "type": "internal_reference", "evidence": "ed-endpoint} For \\(0\\le\\delta\\le5/6\\) and \\(0\\le x\\le1/2\\), define \\(\\mathcal J,t,R_*\\) by Equation~\\eqref{eq:target-cutoff} where \\(\\mathcal J>0\\), and let \\(E_*\\) be \\(E(h)\\) with \\(q=x\\delta\\) and \\(R=R_*\\). Then \\[ -E_*\\ge\\frac{49}{440640}>\\frac1{1", "anchor": { "source": "paper.tex", "line_start": 15996, "line_end": 15996, "char_start": 737531, "char_end": 737555 }, "origin": "source_reference", "context": "statement" }, { "from": "eq:common-high-exponent", "to": "lem:balanced-endpoint", "type": "candidate_proof_dependency", "evidence": "al J=\\frac{j_y}{108}. \\] From \\(R_*=1-\\delta+(\\alpha-\\delta)\\delta P_x/(2\\mathcal J)\\) and Equation~\\eqref{eq:common-high-exponent}, \\[ -E_*= \\frac{1+(3+8y)\\delta}{48} -\\frac{13}{32}\\, \\frac{(5/6-\\delta)\\delta P_x}{\\mathcal J}. \\] Multiplying out gives th", "anchor": { "source": "paper.tex", "line_start": 16020, "line_end": 16020, "char_start": 738140, "char_end": 738171 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:balanced-endpoint-certificate", "to": "lem:balanced-endpoint", "type": "candidate_proof_dependency", "evidence": "tic coefficients in the preceding display multiplied by \\(v\\). Every term on the right of Equation~\\eqref{eq:balanced-endpoint-certificate} is nonnegative: \\(y,\\delta\\ge0\\) and \\(9-13y\\ge5/2\\). The first line is at least \\(49(3+5y)\\), while \\(v=51+41y\\le17(3+5y)\\). He", "anchor": { "source": "paper.tex", "line_start": 16063, "line_end": 16063, "char_start": 739221, "char_end": 739261 }, "origin": "source_reference", "context": "proof" }, { "from": "prop:detector-counts", "to": "sec:conclusion", "type": "internal_reference", "evidence": "ge1/50\\), and the inverse selection has the explicit supply and width margins proved in Proposition~\\ref{prop:detector-counts}. At a zero capacity its unweighted bound differs from \\(1-\\delta\\) by at most a constant times the chosen capacity decrement. S", "anchor": { "source": "paper.tex", "line_start": 16078, "line_end": 16078, "char_start": 739802, "char_end": 739828 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:common-high-exponent", "to": "sec:conclusion", "type": "internal_reference", "evidence": "now the explicit height factors remain outside this exponent. The coefficient of \\(R\\) in Equation~\\eqref{eq:common-high-exponent} is \\(d\\), so at \\(d=h\\) the capacity replacement contributes \\(h\\Delta/4\\). Denote by \\(\\epsilon_{\\rm total}\\) the sum of \\(h\\ep", "anchor": { "source": "paper.tex", "line_start": 16089, "line_end": 16090, "char_start": 740455, "char_end": 740486 }, "origin": "source_reference", "context": "narrative" }, { "from": "prop:detector-counts", "to": "sec:conclusion", "type": "internal_reference", "evidence": "\\min}\\le d\\le1/2\\) and \\(\\delta\\le\\alpha\\), choose \\(t=1\\) and use no selected primes. Proposition~\\ref{prop:detector-counts} gives the ideal exponent \\(1-2\\delta/3\\) outside the floor. Use the slightly larger common exponent \\[ R=\\frac{76}{75}-\\frac23\\d", "anchor": { "source": "paper.tex", "line_start": 16147, "line_end": 16148, "char_start": 742577, "char_end": 742603 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:common-high-exponent", "to": "sec:conclusion", "type": "internal_reference", "evidence": "rs the floor. Its slope is \\(101/150-\\delta/6>0\\) on this range. Using \\(q\\le\\delta/2\\) in Equation~\\eqref{eq:common-high-exponent} yields E(d)\\le E(1/2)+O(\\epsilon) \\le-\\frac{529}{2400}+\\frac{25}{96}\\delta+O(\\epsilon) \\le-\\frac{49}{14400}+O", "anchor": { "source": "paper.tex", "line_start": 16156, "line_end": 16156, "char_start": 742891, "char_end": 742922 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:large-delta-endpoint", "to": "sec:conclusion", "type": "internal_reference", "evidence": "ta/2\\); the last inequality uses \\(\\delta\\le5/6\\). At \\(\\delta=\\alpha\\), the count used in Equation~\\eqref{eq:large-delta-endpoint} and its positive slope already control every \\(d\\le h\\). Together with Section~\\ref{sec:tails}, these cases cover every physical", "anchor": { "source": "paper.tex", "line_start": 16167, "line_end": 16167, "char_start": 743307, "char_end": 743338 }, "origin": "source_reference", "context": "narrative" }, { "from": "sec:tails", "to": "sec:conclusion", "type": "internal_reference", "evidence": "arge-delta-endpoint} and its positive slope already control every \\(d\\le h\\). Together with Section~\\ref{sec:tails}, these cases cover every physical row norm. Only \\(d\\ge1/2\\) uses selected physical prime factors in a row moment; every physica", "anchor": { "source": "paper.tex", "line_start": 16169, "line_end": 16169, "char_start": 743418, "char_end": 743433 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:adaptive-endpoint", "to": "sec:conclusion", "type": "internal_reference", "evidence": "elta\\) from each ideal exponent. In the adaptive range this is precisely the comparison in Equation~\\eqref{eq:adaptive-endpoint}; its only positive capacity allowance is \\(h\\Delta/4\\). The other endpoint ranges have nonpositive ideal exponents before subtra", "anchor": { "source": "paper.tex", "line_start": 16177, "line_end": 16177, "char_start": 743780, "char_end": 743808 }, "origin": "source_reference", "context": "narrative" }, { "from": "sec:principal-signal", "to": "sec:conclusion", "type": "internal_reference", "evidence": "\\(m_{\\rm lo}:=\\Delta\\). The principal contour and approximation margins were established in Section~\\ref{sec:principal-signal}; we now choose all losses together. \\subsection{Order of choices and conclusion} We finish by separating the target-independent", "anchor": { "source": "paper.tex", "line_start": 16187, "line_end": 16187, "char_start": 744232, "char_end": 744258 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:late-height-closure", "to": "sec:conclusion", "type": "internal_reference", "evidence": "numerical choices remain part of this application; the final height selection is supplied by Lemma~\\ref{lem:late-height-closure}. [Order of choices] Under the contradiction in Equation~\\eqref{eq:part-II-bootstr", "anchor": { "source": "paper.tex", "line_start": 16194, "line_end": 16194, "char_start": 744565, "char_end": 744594 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:part-II-bootstrap", "to": "prop:parameter-order", "type": "internal_reference", "evidence": "egin{proposition}[Order of choices] Under the contradiction in Equation~\\eqref{eq:part-II-bootstrap}, there are numbers \\[ 0<\\omega<\\Delta,\\qquad \\sigma>0 \\] and a fixed geometry, moment losses, capacity decrements, slot system,", "anchor": { "source": "paper.tex", "line_start": 16197, "line_end": 16197, "char_start": 744699, "char_end": 744727 }, "origin": "source_reference", "context": "statement" }, { "from": "old-eq:5.14", "to": "prop:parameter-order", "type": "candidate_proof_dependency", "evidence": "}:=\\min\\{m_{\\rm hi},m_{\\rm small}\\}>0. \\] The principal contour margins are \\(m_w,m_z\\) in Equation~\\eqref{old-eq:5.14}. Reserve one half of each of \\(m_{\\rm lo},m_{\\rm high},m_w,m_z\\). On the compact real parameter ranges, the coefficients multip", "anchor": { "source": "paper.tex", "line_start": 16226, "line_end": 16226, "char_start": 745777, "char_end": 745796 }, "origin": "source_reference", "context": "proof" }, { "from": "prop:detector-counts", "to": "prop:parameter-order", "type": "candidate_proof_dependency", "evidence": "/50\\), \\(D_x\\ge37/18\\), and \\(\\mathcal J\\ge35/54\\); the dyadic lengths are bounded, and Proposition~\\ref{prop:detector-counts} gives fixed inverse width and supply margins. The equality \\(\\delta=\\alpha\\) uses the separate no-slot bound above. Choose the", "anchor": { "source": "paper.tex", "line_start": 16232, "line_end": 16232, "char_start": 746155, "char_end": 746181 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:plain", "to": "prop:parameter-order", "type": "candidate_proof_dependency", "evidence": "exponent is less than \\(m_{\\rm high}/8\\). Let \\(\\eta_{\\rm mesh}>0\\) be the mesh supplied by Lemma~\\ref{lem:plain} for these losses and bounded real ranges. Its uniform assertion on \\([3/4,1]\\) applies in particular to the compact closure \\(\\k", "anchor": { "source": "paper.tex", "line_start": 16239, "line_end": 16239, "char_start": 746484, "char_end": 746499 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:compensated-probe-definition", "to": "prop:parameter-order", "type": "candidate_proof_dependency", "evidence": "ows are disjoint for every \\(Z\\), before the ray, excluded-set, or row masks are imposed. Equation~\\eqref{eq:compensated-probe-definition} keeps each window at its original scale in every summand. For each selected row range, \\(U=Z^d\\) has \\(d\\ge1/2\\), so \\[ w_i=\\fr", "anchor": { "source": "paper.tex", "line_start": 16260, "line_end": 16260, "char_start": 747590, "char_end": 747629 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:plain", "to": "prop:parameter-order", "type": "candidate_proof_dependency", "evidence": "dows. Write \\(\\sigma_{\\rm width}\\) for the width decrement denoted \\(\\sigma\\) in the proof of Lemma~\\ref{lem:plain}. That proof uses the pool exponent \\(\\ell_*=\\sigma_{\\rm width}/3\\) and a mesh below \\(\\sigma_{\\rm width}/6\\). At base \\(U\\), ev", "anchor": { "source": "paper.tex", "line_start": 16282, "line_end": 16282, "char_start": 748691, "char_end": 748706 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:principal-scalar", "to": "prop:parameter-order", "type": "candidate_proof_dependency", "evidence": "selected \\(d\\)-range. The marked inverse moment has no mesh condition. The normalizer in Equation~\\eqref{eq:principal-scalar} is nonzero for this actual choice. Put \\(c_i=\\int_0^\\infty W_i(y)y^{-5/6}\\,dy>0\\). The asymptotic already proved in Section~\\re", "anchor": { "source": "paper.tex", "line_start": 16290, "line_end": 16290, "char_start": 749149, "char_end": 749176 }, "origin": "source_reference", "context": "proof" }, { "from": "sec:principal-signal", "to": "prop:parameter-order", "type": "candidate_proof_dependency", "evidence": "choice. Put \\(c_i=\\int_0^\\infty W_i(y)y^{-5/6}\\,dy>0\\). The asymptotic already proved in Section~\\ref{sec:principal-signal} specializes to \\[ S_i(Z)\\sim\\frac{P^{1/6}c_i}{|T|\\log P}>0,\\qquad A_T(Z)\\sim \\left(\\frac{K}{|T|\\ell}\\right)^K \\left(\\pr", "anchor": { "source": "paper.tex", "line_start": 16293, "line_end": 16293, "char_start": 749303, "char_end": 749329 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:detector-dyads", "to": "prop:parameter-order", "type": "candidate_proof_dependency", "evidence": "rs in the principal remainder estimate. Also require \\(e<10^{-3}\\) and the bounds \\(e_0\\) in Lemma~\\ref{lem:detector-dyads}. The slot lengths have already been fixed, so the small powers needed in the amplitude subdivision can be chosen in terms of the", "anchor": { "source": "paper.tex", "line_start": 16321, "line_end": 16321, "char_start": 750394, "char_end": 750418 }, "origin": "source_reference", "context": "proof" }, { "from": "sec:tails", "to": "prop:parameter-order", "type": "candidate_proof_dependency", "evidence": "en costs at most \\(2\\zeta2\\) in Section~\\ref{sec:tails} sufficiently far right that its large-row sum has a saving larger than \\(m_{\\rm high}\\) relative to \\(C(\\beta_*)\\). Its exponent", "anchor": { "source": "paper.tex", "line_start": 16332, "line_end": 16332, "char_start": 750824, "char_end": 750839 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:shared-principal-nonvanishing", "to": "prop:parameter-order", "type": "candidate_proof_dependency", "evidence": "choices, so \\(z_\\infty\\) also precedes the target. Choose the pretarget cutoff needed for Equation~\\eqref{eq:shared-principal-nonvanishing}. Any additional fixed excluded primes required by Proposition~\\ref{prop:probe-errors} after the target is fixed may be added to", "anchor": { "source": "paper.tex", "line_start": 16337, "line_end": 16337, "char_start": 751095, "char_end": 751135 }, "origin": "source_reference", "context": "proof" }, { "from": "prop:probe-errors", "to": "prop:parameter-order", "type": "candidate_proof_dependency", "evidence": "ef{eq:shared-principal-nonvanishing}. Any additional fixed excluded primes required by Proposition~\\ref{prop:probe-errors} after the target is fixed may be added to \\(S\\): the same positive product-tail majorant preserves the shared contraction, and ne", "anchor": { "source": "paper.tex", "line_start": 16338, "line_end": 16338, "char_start": 751199, "char_end": 751222 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:high-bin", "to": "prop:parameter-order", "type": "candidate_proof_dependency", "evidence": "change. For clarity, the nonprincipal real comparison is uniform over all pointwise pairs in Lemma~\\ref{lem:high-bin}. Let \\(\\epsilon_{\\rm real}(d)\\ge0\\) collect the allocated real, mesh, and moment losses for the relevant range. The endpoint in", "anchor": { "source": "paper.tex", "line_start": 16344, "line_end": 16344, "char_start": 751485, "char_end": 751503 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:principal-residue-interface", "to": "prop:parameter-order", "type": "candidate_proof_dependency", "evidence": "and absorb the remaining finite dyadic multiplicities using a power below \\(m_{\\rm high}/8\\). Lemma~\\ref{lem:principal-residue-interface} already includes its own normalizer allowance \\(\\epsilon_{\\rm pr}\\). The four high real allocations above cost less than \\(m_{\\r", "anchor": { "source": "paper.tex", "line_start": 16360, "line_end": 16360, "char_start": 752311, "char_end": 752348 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:part-II-normalized-low", "to": "prop:parameter-order", "type": "candidate_proof_dependency", "evidence": "a common remaining high saving before retained height factors. For the low estimate, apply Equation~\\eqref{eq:part-II-normalized-low} with the pretarget loss \\(\\Delta/2\\), and set \\(\\omega=\\Delta/2\\). Let \\(\\epsilon_{\\rm ht}>0\\) be the minimum of the finitely man", "anchor": { "source": "paper.tex", "line_start": 16369, "line_end": 16370, "char_start": 752821, "char_end": 752854 }, "origin": "source_reference", "context": "proof" }, { "from": "sec:selection", "to": "prop:parameter-order", "type": "candidate_proof_dependency", "evidence": "ypes only enlarges this fixed \\(B_\\eta\\). We use the height ceiling verified at the end of Section~\\ref{sec:selection}. Let \\(A_{{\\rm ht},\\eta}\\le A_\\eta\\) dominate the finitely many detector height orders, enlarging \\(A_\\eta\\) if necessary, and p", "anchor": { "source": "paper.tex", "line_start": 16392, "line_end": 16392, "char_start": 754188, "char_end": 754207 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:detector-dyads", "to": "prop:parameter-order", "type": "candidate_proof_dependency", "evidence": "ht}\\), this proves every literal condition \\((1+T_1)^{A_{\\mathcal A}}\\le U^{\\epsilon/10}\\) in Lemma~\\ref{lem:detector-dyads} throughout the moderate range. The first bound also makes the detector's crude error \\(U^{-189/100+o(1)}T_1^2\\le U^{-187/100+o(1", "anchor": { "source": "paper.tex", "line_start": 16411, "line_end": 16411, "char_start": 754942, "char_end": 754966 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:stage-height-allocation", "to": "prop:parameter-order", "type": "candidate_proof_dependency", "evidence": "al coordinates, including the physical \\(\\Im z\\) coordinate, uses the single allocation in Equation~\\eqref{eq:stage-height-allocation}. Its matrix is fixed by the slot system and the finite retained estimates, and is independent of the external derivative order.", "anchor": { "source": "paper.tex", "line_start": 16420, "line_end": 16420, "char_start": 755355, "char_end": 755389 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:high-bin", "to": "prop:parameter-order", "type": "candidate_proof_dependency", "evidence": "\\eta}\\). No internal moment is reapplied to an externally differentiated profile. Combining Lemma~\\ref{lem:high-bin}, all the row ranges, and the principal remainder estimate therefore gives, for every fixed \\(N\\) and sufficiently large \\(Z\\) in", "anchor": { "source": "paper.tex", "line_start": 16434, "line_end": 16434, "char_start": 756189, "char_end": 756207 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:late-height-hypothesis", "to": "prop:parameter-order", "type": "candidate_proof_dependency", "evidence": "^{B_\\eta}T_1^{-N}. \\] The lower threshold is allowed to depend on \\(\\tau,N,\\eta\\). This is Equation~\\eqref{eq:late-height-hypothesis}. Its low hypothesis is Equation~\\eqref{eq:part-II-normalized-low} with \\(\\omega=\\Delta/2\\), and the two functions do not contain", "anchor": { "source": "paper.tex", "line_start": 16444, "line_end": 16444, "char_start": 756568, "char_end": 756601 }, "origin": "source_reference", "context": "proof" }, { "from": "eq:part-II-normalized-low", "to": "prop:parameter-order", "type": "candidate_proof_dependency", "evidence": "\\(\\tau,N,\\eta\\). This is Equation~\\eqref{eq:late-height-hypothesis}. Its low hypothesis is Equation~\\eqref{eq:part-II-normalized-low} with \\(\\omega=\\Delta/2\\), and the two functions do not contain \\(T_1\\). Apply Lemma~\\ref{lem:late-height-closure} with \\(\\sigma_0", "anchor": { "source": "paper.tex", "line_start": 16445, "line_end": 16446, "char_start": 756634, "char_end": 756667 }, "origin": "source_reference", "context": "proof" }, { "from": "lem:late-height-closure", "to": "prop:parameter-order", "type": "candidate_proof_dependency", "evidence": "normalized-low} with \\(\\omega=\\Delta/2\\), and the two functions do not contain \\(T_1\\). Apply Lemma~\\ref{lem:late-height-closure} with \\(\\sigma_0=7/8\\), \\(C=C_{\\mathrm{II}}\\), this \\(m\\), and the verified ceiling \\(\\tau_{0,\\eta}\\). It chooses \\(\\tau_\\eta\\),", "anchor": { "source": "paper.tex", "line_start": 16447, "line_end": 16447, "char_start": 756752, "char_end": 756781 }, "origin": "source_reference", "context": "proof" }, { "from": "prop:parameter-order", "to": "sec:conclusion", "type": "internal_reference", "evidence": "igma=m/2>0\\). Both \\(\\omega\\) and \\(\\sigma\\) were fixed before the target. Proposition~\\ref{prop:parameter-order} and Equation~\\eqref{eq:shared-principal-nonvanishing} verify the hypotheses of Proposition~\\ref{lem:continuation-criterion} with", "anchor": { "source": "paper.tex", "line_start": 16455, "line_end": 16455, "char_start": 757135, "char_end": 757161 }, "origin": "source_reference", "context": "narrative" }, { "from": "eq:shared-principal-nonvanishing", "to": "sec:conclusion", "type": "internal_reference", "evidence": "ma\\) were fixed before the target. Proposition~\\ref{prop:parameter-order} and Equation~\\eqref{eq:shared-principal-nonvanishing} verify the hypotheses of Proposition~\\ref{lem:continuation-criterion} with \\(\\sigma_0=7/8\\), \\(J_\\eta=J_{\\mathrm{II},\\eta}\\), and", "anchor": { "source": "paper.tex", "line_start": 16456, "line_end": 16456, "char_start": 757175, "char_end": 757215 }, "origin": "source_reference", "context": "narrative" }, { "from": "lem:continuation-criterion", "to": "sec:conclusion", "type": "internal_reference", "evidence": "r-order} and Equation~\\eqref{eq:shared-principal-nonvanishing} verify the hypotheses of Proposition~\\ref{lem:continuation-criterion} with \\(\\sigma_0=7/8\\), \\(J_\\eta=J_{\\mathrm{II},\\eta}\\), and \\(f_\\eta=f_{\\mathrm{II},\\eta}\\). This contradicts \\(\\beta_*>7/8\\); h", "anchor": { "source": "paper.tex", "line_start": 16457, "line_end": 16457, "char_start": 757253, "char_end": 757285 }, "origin": "source_reference", "context": "narrative" }, { "from": "prop:hecke-dirichlet-transfer", "to": "sec:conclusion", "type": "internal_reference", "evidence": "ta=f_{\\mathrm{II},\\eta}\\). This contradicts \\(\\beta_*>7/8\\); hence \\(\\beta_*\\le7/8\\). Proposition~\\ref{prop:hecke-dirichlet-transfer} at \\(\\sigma_0=7/8\\) extends the primitive Hecke conclusion to all finite-order Hecke and Dirichlet \\(L\\)-functions, with the prin", "anchor": { "source": "paper.tex", "line_start": 16460, "line_end": 16461, "char_start": 757452, "char_end": 757487 }, "origin": "source_reference", "context": "narrative" }, { "from": "thm:main", "to": "sec:conclusion", "type": "internal_reference", "evidence": "nite-order Hecke and Dirichlet \\(L\\)-functions, with the principal pole allowed, and proves Theorem~\\ref{thm:main}. {99} \\bibitem{BaierBansal} S. Baier and A. Bansal, \\emph{The large sieve with power moduli for $\\mathbb", "anchor": { "source": "paper.tex", "line_start": 16463, "line_end": 16463, "char_start": 757656, "char_end": 757670 }, "origin": "source_reference", "context": "narrative" }, { "from": "external:MilneCFT", "to": "sec:introduction", "type": "external_reference", "evidence": "rees with the usual finite-order idelic definition: the complex place contributes no nontrivial finite-order continuous character \\cite[Chapter VI, Section 1]{MilneCFT}. For $\\Re s>1$, put \\[ L_F(s,\\eta)=\\sum_{\\mathfrak a\\ne0} \\frac{\\eta(\\mathfrak a)}{(N\\mathfrak a)^s} =\\prod_{\\mathfrak p} \\left(1-\\eta(\\mathfrak p)(N\\mathfrak p)^{-s}\\right)^{-1}. \\]", "anchor": { "source": "paper.tex", "line_start": 57, "line_end": 57, "char_start": 2140, "char_end": 2178 }, "origin": "source_reference", "locator": "Chapter VI, Section 1", "context": "narrative" }, { "from": "external:BCSS", "to": "sec:introduction", "type": "external_reference", "evidence": "alone, the existence of a fixed $\\sigma_0<1$ for which $\\zeta(s)\\ne0$ in $\\Re s>\\sigma_0$ is called the quasi-Riemann hypothesis \\cite[Section 1]{BCSS}. It asks for one gap valid at every height, not merely nonvanishing on $\\Re s=1$. The corresponding uniform formulation over all Dirichlet characters appears in \\cite[p.~288, Equation (1.4) and the f", "anchor": { "source": "paper.tex", "line_start": 79, "line_end": 79, "char_start": 3033, "char_end": 3055 }, "origin": "source_reference", "locator": "Section 1", "context": "narrative" }, { "from": "external:FG", "to": "sec:introduction", "type": "external_reference", "evidence": "very height, not merely nonvanishing on $\\Re s=1$. The corresponding uniform formulation over all Dirichlet characters appears in \\cite[p.~288, Equation (1.4) and the following paragraph]{FG}. The connection between these functions and prime distribution has a long history. Dirichlet's 1837 proof of infinitude of primes in every reduced arithmetic progression introduced the character $L$", "anchor": { "source": "paper.tex", "line_start": 82, "line_end": 82, "char_start": 3217, "char_end": 3278 }, "origin": "source_reference", "locator": "p.~288, Equation (1.4) and the following paragraph", "context": "narrative" }, { "from": "external:Dirichlet", "to": "sec:introduction", "type": "external_reference", "evidence": "of infinitude of primes in every reduced arithmetic progression introduced the character $L$-series that separate residue classes \\cite{Dirichlet}. Riemann's 1859 memoir related the zeros of $\\zeta$ to the distribution of primes and formulated the critical-line conjecture \\cite{Riemann}. Hadamard and de la Vall\\'ee Poussin independently proved", "anchor": { "source": "paper.tex", "line_start": 87, "line_end": 87, "char_start": 3516, "char_end": 3532 }, "origin": "source_reference", "locator": "", "context": "narrative" }, { "from": "external:Riemann", "to": "sec:introduction", "type": "external_reference", "evidence": "et}. Riemann's 1859 memoir related the zeros of $\\zeta$ to the distribution of primes and formulated the critical-line conjecture \\cite{Riemann}. Hadamard and de la Vall\\'ee Poussin independently proved in 1896 that $\\zeta$ has no zero on $\\Re s=1$, obtaining the prime number theorem \\cite{Hadamard,ValleePoussin}. Hecke subsequently developed", "anchor": { "source": "paper.tex", "line_start": 89, "line_end": 89, "char_start": 3659, "char_end": 3673 }, "origin": "source_reference", "locator": "", "context": "narrative" }, { "from": "external:Hadamard", "to": "sec:introduction", "type": "external_reference", "evidence": "and de la Vall\\'ee Poussin independently proved in 1896 that $\\zeta$ has no zero on $\\Re s=1$, obtaining the prime number theorem \\cite{Hadamard,ValleePoussin}. Hecke subsequently developed the character zeta functions and their analytic theory over number fields \\cite{Hecke}. Classical zero-free regions for Hecke $L$-functions approach the line $\\Re s=1$", "anchor": { "source": "paper.tex", "line_start": 92, "line_end": 92, "char_start": 3814, "char_end": 3843 }, "origin": "source_reference", "locator": "", "context": "narrative" }, { "from": "external:ValleePoussin", "to": "sec:introduction", "type": "external_reference", "evidence": "and de la Vall\\'ee Poussin independently proved in 1896 that $\\zeta$ has no zero on $\\Re s=1$, obtaining the prime number theorem \\cite{Hadamard,ValleePoussin}. Hecke subsequently developed the character zeta functions and their analytic theory over number fields \\cite{Hecke}. Classical zero-free regions for Hecke $L$-functions approach the line $\\Re s=1$", "anchor": { "source": "paper.tex", "line_start": 92, "line_end": 92, "char_start": 3814, "char_end": 3843 }, "origin": "source_reference", "locator": "", "context": "narrative" }, { "from": "external:Hecke", "to": "sec:introduction", "type": "external_reference", "evidence": "e{Hadamard,ValleePoussin}. Hecke subsequently developed the character zeta functions and their analytic theory over number fields \\cite{Hecke}. Classical zero-free regions for Hecke $L$-functions approach the line $\\Re s=1$ as the conductor or height increases and can allow one simple real exceptional zero; a precise form for abelian exten", "anchor": { "source": "paper.tex", "line_start": 94, "line_end": 94, "char_start": 3948, "char_end": 3960 }, "origin": "source_reference", "locator": "", "context": "narrative" }, { "from": "external:TZ", "to": "sec:introduction", "type": "external_reference", "evidence": "e conductor or height increases and can allow one simple real exceptional zero; a precise form for abelian extensions is given by \\cite[Theorem 3.1]{TZ}. Such regions do not give uniform exclusion in a fixed half-plane. Zero-density estimates address a different question: they bound the number of zeros to the right of a given vertical line. For examp", "anchor": { "source": "paper.tex", "line_start": 99, "line_end": 99, "char_start": 4178, "char_end": 4200 }, "origin": "source_reference", "locator": "Theorem 3.1", "context": "narrative" }, { "from": "external:GM", "to": "sec:introduction", "type": "external_reference", "evidence": "the right of a given vertical line. For example, Guth and Maynard's large-value estimates improve zero-density bounds for $\\zeta$ \\cite[Theorem 1.2]{GM}, but these zero-density bounds do not exclude every zero. Every finite-order Hecke $L$-function over $F=\\mathbb Q(\\sqrt{-3})$ has no zero in $\\Re s>7/8$. The same hol", "anchor": { "source": "paper.tex", "line_start": 103, "line_end": 103, "char_start": 4485, "char_end": 4507 }, "origin": "source_reference", "locator": "Theorem 1.2", "context": "narrative" }, { "from": "external:Clay", "to": "sec:introduction", "type": "external_reference", "evidence": "Riemann hypothesis or its generalized versions, which place nontrivial zeros on $\\Re s=1/2$. The Riemann hypothesis remains open \\cite{Clay}. Kubota's metaplectic theory and Patterson's cubic theta series provide the automorphic setting \\cite{Kubota,Patterson}. We use the unconditional explicit cusp expansions recorded by Dunn and Radziw", "anchor": { "source": "paper.tex", "line_start": 119, "line_end": 119, "char_start": 5308, "char_end": 5319 }, "origin": "source_reference", "locator": "", "context": "narrative" }, { "from": "external:Kubota", "to": "sec:introduction", "type": "external_reference", "evidence": "othesis remains open \\cite{Clay}. Kubota's metaplectic theory and Patterson's cubic theta series provide the automorphic setting \\cite{Kubota,Patterson}. We use the unconditional explicit cusp expansions recorded by Dunn and Radziwi\\l\\l{} \\cite[Section 5 and Appendix A]{DR}; their GRH-conditional prime asymptotic is not an input. The corresponding fi", "anchor": { "source": "paper.tex", "line_start": 122, "line_end": 122, "char_start": 5417, "char_end": 5440 }, "origin": "source_reference", "locator": "", "context": "narrative" }, { "from": "external:Patterson", "to": "sec:introduction", "type": "external_reference", "evidence": "othesis remains open \\cite{Clay}. Kubota's metaplectic theory and Patterson's cubic theta series provide the automorphic setting \\cite{Kubota,Patterson}. We use the unconditional explicit cusp expansions recorded by Dunn and Radziwi\\l\\l{} \\cite[Section 5 and Appendix A]{DR}; their GRH-conditional prime asymptotic is not an input. The corresponding fi", "anchor": { "source": "paper.tex", "line_start": 122, "line_end": 122, "char_start": 5417, "char_end": 5440 }, "origin": "source_reference", "locator": "", "context": "narrative" }, { "from": "external:DR", "to": "sec:introduction", "type": "external_reference", "evidence": "automorphic setting \\cite{Kubota,Patterson}. We use the unconditional explicit cusp expansions recorded by Dunn and Radziwi\\l\\l{} \\cite[Section 5 and Appendix A]{DR}; their GRH-conditional prime asymptotic is not an input. The corresponding first-moment asymptotic is proved unconditionally in an independent manuscript \\cite[Theorem~1.1]{OpenAICubicFirstMoment}; t", "anchor": { "source": "paper.tex", "line_start": 124, "line_end": 124, "char_start": 5527, "char_end": 5562 }, "origin": "source_reference", "locator": "Section 5 and Appendix A", "context": "narrative" }, { "from": "external:OpenAICubicFirstMoment", "to": "sec:introduction", "type": "external_reference", "evidence": "rime asymptotic is not an input. The corresponding first-moment asymptotic is proved unconditionally in an independent manuscript \\cite[Theorem~1.1]{OpenAICubicFirstMoment}; that result is also not used here. Proposition~\\ref{prop:completed-reflection} derives the reflection used here while retaining the characters' zero-on-nonunit restrictions. The character large-sie", "anchor": { "source": "paper.tex", "line_start": 127, "line_end": 127, "char_start": 5717, "char_end": 5759 }, "origin": "source_reference", "locator": "Theorem~1.1", "context": "narrative" }, { "from": "external:GL", "to": "sec:introduction", "type": "external_reference", "evidence": "-nonunit restrictions. The character large-sieve arguments build on the quadratic Hecke-family estimate of Goldmakher and Louvel \\cite[Definition 1 and Theorem 1.1]{GL}, the higher-order norm recursion of Blomer, Goldmakher, and Louvel \\cite[Theorem 1.3 and Section 3]{BGL}, and Heath-Brown's cubic estimate \\cite[Theorem 2]{HB}. We prove the precise sextic specializa", "anchor": { "source": "paper.tex", "line_start": 133, "line_end": 133, "char_start": 6042, "char_end": 6081 }, "origin": "source_reference", "locator": "Definition 1 and Theorem 1.1", "context": "narrative" }, { "from": "external:BGL", "to": "sec:introduction", "type": "external_reference", "evidence": "Goldmakher and Louvel \\cite[Definition 1 and Theorem 1.1]{GL}, the higher-order norm recursion of Blomer, Goldmakher, and Louvel \\cite[Theorem 1.3 and Section 3]{BGL}, and Heath-Brown's cubic estimate \\cite[Theorem 2]{HB}. We prove the precise sextic specialization needed here in Lemma~\\ref{lem:sextic-large-sieve}, using paired residue characters to carry out that", "anchor": { "source": "paper.tex", "line_start": 135, "line_end": 135, "char_start": 6149, "char_end": 6186 }, "origin": "source_reference", "locator": "Theorem 1.3 and Section 3", "context": "narrative" }, { "from": "external:HB", "to": "sec:introduction", "type": "external_reference", "evidence": "er-order norm recursion of Blomer, Goldmakher, and Louvel \\cite[Theorem 1.3 and Section 3]{BGL}, and Heath-Brown's cubic estimate \\cite[Theorem 2]{HB}. We prove the precise sextic specialization needed here in Lemma~\\ref{lem:sextic-large-sieve}, using paired residue characters to carry out that recursion in the primary-generator convention. An Eise", "anchor": { "source": "paper.tex", "line_start": 136, "line_end": 136, "char_start": 6221, "char_end": 6241 }, "origin": "source_reference", "locator": "Theorem 2", "context": "narrative" }, { "from": "external:GZSextic", "to": "sec:introduction", "type": "external_reference", "evidence": "ters to carry out that recursion in the primary-generator convention. An Eisenstein-integer form is also recorded by Gao and Zhao \\cite[Lemma 2.9]{GZSextic}. The recursive moment arguments are related to the framework of Heath-Brown's quadratic method \\cite[Section 2]{HBQuadratic}. We also use the ordinary Hecke functional equation \\cite[Equation (1.1)]{", "anchor": { "source": "paper.tex", "line_start": 140, "line_end": 140, "char_start": 6494, "char_end": 6520 }, "origin": "source_reference", "locator": "Lemma 2.9", "context": "narrative" }, { "from": "external:HBQuadratic", "to": "sec:introduction", "type": "external_reference", "evidence": "nd Zhao \\cite[Lemma 2.9]{GZSextic}. The recursive moment arguments are related to the framework of Heath-Brown's quadratic method \\cite[Section 2]{HBQuadratic}. We also use the ordinary Hecke functional equation \\cite[Equation (1.1)]{GZ} and prime counting in a fixed ray class \\cite[Theorem 1.1]{TZ}. The planar additive large sieve used in the first stage", "anchor": { "source": "paper.tex", "line_start": 142, "line_end": 142, "char_start": 6616, "char_end": 6645 }, "origin": "source_reference", "locator": "Section 2", "context": "narrative" }, { "from": "external:GZ", "to": "sec:introduction", "type": "external_reference", "evidence": "the framework of Heath-Brown's quadratic method \\cite[Section 2]{HBQuadratic}. We also use the ordinary Hecke functional equation \\cite[Equation (1.1)]{GZ} and prime counting in a fixed ray class \\cite[Theorem 1.1]{TZ}. The planar additive large sieve used in the first stage is classical; compare Huxley's multivariable and number-field inequality \\cite", "anchor": { "source": "paper.tex", "line_start": 143, "line_end": 143, "char_start": 6698, "char_end": 6723 }, "origin": "source_reference", "locator": "Equation (1.1)", "context": "narrative" }, { "from": "external:TZ", "to": "sec:introduction", "type": "external_reference", "evidence": "BQuadratic}. We also use the ordinary Hecke functional equation \\cite[Equation (1.1)]{GZ} and prime counting in a fixed ray class \\cite[Theorem 1.1]{TZ}. The planar additive large sieve used in the first stage is classical; compare Huxley's multivariable and number-field inequality \\cite{Huxley} and the Poisson proof in \\cite[Section 3, Theorem 3]{B", "anchor": { "source": "paper.tex", "line_start": 144, "line_end": 144, "char_start": 6764, "char_end": 6786 }, "origin": "source_reference", "locator": "Theorem 1.1", "context": "narrative" }, { "from": "external:Huxley", "to": "sec:introduction", "type": "external_reference", "evidence": "The planar additive large sieve used in the first stage is classical; compare Huxley's multivariable and number-field inequality \\cite{Huxley} and the Poisson proof in \\cite[Section 3, Theorem 3]{BaierBansal}. We include a direct Eisenstein-lattice proof to record the normalization needed for the reduced-fraction expansion. David, de Faveri", "anchor": { "source": "paper.tex", "line_start": 147, "line_end": 148, "char_start": 6918, "char_end": 6931 }, "origin": "source_reference", "locator": "", "context": "narrative" }, { "from": "external:BaierBansal", "to": "sec:introduction", "type": "external_reference", "evidence": "n the first stage is classical; compare Huxley's multivariable and number-field inequality \\cite{Huxley} and the Poisson proof in \\cite[Section 3, Theorem 3]{BaierBansal}. We include a direct Eisenstein-lattice proof to record the normalization needed for the reduced-fraction expansion. David, de Faveri, Dunn, and Stucky combine Patterson's coefficients, the cubic lar", "anchor": { "source": "paper.tex", "line_start": 148, "line_end": 148, "char_start": 6957, "char_end": 6997 }, "origin": "source_reference", "locator": "Section 3, Theorem 3", "context": "narrative" }, { "from": "external:DFDS", "to": "sec:introduction", "type": "external_reference", "evidence": ", the cubic large sieve, and mollified moments to prove nonvanishing at $s=1/2$ for a positive proportion of a cubic Hecke family \\cite[Theorem 1.1 and Section 1.2]{DFDS}. The zero detector below uses the classical truncated-inverse mechanism; compare \\cite[Appendix C]{MP}, and for a higher-order-character density application \\cite[Corollary 1.6 and Section 5]{BGL}.", "anchor": { "source": "paper.tex", "line_start": 154, "line_end": 154, "char_start": 7312, "char_end": 7352 }, "origin": "source_reference", "locator": "Theorem 1.1 and Section 1.2", "context": "narrative" }, { "from": "external:MP", "to": "sec:introduction", "type": "external_reference", "evidence": "family \\cite[Theorem 1.1 and Section 1.2]{DFDS}. The zero detector below uses the classical truncated-inverse mechanism; compare \\cite[Appendix C]{MP}, and for a higher-order-character density application \\cite[Corollary 1.6 and Section 5]{BGL}. The proof has two stages, each comparing two representations of a completed cubic-theta sum but using a", "anchor": { "source": "paper.tex", "line_start": 156, "line_end": 156, "char_start": 7434, "char_end": 7455 }, "origin": "source_reference", "locator": "Appendix C", "context": "narrative" }, { "from": "external:BGL", "to": "sec:introduction", "type": "external_reference", "evidence": "es the classical truncated-inverse mechanism; compare \\cite[Appendix C]{MP}, and for a higher-order-character density application \\cite[Corollary 1.6 and Section 5]{BGL}. The proof has two stages, each comparing two representations of a completed cubic-theta sum but using a different normalized sum in the continuation argument. Part~\\ref{part:eleven-twelfths} proves", "anchor": { "source": "paper.tex", "line_start": 157, "line_end": 157, "char_start": 7510, "char_end": 7549 }, "origin": "source_reference", "locator": "Corollary 1.6 and Section 5", "context": "narrative" }, { "from": "external:TaoNonresidues", "to": "cor:nonresidues-square-roots", "type": "external_reference", "evidence": ")^A. \\] In particular, $n(p)\\ll_\\delta p^\\delta$ for every $\\delta>0$, proving Vinogradov's least quadratic nonresidue conjecture \\cite[Conjecture 1.1]{TaoNonresidues}. Given an odd prime $p$ and $a\\in\\mathbb F_p$ in binary representation, there is a deterministic algorithm, with running time polynomial in $\\log p$, that returns a square root of $a$ or reports that", "anchor": { "source": "paper.tex", "line_start": 333, "line_end": 333, "char_start": 17363, "char_end": 17400 }, "origin": "source_reference", "locator": "Conjecture 1.1", "context": "statement" }, { "from": "external:BIMS", "to": "cor:nonresidues-square-roots", "type": "external_reference", "evidence": "e in the strictly larger strip $1/16<\\Re s<15/16$. This supplies the weak-GRH hypothesis of Bhargava, Ivanyos, Mittal, and Saxena \\cite[Conjecture 6.3 and Theorem 6.7]{BIMS} with $\\epsilon=7/16$. Their bound for the least quadratic nonresidue gives the asserted inequality, for example with $A=32$; no optimization is needed here. The assertion for each $\\delta>0$ follows", "anchor": { "source": "paper.tex", "line_start": 345, "line_end": 345, "char_start": 17938, "char_end": 17981 }, "origin": "source_reference", "locator": "Conjecture 6.3 and Theorem 6.7", "context": "proof" }, { "from": "external:Galbraith", "to": "cor:nonresidues-square-roots", "type": "external_reference", "evidence": "terministic search; its stopping rule does not require knowing $C$. Use the resulting nonresidue in the Tonelli--Shanks algorithm \\cite[Section 2.9, Algorithm 3 and Lemma 2.9.5]{Galbraith}. All remaining steps are deterministic and polynomial in $\\log p$. Writing $p-1=2^e u$ with $u$ odd only requires removing factors of two, not factoring $u$. \\section{From a common signa", "anchor": { "source": "paper.tex", "line_start": 357, "line_end": 357, "char_start": 18635, "char_end": 18693 }, "origin": "source_reference", "locator": "Section 2.9, Algorithm 3 and Lemma 2.9.5", "context": "proof" }, { "from": "external:MilneCFT", "to": "lem:fixed-numerator-ray", "type": "external_reference", "evidence": "r through a ray group with modulus supported on the primes dividing $6a$; these are the power-residue and ray-group assertions in \\cite[Chapter VIII, (5.3) and (5.5)]{MilneCFT}. For the last statement, an ideal $A$ outside $S$ is coprime to every $\\pi_{\\mathfrak p}$ and to every unit. Its symbol therefore has no zero there, and the power of each $\\chi_A(\\pi_{\\mathfrak p})", "anchor": { "source": "paper.tex", "line_start": 689, "line_end": 689, "char_start": 33574, "char_end": 33620 }, "origin": "source_reference", "locator": "Chapter VIII, (5.3) and (5.5)", "context": "proof" }, { "from": "external:DR", "to": "lem:fixed-gauss-phase", "type": "external_reference", "evidence": "\\chi_q(p)^3. \\] We use cubic reciprocity in the following precise form: for coprime primary $a,b$ one has $(a/b)_3=(b/a)_3$; see \\cite[Equation (1.4)]{DR}. Thus the quotient $\\chi_b(a)/\\chi_a(b)$ is a sign. At $p,q$ its cube is $\\chi_q(p)^3\\chi_p(q)^3$, so this sign equals $\\mathfrak r(p,q)$. Multiplicativity in both arguments and the bicharacter pro", "anchor": { "source": "paper.tex", "line_start": 998, "line_end": 998, "char_start": 46942, "char_end": 46967 }, "origin": "source_reference", "locator": "Equation (1.4)", "context": "proof" }, { "from": "external:GZ", "to": "lem:hecke-strip-growth", "type": "external_reference", "evidence": "si), \\quad |\\varepsilon(\\psi)|=1. \\] This is the functional equation for primitive characters of trivial infinite type stated in \\cite[Equation (1.1)]{GZ}; a nonprincipal $L(s,\\psi)$ is entire. Absolute Euler convergence bounds $L(11/10+it,\\psi)$ uniformly in $Q,t$. Applying the functional equation and Stirling's formula on $\\Re s=-1/10$ gives \\[ |L", "anchor": { "source": "paper.tex", "line_start": 1445, "line_end": 1445, "char_start": 68054, "char_end": 68079 }, "origin": "source_reference", "locator": "Equation (1.1)", "context": "proof" }, { "from": "external:DR", "to": "sec:completed-reflection", "type": "external_reference", "evidence": ". The theta function and its Fourier coefficients at three fixed cusps are the unconditional input from Dunn and Radziwi{\\l}{\\l}~\\cite[Section~5 and Appendix~A]{DR}. The original coefficient and cusp calculations are in Patterson~\\cite[Theorem~8.1 and Sections~7--8]{Patterson}; the formal input here remains the formulas in~\\cite{DR}. The finite transform below", "anchor": { "source": "paper.tex", "line_start": 1658, "line_end": 1659, "char_start": 77949, "char_end": 77984 }, "origin": "source_reference", "locator": "Section~5 and\nAppendix~A", "context": "narrative" }, { "from": "external:Patterson", "to": "sec:completed-reflection", "type": "external_reference", "evidence": "om Dunn and Radziwi{\\l}{\\l}~\\cite[Section~5 and Appendix~A]{DR}. The original coefficient and cusp calculations are in Patterson~\\cite[Theorem~8.1 and Sections~7--8]{Patterson}; the formal input here remains the formulas in~\\cite{DR}. The finite transform below derives the masked formula, including its local factors and phase. The later mean-square argument uses this phas", "anchor": { "source": "paper.tex", "line_start": 1660, "line_end": 1660, "char_start": 78051, "char_end": 78098 }, "origin": "source_reference", "locator": "Theorem~8.1 and Sections~7--8", "context": "narrative" }, { "from": "external:DR", "to": "sec:completed-reflection", "type": "external_reference", "evidence": "cusp calculations are in Patterson~\\cite[Theorem~8.1 and Sections~7--8]{Patterson}; the formal input here remains the formulas in~\\cite{DR}. The finite transform below derives the masked formula, including its local factors and phase. The later mean-square argument uses this phase only after fixing its finite cusp sectors. \\subsection", "anchor": { "source": "paper.tex", "line_start": 1661, "line_end": 1661, "char_start": 78146, "char_end": 78155 }, "origin": "source_reference", "locator": "", "context": "narrative" }, { "from": "external:DR", "to": "prop:completed-reflection", "type": "external_reference", "evidence": "\\{g\\in\\Gamma:g\\equiv I\\pmod3\\}$, and $\\Gamma_2=\\langle\\mathrm{SL}_2(\\mathbb Z),\\Gamma_1(3)\\rangle$. The unconditional formulas of \\cite[Equations (5.4), (5.7), (5.9), and (5.12)--(5.17)]{DR} give a smooth cubic theta function $\\theta$ on $\\mathbb H^3=\\mathbb C\\times\\mathbb R_{>0}$ with \\[ \\theta(gw)=\\kappa(g)\\theta(w)\\quad(g\\in\\Gamma_2),\\qquad \\kappa\\!a&b\\\\c&\\delta'\\end{", "anchor": { "source": "paper.tex", "line_start": 1894, "line_end": 1895, "char_start": 87972, "char_end": 88032 }, "origin": "source_reference", "locator": "Equations (5.4), (5.7), (5.9), and (5.12)--(5.17)", "context": "proof" }, { "from": "external:DR", "to": "prop:completed-reflection", "type": "external_reference", "evidence": "on} Here $C_H(v)$ is independent of $z$. The three functions $\\Theta_0,\\Theta_1,\\Theta_2$ are respectively $F_1,F_{19},F_{10}$ in~\\cite{DR}. Its Appendix~A, rows $1,19,10$, expresses their coefficients in terms of $\\tau,\\tau_1,\\tau_2$ in its Equations (5.7), (5.13), and (5.14). In their notation the conjugate expansion uses $\\overline{", "anchor": { "source": "paper.tex", "line_start": 1942, "line_end": 1942, "char_start": 89942, "char_end": 89951 }, "origin": "source_reference", "locator": "", "context": "proof" }, { "from": "external:DR", "to": "prop:completed-reflection", "type": "external_reference", "evidence": "only the cubic symbol, including at primes in $S$. To make their domain explicit, for every primary squarefree $n$ define, as in~\\cite[Equation (1.7)]{DR}, \\widetilde g_3(n)=q_n^{-1/2}\\sum_{v\\bmod n} \\left(\\frac vn\\right)_3\\check e(v/n), \\qquad \\widetilde g_3(1)=1. The ord", "anchor": { "source": "paper.tex", "line_start": 1955, "line_end": 1955, "char_start": 90618, "char_end": 90643 }, "origin": "source_reference", "locator": "Equation (1.7)", "context": "proof" }, { "from": "external:DR", "to": "prop:completed-reflection", "type": "external_reference", "evidence": "mary and $n$ squarefree. The infinity support has $\\lambda^3\\mu\\in\\mathcal O$; every other ramified exponent in Equation (5.7) of~\\cite{DR} makes $x$ divisible by $\\lambda$, and the negative sign in the remaining pair gives $x\\equiv-1\\pmod3$. At the isolated index the coefficient in $\\bar\\theta$ is \\label{eq:reflection-i", "anchor": { "source": "paper.tex", "line_start": 1988, "line_end": 1989, "char_start": 92485, "char_end": 92494 }, "origin": "source_reference", "locator": "", "context": "proof" }, { "from": "external:DR", "to": "prop:completed-reflection", "type": "external_reference", "evidence": "n\\right)_3^{-1}\\gamma_2(n) =\\chi_n(\\lambda)^{-2}\\gamma_2(n)\\qquad((n,S)=1). The cubic supplementary laws in~\\cite[Equation (1.5)]{DR} make $(\\lambda/x)_3$ periodic on all primary $x$ modulo $9$. On primary elements prime to $S$ it equals $\\chi_x(\\lambda)^2$. A character in $\\mathcal X$ is periodic on primary elements prime to $\\m", "anchor": { "source": "paper.tex", "line_start": 2003, "line_end": 2003, "char_start": 93277, "char_end": 93302 }, "origin": "source_reference", "locator": "Equation (1.5)", "context": "proof" }, { "from": "external:DR", "to": "prop:completed-reflection", "type": "external_reference", "evidence": "This uses every additive frequency, including nonunit $h_0$ and zero local frequencies. We do not invoke the twisted formula in \\cite[Lemma~5.2 and Corollary~5.1]{DR}, whose unit-Fourier-support hypothesis need not hold for these masks. The theta function with this multiplier on its infinity coefficient $x=\\lambda^3\\mu\\in\\mathcal O$ is the finite sum \\begin{equat", "anchor": { "source": "paper.tex", "line_start": 2041, "line_end": 2041, "char_start": 94922, "char_end": 94960 }, "origin": "source_reference", "locator": "Lemma~5.2 and Corollary~5.1", "context": "proof" }, { "from": "external:DR", "to": "prop:completed-reflection", "type": "external_reference", "evidence": "J(s)=\\int_0^\\infty \\left.\\partial_{\\bar z}\\mathcal F(z,v)\\right|_{z=0} v^{2s-1}\\,dv. \\] The Bessel identity used in~\\cite[Lemma~5.3]{DR} is \\[ \\int_0^\\infty K_{1/3}(y)y^{w-1}\\,dy =2^{w-2}\\Gamma((w-1/3)/2)\\Gamma((w+1/3)/2) \\qquad(\\Re w>1/3). \\] On $\\Re s>1$ the coefficient sums converge absolutely, so we may differentiate and", "anchor": { "source": "paper.tex", "line_start": 2288, "line_end": 2288, "char_start": 106357, "char_end": 106377 }, "origin": "source_reference", "locator": "Lemma~5.3", "context": "proof" }, { "from": "external:DR", "to": "prop:completed-reflection", "type": "external_reference", "evidence": "translated term $z_h=a/c$, use the matrix $g=GH$ constructed above. The coordinate action of $g^{-1}$ on hyperbolic space, from \\cite[Equation (5.1)]{DR}, gives \\[ g^{-1}(z+a/c,v)= \\left(-\\frac{\\delta'}c-\\frac{\\bar z}{c^2(v^2+|z|^2)}, \\frac{v}{q_c(v^2+|z|^2)}\\right). \\] At $z=0$, the derivative of its first coordinate with respect to", "anchor": { "source": "paper.tex", "line_start": 2315, "line_end": 2315, "char_start": 107508, "char_end": 107533 }, "origin": "source_reference", "locator": "Equation (5.1)", "context": "proof" }, { "from": "external:GL", "to": "sec:completed-reflection", "type": "external_reference", "evidence": "_a(b)^3\\right|^2 \\ll_\\epsilon (UV)^\\epsilon(U+V)\\sum_b|c_b|^2. This is Goldmakher--Louvel's quadratic large sieve \\cite[Definition 1 and Theorem 1.1]{GL}. To verify its family hypotheses, let \\(\\vartheta_a((x))\\) denote the quadratic residue symbol evaluated at the primary generator of \\((x)\\). The adjustment of \\(x\\) to its primary generator contri", "anchor": { "source": "paper.tex", "line_start": 2651, "line_end": 2651, "char_start": 123881, "char_end": 123920 }, "origin": "source_reference", "locator": "Definition 1 and Theorem 1.1", "context": "narrative" }, { "from": "external:Huxley", "to": "sec:probe", "type": "external_reference", "evidence": "he required mean square. The planar additive large sieve is classical; compare Huxley's multivariable and number-field inequality \\cite{Huxley} and the Poisson proof in \\cite[Section~3, Theorem~3]{BaierBansal}. We include the lattice proof to record the normalization used here. [Planar additive large sieve] \\label{lem:planar-ad", "anchor": { "source": "paper.tex", "line_start": 3493, "line_end": 3494, "char_start": 163677, "char_end": 163690 }, "origin": "source_reference", "locator": "", "context": "narrative" }, { "from": "external:BaierBansal", "to": "sec:probe", "type": "external_reference", "evidence": "itive large sieve is classical; compare Huxley's multivariable and number-field inequality \\cite{Huxley} and the Poisson proof in \\cite[Section~3, Theorem~3]{BaierBansal}. We include the lattice proof to record the normalization used here. [Planar additive large sieve] Let $0<\\delta\\le1$, let $Q\\ge1$, and let $z_1,\\ldots", "anchor": { "source": "paper.tex", "line_start": 3494, "line_end": 3494, "char_start": 163716, "char_end": 163756 }, "origin": "source_reference", "locator": "Section~3, Theorem~3", "context": "narrative" }, { "from": "external:GZ", "to": "lem:buffered-bins", "type": "external_reference", "evidence": "on_1/2}\\). For the last assertion, the primitive functional equation recalled in the proof of Lemma~\\ref{lem:hecke-strip-growth} \\cite[Equation (1.1)]{GZ} relates the value at \\(1-a-6e+it\\) to the conjugate primitive function at \\(a+6e-it\\). The conjugate presentation is in \\(\\mathcal X_u\\). The conductor and gamma quotient contribute at most \\[ Q_\\", "anchor": { "source": "paper.tex", "line_start": 4356, "line_end": 4356, "char_start": 199080, "char_end": 199105 }, "origin": "source_reference", "locator": "Equation (1.1)", "context": "proof" }, { "from": "external:MP", "to": "prop:detector-witness", "type": "external_reference", "evidence": "oposition} The truncated-inverse and Gamma-integral construction is a form of the classical zero detector; compare \\cite[Appendix C]{MP} and \\cite[Section~13.1]{GM}. The buffered row-dependent rectangle and the simultaneous lower bounds needed here are established below. Choose the zero \\(\\rho\\) supplied by the bin and set \\[ D_*=U^", "anchor": { "source": "paper.tex", "line_start": 4551, "line_end": 4552, "char_start": 207501, "char_end": 207522 }, "origin": "source_reference", "locator": "Appendix C", "context": "proof" }, { "from": "external:GM", "to": "prop:detector-witness", "type": "external_reference", "evidence": "The truncated-inverse and Gamma-integral construction is a form of the classical zero detector; compare \\cite[Appendix C]{MP} and \\cite[Section~13.1]{GM}. The buffered row-dependent rectangle and the simultaneous lower bounds needed here are established below. Choose the zero \\(\\rho\\) supplied by the bin and set \\[ D_*=U^t,\\qquad Y_*=U^{20}. \\] Let", "anchor": { "source": "paper.tex", "line_start": 4552, "line_end": 4552, "char_start": 207527, "char_end": 207550 }, "origin": "source_reference", "locator": "Section~13.1", "context": "proof" }, { "from": "external:BGL", "to": "sec:sextic-row-count", "type": "external_reference", "evidence": "quired squarefree sextic large sieve in the primary convention, using the norm-recursion method of Blomer, Goldmakher, and Louvel~\\cite[Section 3]{BGL}. We then fix the part of a physical row whose prime valuations are at least two and apply the sieve to its squarefree factor. The size of the fixed part also gives an independent upper bound for the", "anchor": { "source": "paper.tex", "line_start": 4693, "line_end": 4693, "char_start": 213123, "char_end": 213144 }, "origin": "source_reference", "locator": "Section 3", "context": "narrative" }, { "from": "external:BGL", "to": "lem:sextic-large-sieve", "type": "external_reference", "evidence": "wise depend on $k$. We follow the higher-order large-sieve recursion of Blomer, Goldmakher, and Louvel \\cite[Section~3]{BGL}, giving the local verification for the present primary normalization. In the finite Fourier calculation the individual residue characters act on elements; their unit-trivial quotient is the characte", "anchor": { "source": "paper.tex", "line_start": 4729, "line_end": 4729, "char_start": 214878, "char_end": 214899 }, "origin": "source_reference", "locator": "Section~3", "context": "proof" }, { "from": "external:BGL", "to": "lem:sextic-large-sieve", "type": "external_reference", "evidence": "d{split} The maxima may be restricted to nonempty ranges. This is the combination of the gcd and Poisson steps in \\cite[Lemmas~3.2--3.3]{BGL}; every sequence used here is common within its norm sum, and $S$ remains fixed. \\medskip\\noindentThe initial and power bounds. Row inclusion gives $D_j\\le E_j$. Hilbert space duality and sex", "anchor": { "source": "paper.tex", "line_start": 5007, "line_end": 5007, "char_start": 227776, "char_end": 227803 }, "origin": "source_reference", "locator": "Lemmas~3.2--3.3", "context": "proof" }, { "from": "external:GL", "to": "lem:sextic-large-sieve", "type": "external_reference", "evidence": "uation~\\eqref{eq:sextic-sieve-norm-initial}. We need the following near-monotonicity, also used in the large-sieve recursions of \\cite[Lemma~4.4 and Remark~5]{GL} and \\cite[Lemma~3.1]{BGL}: D_j(M_1,N)\\ll_S D_j(M_2,N) \\quad\\text{if }M_2\\ge C M_1\\log(2M_1N),\\quad M_1,M_2,N\\ge1. Here is a dire", "anchor": { "source": "paper.tex", "line_start": 5037, "line_end": 5037, "char_start": 229197, "char_end": 229230 }, "origin": "source_reference", "locator": "Lemma~4.4 and Remark~5", "context": "proof" }, { "from": "external:BGL", "to": "lem:sextic-large-sieve", "type": "external_reference", "evidence": "tial}. We need the following near-monotonicity, also used in the large-sieve recursions of \\cite[Lemma~4.4 and Remark~5]{GL} and \\cite[Lemma~3.1]{BGL}: D_j(M_1,N)\\ll_S D_j(M_2,N) \\quad\\text{if }M_2\\ge C M_1\\log(2M_1N),\\quad M_1,M_2,N\\ge1. Here is a direct verification. The asse", "anchor": { "source": "paper.tex", "line_start": 5037, "line_end": 5037, "char_start": 229235, "char_end": 229256 }, "origin": "source_reference", "locator": "Lemma~3.1", "context": "proof" }, { "from": "external:BGL", "to": "lem:sextic-large-sieve", "type": "external_reference", "evidence": "1}+4d_N\\le P/2$, proving Equation~\\eqref{eq:sextic-sieve-norm-monotone}. We finish with the power decomposition and recursion of \\cite[Lemmas~3.4--3.5]{BGL}. Suppose, simultaneously for $j\\in J$, that for every positive loss D_j(M,N)\\ll_{S,\\varepsilon}(MN)^\\varepsilon \\{M^\\alpha+N+(MN)^{2/3", "anchor": { "source": "paper.tex", "line_start": 5067, "line_end": 5067, "char_start": 230582, "char_end": 230609 }, "origin": "source_reference", "locator": "Lemmas~3.4--3.5", "context": "proof" }, { "from": "external:BGL", "to": "sec:sextic-row-count", "type": "external_reference", "evidence": "factor of the physical row. The separation of squarefree and powerful row factors is also used in the proof of Corollary 1.4 of \\cite{BGL}; here we keep the actual norm of the powerful factor because it controls both the moment and the number of rows. [Sextic-sieve row envelope] Fix the a", "anchor": { "source": "paper.tex", "line_start": 5178, "line_end": 5178, "char_start": 235938, "char_end": 235948 }, "origin": "source_reference", "locator": "", "context": "narrative" }, { "from": "external:TZ", "to": "sec:shared-analytic-estimates", "type": "external_reference", "evidence": "d to a nonidentity Frobenius class supplies a prime outside the fixed set $S$ where the associated sextic character is nontrivial \\cite[Theorem 1.1]{TZ}. By the Kummer description in Lemma~\\ref{lem:fixed-numerator-ray}, this is $\\chi_\\bullet(u)$. Therefore $u=1$ is the sole principal numerator row. A denominator attached to a bounded unit row may b", "anchor": { "source": "paper.tex", "line_start": 5512, "line_end": 5512, "char_start": 251967, "char_end": 251989 }, "origin": "source_reference", "locator": "Theorem 1.1", "context": "narrative" }, { "from": "external:TZ", "to": "lem:ray-prime-normalizer", "type": "external_reference", "evidence": ":=\\#\\{p\\in1_T:q_p\\le x\\} \\sim\\frac{\\operatorname{Li}(x)}{|T|}. \\] This is the fixed-extension consequence of Chebotarev in \\cite[Theorem 1.1]{TZ}; the extension and its conductor are fixed as $x\\to\\infty$. If $\\operatorname{supp}W\\subset[a,b]\\subset(0,\\infty)$, the error $o(x/\\log x)$ is uniform for $aP\\le x\\le bP$ as $P\\to\\infty$. Stieltjes", "anchor": { "source": "paper.tex", "line_start": 7027, "line_end": 7027, "char_start": 322305, "char_end": 322327 }, "origin": "source_reference", "locator": "Theorem 1.1", "context": "proof" }, { "from": "external:HB", "to": "sec:marked", "type": "external_reference", "evidence": "{*}|c_b|^2, where both stars mean squarefree in \\(\\mathcal O\\), and the coefficients are arbitrary complex numbers \\cite[Theorem 2]{HB}. The indices need not have squarefree rational norm, and there is no exclusion of rational prime factors. Conjugation and cubic reciprocity allow the opposite orientation. In this sieve and Equati", "anchor": { "source": "paper.tex", "line_start": 7499, "line_end": 7499, "char_start": 344535, "char_end": 344555 }, "origin": "source_reference", "locator": "Theorem 2", "context": "narrative" }, { "from": "external:GZ", "to": "prop:probe-errors", "type": "external_reference", "evidence": "torname{rad}(u)} \\le q_u\\prod_{p\\in J_0}q_p^{-(j_p-1)}. The functional equation for the primitive numerator \\cite[Equation (1.1)]{GZ}, the assumed reflected bound, and Stirling give a cost $q_{\\mathfrak f_u}^{A_*}U^\\epsilon(3+T_1)^C$, where \\[ A_*=1/2-w_r=a-1/2+6e>0. \\] At a selected $p\\mid u$ the primitive character already has v", "anchor": { "source": "paper.tex", "line_start": 9012, "line_end": 9012, "char_start": 412620, "char_end": 412645 }, "origin": "source_reference", "locator": "Equation (1.1)", "context": "proof" }, { "from": "external:HBQuadratic", "to": "sec:inverse", "type": "external_reference", "evidence": "n moving modulus cannot be treated as part of the fixed arithmetic data. For the recursive Poisson large-sieve framework, compare \\cite[Section~2]{HBQuadratic}, \\cite[Section~4]{HB}, and \\cite[Sections~4--7]{GL}; the present marked recursion is proved below. [Canonical marked estimate] Fix bounded nonnegative ranges", "anchor": { "source": "paper.tex", "line_start": 9293, "line_end": 9293, "char_start": 425338, "char_end": 425367 }, "origin": "source_reference", "locator": "Section~2", "context": "narrative" }, { "from": "external:HB", "to": "sec:inverse", "type": "external_reference", "evidence": "ted as part of the fixed arithmetic data. For the recursive Poisson large-sieve framework, compare \\cite[Section~2]{HBQuadratic}, \\cite[Section~4]{HB}, and \\cite[Sections~4--7]{GL}; the present marked recursion is proved below. [Canonical marked estimate] Fix bounded nonnegative ranges for \\(M,N,V,z_0\\), a", "anchor": { "source": "paper.tex", "line_start": 9293, "line_end": 9293, "char_start": 425369, "char_end": 425389 }, "origin": "source_reference", "locator": "Section~4", "context": "narrative" }, { "from": "external:GL", "to": "sec:inverse", "type": "external_reference", "evidence": "rithmetic data. For the recursive Poisson large-sieve framework, compare \\cite[Section~2]{HBQuadratic}, \\cite[Section~4]{HB}, and \\cite[Sections~4--7]{GL}; the present marked recursion is proved below. [Canonical marked estimate] Fix bounded nonnegative ranges for \\(M,N,V,z_0\\), a bound for the number of slots,", "anchor": { "source": "paper.tex", "line_start": 9294, "line_end": 9294, "char_start": 425395, "char_end": 425419 }, "origin": "source_reference", "locator": "Sections~4--7", "context": "narrative" }, { "from": "external:TZ", "to": "sec:plain", "type": "external_reference", "evidence": "},\\ p\\notin\\mathcal S\\}. \\] The prime ideal theorem in the fixed field, equivalently its fixed ray-class form \\cite[Theorem 1.1]{TZ}, gives \\(|\\mathcal P|=Z^{\\ell_*+o(1)}\\). Because the live slot lengths are at most \\(\\eta\\) and their relative annular supports are fixed, the single inequality \\(Z^{\\ell_*-\\eta}>2\\max_i b_i\\) makes", "anchor": { "source": "paper.tex", "line_start": 13464, "line_end": 13464, "char_start": 624839, "char_end": 624861 }, "origin": "source_reference", "locator": "Theorem 1.1", "context": "narrative" }, { "from": "prop:parameter-order", "to": "thm:main", "type": "proof_dependency", "evidence": "Final high bound and principal contraction verify the continuation criterion at 7/8. Source: paper.tex lines 16455–16459.", "anchor": { "source": "paper.tex", "line_start": 16455, "line_end": 16459 }, "origin": "source_audit_annotation", "verification_status": "source_grounded_stated_dependency; mathematical_input_not_independently_certified", "supporting_source_refs": { "native_labels": [ "prop:parameter-order", "eq:shared-principal-nonvanishing", "lem:continuation-criterion" ], "bibliography_keys": [] }, "evidence_scope": "explicit_source_application", "supporting_quote": "verify the hypotheses of" }, { "from": "lem:continuation-criterion", "to": "thm:main", "type": "proof_dependency", "evidence": "Continuation contradicts beta*>7/8. 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Source: paper.tex lines 3701–3719.", "anchor": { "source": "paper.tex", "line_start": 3701, "line_end": 3719 }, "origin": "source_audit_annotation", "verification_status": "source_grounded_stated_dependency; mathematical_input_not_independently_certified", "supporting_source_refs": { "native_labels": [ "eq:low-separated", "lem:balanced-additive-norm" ], "bibliography_keys": [] }, "evidence_scope": "explicit_source_application" }, { "from": "lem:sextic-large-sieve", "to": "prop:sextic-row-count", "type": "proof_dependency", "evidence": "Apply the sieve to a fixed squarefree residual row-factor. 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Source: paper.tex lines 13039–13052.", "anchor": { "source": "paper.tex", "line_start": 13039, "line_end": 13052 }, "origin": "source_audit_annotation", "verification_status": "source_grounded_stated_dependency; mathematical_input_not_independently_certified", "supporting_source_refs": { "native_labels": [ "lem:centered-lattice-cancellation" ], "bibliography_keys": [] }, "evidence_scope": "explicit_source_application" }, { "from": "lem:logarithmic-control", "to": "lem:plain", "type": "proof_dependency", "evidence": "Obtain prime bounds above the dynamic zero supremum. Source: paper.tex lines 12831–12845.", "anchor": { "source": "paper.tex", "line_start": 12831, "line_end": 12845 }, "origin": "source_audit_annotation", "verification_status": "source_grounded_stated_dependency; mathematical_input_not_independently_certified", "supporting_source_refs": { "native_labels": [ "lem:logarithmic-control" ], "bibliography_keys": [] }, "evidence_scope": "explicit_source_application" }, { "from": "lem:marked", "to": "prop:detector-counts", "type": "proof_dependency", "evidence": "Inversewitness selected prime capacity is read from its two inequalities. Source: paper.tex lines 15224–15243.", "anchor": { "source": "paper.tex", "line_start": 15224, "line_end": 15243 }, "origin": "source_audit_annotation", "verification_status": "source_grounded_stated_dependency; mathematical_input_not_independently_certified", "supporting_source_refs": { "native_labels": [ "prop:detector-witness", "lem:marked", "lem:plain" ], "bibliography_keys": [] }, "evidence_scope": "explicit_source_application" }, { "from": "lem:plain", "to": "prop:detector-counts", "type": "proof_dependency", "evidence": "Plainwitness selected prime capacity uses its affine inequality and zero-slot endpoint. Source: paper.tex lines 15224–15243.", "anchor": { "source": "paper.tex", "line_start": 15224, "line_end": 15243 }, "origin": "source_audit_annotation", "verification_status": "source_grounded_stated_dependency; mathematical_input_not_independently_certified", "supporting_source_refs": { "native_labels": [ "prop:detector-witness", "lem:marked", "lem:plain" ], "bibliography_keys": [] }, "evidence_scope": "explicit_source_application" }, { "from": "lem:inverse-amplification", "to": "prop:detector-counts", "type": "proof_dependency", "evidence": "Handle long inverse witnesses and the zero-capacity endpoint. Source: paper.tex lines 15245–15271.", "anchor": { "source": "paper.tex", "line_start": 15245, "line_end": 15271 }, "origin": "source_audit_annotation", "verification_status": "source_grounded_stated_dependency; mathematical_input_not_independently_certified", "supporting_source_refs": { "native_labels": [ "lem:inverse-amplification", "eq:amplified-note" ], "bibliography_keys": [] }, "evidence_scope": "explicit_source_application" }, { "from": "prop:detector-witness", "to": "prop:detector-counts", "type": "proof_dependency", "evidence": "Saturated pair supplies common r,m and rowwise witness lower bounds. Source: paper.tex lines 15224–15231.", "anchor": { "source": "paper.tex", "line_start": 15224, "line_end": 15231 }, "origin": "source_audit_annotation", "verification_status": "source_grounded_stated_dependency; mathematical_input_not_independently_certified", "supporting_source_refs": { "native_labels": [ "prop:detector-witness" ], "bibliography_keys": [] }, "evidence_scope": "explicit_source_application" }, { "from": "prop:probe-errors", "to": "lem:high-bin", "type": "proof_dependency", "evidence": "Decompose the local correction only on retained contours. Source: paper.tex lines 15770–15792.", "anchor": { "source": "paper.tex", "line_start": 15770, "line_end": 15792 }, "origin": "source_audit_annotation", "verification_status": "source_grounded_stated_dependency; mathematical_input_not_independently_certified", "supporting_source_refs": { "native_labels": [ "prop:probe-errors", "eq:dynamic-compensated-decomposition", "lem:deleted-euler-factors", "old-eq:5.13e", "old-eq:5.13d" ], "bibliography_keys": [] }, "evidence_scope": "explicit_source_application" }, { "from": "lem:bin-contour-accounting", "to": "lem:high-bin", "type": "proof_dependency", "evidence": "Provide fixed-bin contour move and complete exponent account. Source: paper.tex lines 15759–15773.", "anchor": { "source": "paper.tex", "line_start": 15759, "line_end": 15773 }, "origin": "source_audit_annotation", "verification_status": "source_grounded_stated_dependency; mathematical_input_not_independently_certified", "supporting_source_refs": { "native_labels": [ "lem:bin-contour-accounting", "eq:part-II-bin-ceiling", "sec:selection", "eq:stage-height-allocation", "prop:probe-errors", "eq:dynamic-compensated-decomposition" ], "bibliography_keys": [] }, "evidence_scope": "explicit_source_application" }, { "from": "prop:detector-counts", "to": "lem:balanced-endpoint", "type": "proof_dependency", "evidence": "Its two affine row counts are balanced at the selected cutoff. Source: paper.tex lines 15949–15992.", "anchor": { "source": "paper.tex", "line_start": 15949, "line_end": 15992 }, "origin": "source_audit_annotation", "verification_status": "source_grounded_stated_dependency; mathematical_input_not_independently_certified", "supporting_source_refs": { "native_labels": [ "prop:detector-counts", "eq:plain-zero-loss-capacity", "eq:capacity-comparison", "eq:no-slot-inverse-count" ], "bibliography_keys": [] }, "evidence_scope": "explicit_source_application" }, { "from": "lem:high-bin", "to": "prop:parameter-order", "type": "proof_dependency", "evidence": "Combine retained bins with outer ranges and principal remainders. Source: paper.tex lines 16432–16451.", "anchor": { "source": "paper.tex", "line_start": 16432, "line_end": 16451 }, "origin": "source_audit_annotation", "verification_status": "source_grounded_stated_dependency; mathematical_input_not_independently_certified", "supporting_source_refs": { "native_labels": [ "lem:high-bin", "eq:late-height-hypothesis", "eq:part-II-normalized-low", "lem:late-height-closure" ], "bibliography_keys": [] }, "evidence_scope": "explicit_source_application" }, { "from": "lem:balanced-endpoint", "to": "prop:parameter-order", "type": "proof_dependency", "evidence": "Lemma20.2 certifies the ideal endpoint margin; the following exponent comparison adds the allocated losses and compares it with the rightmost-zero scale. The final proof uses this endpoint inequality to reserve common margins. Source: paper.tex lines15994–16108 and16340–16366.", "anchor": { "source": "paper.tex", "line_start": 16340, "line_end": 16366 }, "origin": "source_audit_annotation", "verification_status": "source_grounded_stated_numerical_dependency; full_analytic_input_not_certified", "supporting_source_refs": { "native_labels": [ "lem:high-bin", "lem:principal-residue-interface" ], "bibliography_keys": [] }, "evidence_scope": "source_grounded_numerical_chain", "supporting_anchors": [ { "source": "paper.tex", "line_start": 15994, "line_end": 16072 }, { "source": "paper.tex", "line_start": 16074, "line_end": 16108 }, { "source": "paper.tex", "line_start": 16340, "line_end": 16366 } ] }, { "from": "prop:probe-low", "to": "prop:parameter-order", "type": "proof_dependency", "evidence": "Proposition15.3 is first divided by the common positive normalizer to obtain the normalized low estimate, which the final order-of-choices proof applies with lossDelta/2. Source: paper.tex lines15633–15647 and16369–16372.", "anchor": { "source": "paper.tex", "line_start": 16369, "line_end": 16372 }, "origin": "source_audit_annotation", "verification_status": "source_grounded_stated_dependency; mathematical_input_not_independently_certified", "supporting_source_refs": { "native_labels": [ "eq:part-II-normalized-low", "prop:probe-low" ], "bibliography_keys": [] }, "evidence_scope": "explicit_two_step_source_application", "supporting_quote": "For the low estimate, apply", "supporting_anchors": [ { "source": "paper.tex", "line_start": 15633, "line_end": 15647 }, { "source": "paper.tex", "line_start": 16369, "line_end": 16372 } ] }, { "from": "lem:late-height-closure", "to": "prop:parameter-order", "type": "proof_dependency", "evidence": "Choose target-dependent tau then external N after retained orders, preserving common sigma. Source: paper.tex lines 16444–16452.", "anchor": { "source": "paper.tex", "line_start": 16444, "line_end": 16452 }, "origin": "source_audit_annotation", "verification_status": "source_grounded_stated_dependency; mathematical_input_not_independently_certified", "supporting_source_refs": { "native_labels": [ "eq:late-height-hypothesis", "eq:part-II-normalized-low", "lem:late-height-closure" ], "bibliography_keys": [] }, "evidence_scope": "explicit_source_application" }, { "from": "lem:principal-residue-interface", "to": "prop:parameter-order", "type": "proof_dependency", "evidence": "Principalremainder estimates contribute separately retained margins. Source: paper.tex lines 16357–16368.", "anchor": { "source": "paper.tex", "line_start": 16357, "line_end": 16368 }, "origin": "source_audit_annotation", "verification_status": "source_grounded_stated_dependency; mathematical_input_not_independently_certified", "supporting_source_refs": { "native_labels": [ "lem:principal-residue-interface" ], "bibliography_keys": [] }, "evidence_scope": "explicit_source_application" }, { "from": "external:BaierBansal", "to": "manuscript", "type": "background", "evidence": "Comparison with an existing planar/number-field Poisson proof; the paper gives its own Eisenstein-lattice bound. Source: paper.tex lines 146–150.", "anchor": { "source": "paper.tex", "line_start": 146, "line_end": 150 }, "origin": "source_audit_annotation", "verification_status": "source_attribution", "supporting_source_refs": { "native_labels": [], "bibliography_keys": [ "Huxley", "BaierBansal" ] }, "evidence_scope": "explicit_source_application" }, { "from": "external:BIMS", "to": "cor:nonresidues-square-roots", "type": "proof_dependency", "evidence": "Fixedhalf-plane implies polylogarithmic least quadratic nonresidue. Source: paper.tex lines 339–347.", "anchor": { "source": "paper.tex", "line_start": 339, "line_end": 347 }, "origin": "source_audit_annotation", "locator": "Conjecture6.3 and Theorem6.7", "verification_status": "source_grounded_stated_external_input; external_theorem_not_independently_certified", "supporting_source_refs": { "native_labels": [ "thm:main" ], "bibliography_keys": [ "BIMS" ] }, "evidence_scope": "explicit_source_application" }, { "from": "external:BCSS", "to": "manuscript", "type": "background", "evidence": "Definition/context of the quasi-Riemann hypothesis. Source: paper.tex lines 77–81.", "anchor": { "source": "paper.tex", "line_start": 77, "line_end": 81 }, "origin": "source_audit_annotation", "verification_status": "source_attribution", "supporting_source_refs": { "native_labels": [], "bibliography_keys": [ "BCSS" ] }, "evidence_scope": "explicit_source_application" }, { "from": "external:BGL", "to": "lem:sextic-large-sieve", "type": "proof_dependency", "evidence": "Higher-order norm-recursion interfaces checked in the paper for sextic symbols. Source: paper.tex lines 4727–4736.", "anchor": { "source": "paper.tex", "line_start": 4727, "line_end": 4736 }, "origin": "source_audit_annotation", "locator": "Theorem1.3 and Section3; Lemmas3.1–3.5", "verification_status": "source_grounded_stated_external_input; external_theorem_not_independently_certified", "supporting_source_refs": { "native_labels": [], "bibliography_keys": [ "BGL" ] }, "evidence_scope": "explicit_source_application" }, { "from": "external:BGL", "to": "lem:sextic-large-sieve", "type": "proof_dependency", "evidence": "gcd plus Poisson interface corresponding to the derived recursion. Source: paper.tex lines 4999–5009.", "anchor": { "source": "paper.tex", "line_start": 4999, "line_end": 5009 }, "origin": "source_audit_annotation", "locator": "Lemmas3.2–3.3", "verification_status": "source_grounded_stated_external_input; external_theorem_not_independently_certified", "supporting_source_refs": { "native_labels": [], "bibliography_keys": [ "BGL" ] }, "evidence_scope": "explicit_source_application" }, { "from": "external:BGL", "to": "lem:sextic-large-sieve", "type": "proof_dependency", "evidence": "Power decomposition and provisional-exponent improvement. Source: paper.tex lines 5064–5074.", "anchor": { "source": "paper.tex", "line_start": 5064, "line_end": 5074 }, "origin": "source_audit_annotation", "locator": "Lemmas3.4–3.5", "verification_status": "source_grounded_stated_external_input; external_theorem_not_independently_certified", "supporting_source_refs": { "native_labels": [ "eq:sextic-sieve-norm-monotone" ], "bibliography_keys": [ "BGL" ] }, "evidence_scope": "explicit_source_application" }, { "from": "external:Clay", "to": "manuscript", "type": "background", "evidence": "Critical-line Riemann-hypothesis context. Source: paper.tex lines 117–121.", "anchor": { "source": "paper.tex", "line_start": 117, "line_end": 121 }, "origin": "source_audit_annotation", "verification_status": "source_attribution", "supporting_source_refs": { "native_labels": [], "bibliography_keys": [ "Clay" ] }, "evidence_scope": "explicit_source_application" }, { "from": "external:DFDS", "to": "manuscript", "type": "background", "evidence": "Related cubic-family central-point nonvanishing result. Source: paper.tex lines 152–156.", "anchor": { "source": "paper.tex", "line_start": 152, "line_end": 156 }, "origin": "source_audit_annotation", "verification_status": "source_attribution", "supporting_source_refs": { "native_labels": [], "bibliography_keys": [ "DFDS", "MP" ] }, "evidence_scope": "explicit_source_application" }, { "from": "external:Dirichlet", "to": "manuscript", "type": "background", "evidence": "Historical introduction of character L-series. Source: paper.tex lines 85–89.", "anchor": { "source": "paper.tex", "line_start": 85, "line_end": 89 }, "origin": "source_audit_annotation", "verification_status": "source_attribution", "supporting_source_refs": { "native_labels": [], "bibliography_keys": [ "Dirichlet", "Riemann" ] }, "evidence_scope": "explicit_source_application" }, { "from": "external:DR", "to": "lem:fixed-gauss-phase", "type": "proof_dependency", "evidence": "Cubic reciprocity on primary coprime inputs. Source: paper.tex lines 994–1003.", "anchor": { "source": "paper.tex", "line_start": 994, "line_end": 1003 }, "origin": "source_audit_annotation", "locator": "Equation1.4", "verification_status": "source_grounded_stated_external_input; external_theorem_not_independently_certified", "supporting_source_refs": { "native_labels": [], "bibliography_keys": [ "DR" ] }, "evidence_scope": "explicit_source_application" }, { "from": "external:DR", "to": "prop:completed-reflection", "type": "proof_dependency", "evidence": "Cubic supplementarylaw makes the fixed mask periodic modulo9. Source: paper.tex lines 2001–2010.", "anchor": { "source": "paper.tex", "line_start": 2001, "line_end": 2010 }, "origin": "source_audit_annotation", "locator": "Equation1.5", "verification_status": "source_grounded_stated_external_input; external_theorem_not_independently_certified", "supporting_source_refs": { "native_labels": [], "bibliography_keys": [ "DR" ] }, "evidence_scope": "explicit_source_application" }, { "from": "external:DR", "to": "prop:completed-reflection", "type": "proof_dependency", "evidence": "Thetaautomorphy and three cuspfamilies with explicit whole-index coefficient support. Source: paper.tex lines 1888–2010.", "anchor": { "source": "paper.tex", "line_start": 1888, "line_end": 2010 }, "origin": "source_audit_annotation", "locator": "Equations5.4,5.7,5.9,5.12–5.17 and AppendixA rows1,19,10", "verification_status": "source_grounded_stated_external_input; external_theorem_not_independently_certified", "supporting_source_refs": { "native_labels": [ "old-eq:4.3" ], "bibliography_keys": [ "DR" ] }, "evidence_scope": "explicit_source_application" }, { "from": "external:DR", "to": "prop:completed-reflection", "type": "proof_dependency", "evidence": "Translatedtheta identity under its unit-Fourier-support hypothesis. Source: paper.tex lines 2037–2049.", "anchor": { "source": "paper.tex", "line_start": 2037, "line_end": 2049 }, "origin": "source_audit_annotation", "locator": "Lemma5.2 and Corollary5.1", "verification_status": "source_grounded_stated_external_input; external_theorem_not_independently_certified", "supporting_source_refs": { "native_labels": [], "bibliography_keys": [ "DR" ] }, "evidence_scope": "explicit_source_application" }, { "from": "external:DR", "to": "prop:completed-reflection", "type": "proof_dependency", "evidence": "Besselintegral identity used for the direct/reflected Mellin calculations. Source: paper.tex lines 2286–2307.", "anchor": { "source": "paper.tex", "line_start": 2286, "line_end": 2307 }, "origin": "source_audit_annotation", "locator": "Lemma5.3", "verification_status": "source_grounded_stated_external_input; external_theorem_not_independently_certified", "supporting_source_refs": { "native_labels": [ "eq:reflection-infinity-coefficient", "eq:reflection-cubic-sextic-conversion" ], "bibliography_keys": [ "DR" ] }, "evidence_scope": "explicit_source_application" }, { "from": "external:DR", "to": "prop:completed-reflection", "type": "proof_dependency", "evidence": "Hyperbolic coordinate action used in the cusp chain rule. Source: paper.tex lines 2311–2330.", "anchor": { "source": "paper.tex", "line_start": 2311, "line_end": 2330 }, "origin": "source_audit_annotation", "locator": "Equation5.1", "verification_status": "source_grounded_stated_external_input; external_theorem_not_independently_certified", "supporting_source_refs": { "native_labels": [], "bibliography_keys": [ "DR" ] }, "evidence_scope": "explicit_source_application" }, { "from": "external:FG", "to": "manuscript", "type": "background", "evidence": "Uniform Dirichlet quasi-Riemann formulation. Source: paper.tex lines 80–84.", "anchor": { "source": "paper.tex", "line_start": 80, "line_end": 84 }, "origin": "source_audit_annotation", "verification_status": "source_attribution", "supporting_source_refs": { "native_labels": [], "bibliography_keys": [ "FG" ] }, "evidence_scope": "explicit_source_application" }, { "from": "external:Galbraith", "to": "cor:nonresidues-square-roots", "type": "proof_dependency", "evidence": "Tonelli–Shanks once a deterministic nonresidue is found. Source: paper.tex lines 349–360.", "anchor": { "source": "paper.tex", "line_start": 349, "line_end": 360 }, "origin": "source_audit_annotation", "locator": "Section2.9 Algorithm3 and Lemma2.9.5", "verification_status": "source_grounded_stated_external_input; external_theorem_not_independently_certified", "supporting_source_refs": { "native_labels": [], "bibliography_keys": [ "Galbraith" ] }, "evidence_scope": "explicit_source_application" }, { "from": "external:GZ", "to": "lem:hecke-strip-growth", "type": "proof_dependency", "evidence": "Functional equation plusgammafactor conductor scaling for finite-order Hecke L-functions. Source: paper.tex lines 1436–1458.", "anchor": { "source": "paper.tex", "line_start": 1436, "line_end": 1458 }, "origin": "source_audit_annotation", "locator": "Equation1.1", "verification_status": "source_grounded_stated_external_input; external_theorem_not_independently_certified", "supporting_source_refs": { "native_labels": [], "bibliography_keys": [ "GZ" ] }, "evidence_scope": "explicit_source_application" }, { "from": "external:GZ", "to": "prop:probe-errors", "type": "proof_dependency", "evidence": "Reflect a local nonprincipal numerator while allocating selected conductor deficits. Source: paper.tex lines 9007–9023.", "anchor": { "source": "paper.tex", "line_start": 9007, "line_end": 9023 }, "origin": "source_audit_annotation", "locator": "Equation1.1", "verification_status": "source_grounded_stated_external_input; external_theorem_not_independently_certified", "supporting_source_refs": { "native_labels": [], "bibliography_keys": [ "GZ" ] }, "evidence_scope": "explicit_source_application" }, { "from": "external:GZSextic", "to": "manuscript", "type": "background", "evidence": "Related recorded Eisenstein sextic sieve form, only cited for comparison. Source: paper.tex lines 138–142.", "anchor": { "source": "paper.tex", "line_start": 138, "line_end": 142 }, "origin": "source_audit_annotation", "verification_status": "source_attribution", "supporting_source_refs": { "native_labels": [], "bibliography_keys": [ "GZSextic", "HBQuadratic" ] }, "evidence_scope": "explicit_source_application" }, { "from": "external:GL", "to": "lem:completed-quadratic-reduction", "type": "proof_dependency", "evidence": "Quadratic residue-family norm bound(U+V)(UV)^epsilon; paper checks family hypotheses. Source: paper.tex lines 2638–2671.", "anchor": { "source": "paper.tex", "line_start": 2638, "line_end": 2671 }, "origin": "source_audit_annotation", "locator": "Definition1 and Theorem1.1", "verification_status": "source_grounded_stated_external_input; external_theorem_not_independently_certified", "supporting_source_refs": { "native_labels": [ "eq:quadratic-large-sieve", "eq:reciprocity-four-class" ], "bibliography_keys": [ "GL" ] }, "evidence_scope": "explicit_source_application" }, { "from": "external:GM", "to": "manuscript", "type": "background", "evidence": "Zero-density contrast and a comparison for classical zero detection; no imported density bound used to exclude all zeros. Source: paper.tex lines 101–105.", "anchor": { "source": "paper.tex", "line_start": 101, "line_end": 105 }, "origin": "source_audit_annotation", "verification_status": "source_attribution", "supporting_source_refs": { "native_labels": [], "bibliography_keys": [ "GM" ] }, "evidence_scope": "explicit_source_application" }, { "from": "external:Hadamard", "to": "manuscript", "type": "background", "evidence": "Historical prime-number theorem/nonvanishing on Re(s)=1. Source: paper.tex lines 90–94.", "anchor": { "source": "paper.tex", "line_start": 90, "line_end": 94 }, "origin": "source_audit_annotation", "verification_status": "source_attribution", "supporting_source_refs": { "native_labels": [], "bibliography_keys": [ "Hadamard", "ValleePoussin", "Hecke" ] }, "evidence_scope": "explicit_source_application" }, { "from": "external:HBQuadratic", "to": "manuscript", "type": "background", "evidence": "Recursive Poisson moment framework; the marked recursion is proved internally. Source: paper.tex lines 140–144.", "anchor": { "source": "paper.tex", "line_start": 140, "line_end": 144 }, "origin": "source_audit_annotation", "verification_status": "source_attribution", "supporting_source_refs": { "native_labels": [], "bibliography_keys": [ "GZSextic", "HBQuadratic", "GZ", "TZ" ] }, "evidence_scope": "explicit_source_application" }, { "from": "external:HB", "to": "lem:hybrid-energy", "type": "proof_dependency", "evidence": "Cubic squarefree large-sieve norm U+V+(UV)^(2/3), arbitrary complex coefficients. Source: paper.tex lines 7485–7508.", "anchor": { "source": "paper.tex", "line_start": 7485, "line_end": 7508 }, "origin": "source_audit_annotation", "locator": "Theorem2", "verification_status": "source_grounded_stated_external_input; external_theorem_not_independently_certified", "supporting_source_refs": { "native_labels": [ "lem:completed-quadratic-reduction", "eq:quadratic-large-sieve" ], "bibliography_keys": [ "HB" ] }, "evidence_scope": "explicit_source_application" }, { "from": "external:Hecke", "to": "manuscript", "type": "background", "evidence": "Historical character zeta analytic theory. Source: paper.tex lines 92–96.", "anchor": { "source": "paper.tex", "line_start": 92, "line_end": 96 }, "origin": "source_audit_annotation", "verification_status": "source_attribution", "supporting_source_refs": { "native_labels": [], "bibliography_keys": [ "Hadamard", "ValleePoussin", "Hecke" ] }, "evidence_scope": "explicit_source_application" }, { "from": "external:Huxley", "to": "manuscript", "type": "background", "evidence": "Comparison to classical planar/number-field large sieve; direct proof supplied. Source: paper.tex lines 145–149.", "anchor": { "source": "paper.tex", "line_start": 145, "line_end": 149 }, "origin": "source_audit_annotation", "verification_status": "source_attribution", "supporting_source_refs": { "native_labels": [], "bibliography_keys": [ "Huxley", "BaierBansal" ] }, "evidence_scope": "explicit_source_application" }, { "from": "external:Kubota", "to": "manuscript", "type": "background", "evidence": "Metaplectic setting. Source: paper.tex lines 120–124.", "anchor": { "source": "paper.tex", "line_start": 120, "line_end": 124 }, "origin": "source_audit_annotation", "verification_status": "source_attribution", "supporting_source_refs": { "native_labels": [], "bibliography_keys": [ "Kubota", "Patterson", "DR" ] }, "evidence_scope": "explicit_source_application" }, { "from": "external:MP", "to": "manuscript", "type": "background", "evidence": "Comparison for the truncated-inverse zero-detector mechanism. Source: paper.tex lines 154–158.", "anchor": { "source": "paper.tex", "line_start": 154, "line_end": 158 }, "origin": "source_audit_annotation", "verification_status": "source_attribution", "supporting_source_refs": { "native_labels": [], "bibliography_keys": [ "DFDS", "MP", "BGL" ] }, "evidence_scope": "explicit_source_application" }, { "from": "external:MilneCFT", "to": "thm:main", "type": "proof_dependency", "evidence": "Finite-order idelicHecke characters have no nontrivial continuous complex-place part. Source: paper.tex lines 50–58.", "anchor": { "source": "paper.tex", "line_start": 50, "line_end": 58 }, "origin": "source_audit_annotation", "locator": "ChapterVI Section1", "verification_status": "source_grounded_stated_external_input; external_theorem_not_independently_certified", "supporting_source_refs": { "native_labels": [], "bibliography_keys": [ "MilneCFT" ] }, "evidence_scope": "explicit_source_application" }, { "from": "external:MilneCFT", "to": "lem:fixed-numerator-ray", "type": "proof_dependency", "evidence": "Kummerdescription for fixed-numerator residuecharacters. Source: paper.tex lines 676–699.", "anchor": { "source": "paper.tex", "line_start": 676, "line_end": 699 }, "origin": "source_audit_annotation", "locator": "ChapterVIII (5.3) and(5.5)", "verification_status": "source_grounded_stated_external_input; external_theorem_not_independently_certified", "supporting_source_refs": { "native_labels": [], "bibliography_keys": [ "MilneCFT" ] }, "evidence_scope": "explicit_source_application" }, { "from": "external:OpenAICubicFirstMoment", "to": "manuscript", "type": "excluded_input", "evidence": "The introduction explicitly says this independent first-moment theorem is not used. Source: paper.tex lines124–128.", "anchor": { "source": "paper.tex", "line_start": 124, "line_end": 128 }, "origin": "source_audit_annotation", "locator": "Theorem1.1", "verification_status": "explicit_source_exclusion", "supporting_source_refs": { "native_labels": [ "prop:completed-reflection" ], "bibliography_keys": [ "DR", "OpenAICubicFirstMoment" ] }, "evidence_scope": "explicit_source_application", "supporting_quote": "that result is also not used" }, { "from": "external:Patterson", "to": "manuscript", "type": "background", "evidence": "Original theta/coefficient calculations underlying DR; the paper says its formal input remains DR. Source: paper.tex lines 120–124.", "anchor": { "source": "paper.tex", "line_start": 120, "line_end": 124 }, "origin": "source_audit_annotation", "verification_status": "source_attribution", "supporting_source_refs": { "native_labels": [], "bibliography_keys": [ "Kubota", "Patterson", "DR" ] }, "evidence_scope": "explicit_source_application" }, { "from": "external:Riemann", "to": "manuscript", "type": "background", "evidence": "Historical connection of zeta zeros and primes. Source: paper.tex lines 87–91.", "anchor": { "source": "paper.tex", "line_start": 87, "line_end": 91 }, "origin": "source_audit_annotation", "verification_status": "source_attribution", "supporting_source_refs": { "native_labels": [], "bibliography_keys": [ "Dirichlet", "Riemann" ] }, "evidence_scope": "explicit_source_application" }, { "from": "external:TaoNonresidues", "to": "manuscript", "type": "background", "evidence": "Vinogradov least nonresidue conjecture contextual citation. Source: paper.tex lines 331–335.", "anchor": { "source": "paper.tex", "line_start": 331, "line_end": 335 }, "origin": "source_audit_annotation", "verification_status": "source_attribution", "supporting_source_refs": { "native_labels": [], "bibliography_keys": [ "TaoNonresidues" ] }, "evidence_scope": "explicit_source_application" }, { "from": "external:TZ", "to": "lem:principal-residue-interface", "type": "proof_dependency", "evidence": "Chebotarev detects a nontrivial sexticFrobenius class, separating the sole principal numerator row. Source: paper.tex lines 5500–5516.", "anchor": { "source": "paper.tex", "line_start": 5500, "line_end": 5516 }, "origin": "source_audit_annotation", "locator": "Theorem1.1", "verification_status": "source_grounded_stated_external_input; external_theorem_not_independently_certified", "supporting_source_refs": { "native_labels": [ "lem:fixed-numerator-ray" ], "bibliography_keys": [ "TZ" ] }, "evidence_scope": "explicit_source_application" }, { "from": "external:TZ", "to": "lem:ray-prime-normalizer", "type": "proof_dependency", "evidence": "Fixed-ray prime normalizer asymptotic, with fixed conductor before P→infinity. Source: paper.tex lines 7019–7040.", "anchor": { "source": "paper.tex", "line_start": 7019, "line_end": 7040 }, "origin": "source_audit_annotation", "locator": "Theorem1.1", "verification_status": "source_grounded_stated_external_input; external_theorem_not_independently_certified", "supporting_source_refs": { "native_labels": [ "eq:ray-prime-normalizer" ], "bibliography_keys": [ "TZ" ] }, "evidence_scope": "explicit_source_application" }, { "from": "external:TZ", "to": "lem:plain", "type": "proof_dependency", "evidence": "Fixedfield prime pool of Z^(ell*+o(1)) size, disjoint from live short slots. Source: paper.tex lines 13450–13475.", "anchor": { "source": "paper.tex", "line_start": 13450, "line_end": 13475 }, "origin": "source_audit_annotation", "locator": "Theorem1.1", "verification_status": "source_grounded_stated_external_input; external_theorem_not_independently_certified", "supporting_source_refs": { "native_labels": [], "bibliography_keys": [ "TZ" ] }, "evidence_scope": "explicit_source_application" }, { "from": "external:ValleePoussin", "to": "manuscript", "type": "background", "evidence": "Historical prime-number theorem/nonvanishing on Re(s)=1. Source: paper.tex lines 90–94.", "anchor": { "source": "paper.tex", "line_start": 90, "line_end": 94 }, "origin": "source_audit_annotation", "verification_status": "source_attribution", "supporting_source_refs": { "native_labels": [], "bibliography_keys": [ "Hadamard", "ValleePoussin", "Hecke" ] }, "evidence_scope": "explicit_source_application" }, { "from": "external:DR", "to": "manuscript", "type": "excluded_input", "evidence": "The introduction explicitly excludes the GRH-conditional prime asymptotic while importing only unconditional cusp formulas. Source: paper.tex lines120–128. DRnative source lines213–221 show the GRH hypothesis of its main prime asymptotic.", "anchor": { "source": "paper.tex", "line_start": 120, "line_end": 128 }, "origin": "source_audit_annotation", "locator": "GRH-conditional prime asymptotic, not the unconditional cusp formulas", "verification_status": "explicit_source_exclusion", "supporting_source_refs": { "native_labels": [ "prop:completed-reflection" ], "bibliography_keys": [ "Kubota", "Patterson", "DR", "OpenAICubicFirstMoment" ] }, "evidence_scope": "explicit_source_application", "supporting_quote": "prime asymptotic is not an input" } ], "coverage": { "mode": "native_latex_index", "claims": 67, "sections": 20, "external_references": 28, "source_bibliography_entries": 28, "unresolved_bibliography": 0, "unresolved": 0 }, "limitations": [ "Reference edges are not certified proof dependencies. Edge direction is prerequisite → consumer.", "Numbers reflect simple source counters; labels and line anchors are authoritative.", "Extractive statements and proof openings are marked; they are not reconstructed proof gists.", "Macros, included files and custom counters require an expanded source or annotated fallback." ] }