{
  "packet_id": "reflection_external",
  "scope_summary": "Audited Proposition 5.1 (completed cubic reflection) in the pinned paper.tex span 1648-2463 against the \u00a74.1-4.2 conventions and the included Dunn-Radziwill (DR) spans: finite Fourier inversion with exponent-zero masks (C_{p,j}), the L-periodic mask phi and its Fourier normalization, the whole-index mask on nb^3 with shared primes between n and b (and with moving primes p in P), the identification of the direct Mellin integral with T=L_0 D (constants i/4, (2pi)^{-2s}, 27^s, 3^{5/2}, alpha-bar(lambda^{-3})=-i), the theta conjugation convention (coefficients overline{d_j(-mu)}, constant term sigma v^{2/3} killed by d/d z-bar), the g^{-1} coordinate action and chain rule, the three cusp-matrix cases and kappa(G), the CRT additive phase and epsilon_p, the local Gauss/Ramanujan sums producing omega_{p,j} B_p, the scalar -i/81 and K, the gamma quotient R(t), branch compatibility for inactive exponent-zero primes, and consistency with DR Lemma 5.2 / Appendix table (k_2(E)=19, k_3(E)=10). Not reconstructed: Patterson's coefficient formulas, Patterson's extension of the Kubota symbol to Gamma_2, equation (pointwise-mellin-tail), and later sections.",
  "audited_steps": [
    {
      "anchor": "paper.tex eq:reflection-local-fourier (C_{p,j})",
      "result": "checked",
      "evidence": "chi_p(x)^0=1_{p\u2224x}: sum over units of e(-hx/p) is -1 (h\u22600) and q_p-1 (h=0), giving C_{p,0}=-q_p^{-1}, 1-q_p^{-1}. For j\u22600, x=h^{-1}y gives q_p^{-1/2}tau^-_{p,j}chi_p(h)^{-j}; h=0 gives 0 by orthogonality. Pairing e(hx/p) nondegenerate since the e-dual of O is O (self-duality shown in \u00a74.1)."
    },
    {
      "anchor": "paper.tex phi, hat-phi definition and L-periodicity",
      "result": "checked",
      "evidence": "Inversion phi(x)=sum_{h0} hatphi(h0) e(h0 x/L) with hatphi=q_L^{-1} sum phi(x)e(-h0x/L); valid since O/(L) has q_L elements and characters x\u21a6e(hx/L) are distinct for distinct h mod L. Periodicity: (lambda/x)_3 periodic mod 9 (DR supprecip), Psi_0 periodic mod its conductor on E-units, zero branch periodic mod E, and 9|18|L, E|L."
    },
    {
      "anchor": "paper.tex eq:reflection-translated-theta and whole-index mask",
      "result": "checked",
      "evidence": "check-e(mu z_h)=e(lambda^3 mu (h0/L+sum h_p/p))=e(x(...)) with x=lambda^3 mu\u2208O, so sum_h C(h) check-e(mu z_h)=phi(x)prod chi_p(x)^{j_p}. On x=nb^3, (nb^3,E)=1 iff (n,E)=(b,E)=1; chi_{nb^3}(lambda)^2=chi_n(lambda)^2 chi_b(lambda)^6=chi_n(lambda)^2; chi_p(nb^3)^{j}=chi_p(n)^j chi_p(b)^{3j} with both sides zero iff p|nb, including j=0. Representation x=nb^3 (n sf, both primary) unique by v_p(x) mod 3."
    },
    {
      "anchor": "paper.tex eq:reflection-infinity-coefficient / eq:reflection-cubic-sextic-conversion",
      "result": "checked (conditional on DR cubiccoeff)",
      "evidence": "DR: overline{tau(lambda^{-3}cd^3)}=g(1,c)|d/c|3^{5/2}=3^{5/2}|d| g~(c) (consistent with DR Remark FSremark). gamma_2(n)=q_n^{-1/2}sum chi_n(v)^2 check-e(v/(lambda n)); v=lambda y gives gamma_2(n)=(lambda/n)_3 g~_3(n). Also chi_p^2\u2261x^{(P-1)/3}=(x/p)_3 matches DR definition."
    },
    {
      "anchor": "paper.tex eq:reflection-direct-mellin",
      "result": "checked",
      "evidence": "d/dz-bar check-e(mu z)=2 pi i mu-bar check-e; int v^{2s}K_{1/3}(4pi|mu|v)dv=(4pi|mu|)^{-2s-1}2^{2s-1}Gamma(s+1/3)Gamma(s+2/3); product gives (i/4)(2pi)^{-2s} alpha-bar(mu) q_mu^{-s}. With q_{lambda^{-3}}=1/27 and alpha-bar(lambda^{-3})=-i (lambda=i\u221a3), (i/4)(-i)3^{5/2}27^s(2pi)^{-2s}=(3^{5/2}/4)(27/(2pi)^2)^s and |b| q_b^{-3s}=q_b^{-3s+1/2} reproduces L_0 D."
    },
    {
      "anchor": "paper.tex g^{-1} coordinate action and chain rule",
      "result": "checked",
      "evidence": "From DR grpaction with gamma=g^{-1}=((delta',-b),(-c,a)): c'(z+a/c)+d'=-cz, numerator -delta' c-bar(|z|^2+v^2)-c-bar z-bar/c, giving (-delta'/c - z-bar/(c^2(v^2+|z|^2)), v/(q_c(v^2+|z|^2))). At z=0: d/dz-bar z'=-(cv)^{-2}, d/dz-bar z'-bar=0, d/dz-bar v'=0. Constant term C_H(v') removed by d/dz'."
    },
    {
      "anchor": "paper.tex eq:reflection-reflected-mellin scalar",
      "result": "checked",
      "evidence": "-2pi i c^{-2} q_c^{2-2s} mu * (4pi|mu|)^{2s-3}2^{1-2s}Gamma(4/3-s)Gamma(5/3-s) = -(i/4)(2pi)^{2s-2} alpha-bar(c)^2 q_c^{1-2s} alpha(mu) q_mu^{s-1}; v=1/(q_c u) substitution gives q_c^{3-2s}\u00b7q_c^{-1}=q_c^{2-2s}; c^{-2}=alpha-bar(c)^2/q_c."
    },
    {
      "anchor": "paper.tex ratio -i/81, K, R(t)",
      "result": "checked",
      "evidence": "(2pi)^{4s-2}27^{-s}3^{-5/2} at s=1/2-t equals ((2pi)^4/27)^{-t} 3^{-4}; q_c^{1-2s}q_mu^{s-1}=q_mu^{-1/2}(q_mu/q_c^2)^{-t}; Gamma(3/2-s\u00b11/6)/Gamma(s+1/3)Gamma(s+2/3)=prod Gamma(1+t\u00b11/6)/Gamma(1-t\u00b11/6)=R(t); ds=-dt with orientation reversal gives no sign. Stirling: |R(sigma+iu)|\u224d|u|^{(2sigma+1)-(1-2sigma)}=|u|^{4sigma}; first numerator pole at t=-5/6."
    },
    {
      "anchor": "paper.tex cusp-matrix cases eq:reflection-cusp-case-zero / middle / third",
      "result": "checked",
      "evidence": "Case 3|c: G=g\u2208Gamma_1(3) (a\u22611, delta'\u2261a^{-1}\u22611 mod 3, b\u22610 mod lambda^{12}), kappa=(c_F r/a)_3=(c_F/a)_3(a/r)_3 by reciprocity. Case v_lambda(c)=1: G=gL(-u_0)=((a-u_0 b, b),(c-u_0 delta', delta'))\u2208Gamma_1(3); a(c-u_0 delta')=cA-u_0 verified; (A/a)_3=(-u_0 b/a)_3=(u_0/a)_3(c/a)_3^{-1}. Case (lambda,c)=1: G=((-b, a+bu_0),(-delta', c+delta'u_0)), -bc=1-a delta'\u22611 mod 9, (delta'/-bc)_3=1 via supprecip+reciprocity, kappa=(delta'/c)_3^{-1}=(a/c)_3. All give kappa(G)=kappa_F prod_{p|r} chi_p(a)^2 with kappa_F fixed by h0 and r mod M^2."
    },
    {
      "anchor": "paper.tex eq:reflection-additive-phase and epsilon_p",
      "result": "checked",
      "evidence": "check-e(-delta' mu/c)=e(-lambda delta' x/(lambda^4 c))=e(-delta' x/(lambda^3 c_F r)). CRT: local numerator at p is -delta'(D_F r/p)^{-1} with delta'\u2261(sigma_p h_p)^{-1}, D_F r/p=lambda^3 c/p, giving -((lambda^3c/p)sigma_p)^{-1}h_p^{-1}=epsilon_p h_p^{-1}. a\u2261lambda^2 c_F (r/p) h_p=sigma_p h_p mod p verified from eq:reflection-cusp-fraction."
    },
    {
      "anchor": "paper.tex local sums producing omega_{p,j}B_p",
      "result": "checked",
      "evidence": "Conjugated multiplier gives chi_p(sigma_p)^{-2}chi_p(h)^{-2}. j\u22600: q_p^{-1/2}tau^-_{p,j} sum_{y}chi_p(y)^{j+2}e(epsilon_p x y/p)=q_p^{1/2}tau^+_{p,j+2}chi_p(epsilon_p x)^{-j-2} (zero on both sides if p|x, j\u22604); j=4: Ramanujan sum -1+q_p 1_{p|x}; active j=0: -q_p^{-1}\u00b7q_p^{1/2}tau^+_{p,2}chi_p(epsilon_p x)^{-2}; inactive j=0: C_{p,0}(0)=1-q_p^{-1} with no multiplier factor since p\u2224r. Cross-prime phase: epsilon_p=-lambda^{-5}(c/p)^{-2} gives exponent 2j+2 (j=4: -2+... \u2261-2 mod 6)."
    },
    {
      "anchor": "paper.tex branch compatibility statement (end of item 3)",
      "result": "checked",
      "evidence": "Adding p with j_p=0 inactive leaves h0, r, hence c_F, c, a, delta', H, kappa_F, vartheta, epsilon_q, sigma_q unchanged; only C_{p,0}(0)=1-q_p^{-1} multiplies C(h). Direct side changes by mask 1_{p\u2224nb}, matched by (inactive branch)+(active branch) decomposition of the full inversion."
    },
    {
      "anchor": "Consistency with DR Lemma functwisteq / Appendix table",
      "result": "checked",
      "evidence": "DR translation by -lambda^4 d/r carries multiplier (lambda^4 a/r)_3=((-numerator)^{-1}/r)_3=(numerator/r)_3^{-1}, matching paper's conj kappa factor chi_p(a)^{-2}; new argument lambda^4 a/r \u2261 -a_paper^{-1}/c matches -delta'/c. Paper's H=((u_0,-1),(1,0)) with u_0=omega equals gamma_2 E, and E L(-omega)L(-omega^2)=E L(1)\u2208SL_2(Z), consistent with k_2(E)=19, k_3(E)=10; DR autcor factor (r-bar/r)^{-ell} at ell=-1 matches alpha-bar(c)^2."
    },
    {
      "anchor": "paper.tex old-eq:4.3 support/size bound",
      "result": "checked (conditional on DR tables)",
      "evidence": "tau: 3^{n/2+2}|d| with k=3n-4 gives 3^{k/6+8/3}|d|; 3^{n/2+5/2}|d| with k=3n-3 gives 3^{k/6+3}|d|\u226427\u00b73^{k/6}|d|; tau_1,tau_2: 9|d| at k=-4 vs 27\u00b73^{-2/3}\u224812.98. Unimodularity of twisted g~_3 for squarefree primary n (incl. n=-2, residue field F_4) via y=tv."
    }
  ],
  "objections": [
    {
      "anchor": "paper.tex 'The fixed theta input' and eq:reflection-cusp-expansion; DR (chidef), (cubiccoeff), (cdcond)",
      "classification": "unverified_step",
      "claim": "The signs of all B_p exponents (-j_p-2), the direction chi_p(a)^{-2} in the conjugated multiplier, and the direct-side coefficient 3^{5/2}|b| g~_3(n) (as opposed to its conjugate) rest entirely on DR/Patterson conventions not derivable from the packet: theta(gamma w)=chi(gamma)theta(w) with chi=(c/a)_3 (not its conjugate), Patterson's homomorphic extension to Gamma_2 with chi|_{SL_2(Z)}=1, the meaning of g(1,c), and the absence of a coprimality condition on (c,d) in (cdcond).",
      "evidence": "Paper: 'kappa((a,b;c,delta'))=(c/a)_3' and 'd_I(lambda^{-3}nb^3)=3^{5/2}|b| g~_3(n)'; DR: 'overline{g(1,c)}|d/c|3^{n/2+5/2}' with c,d\u22611 mod 3, mu^2(c)=1 and no (c,d)=1.",
      "derivation": "If the multiplier were conjugated, every chi_p(h)^{-2} in the local sums would become chi_p(h)^{+2}, changing B_p to chi_p(x)^{-j_p+2} and shifting the Ramanujan branch from j=4 to j=2; the internal computation is consistent with DR Lemma 5.2 ((lambda^4 a/r)_3 multiplier) and DR Remark FSremark (coefficient \u221d g~(c)|d|), so no mismatch is exhibited, but these are external inputs.",
      "hypotheses_checked": [
        "paper's kappa matches DR (chidef)",
        "paper's d_I matches DR cubiccoeff via overline{tau(mu)}=g(1,c)|d/c|3^{5/2}",
        "DR Remark FSremark confirms coefficient \u221d g~(c)|d|"
      ],
      "possible_repair": "none needed if DR/Patterson conventions hold as quoted; verify Patterson Thm 8.1 and \u00a72 of Pat1 directly.",
      "dependency_impact": "Global phase of eq:reflection-row-phase and the exponents in item (2) B_p; all downstream uses of the column couplings 2j+2."
    },
    {
      "anchor": "paper.tex 'Shift the integral ... Equation (eq:pointwise-mellin-tail)'",
      "classification": "unverified_step",
      "claim": "The contour shift of the completed sum from Re s=a>1 to Re s=1/2-sigma<0 cites eq:pointwise-mellin-tail, which is not in the packet; the horizontal-join vanishing is therefore unverified here.",
      "evidence": "'Equation (pointwise-mellin-tail) applied to the weighted logarithmic derivatives of V makes the horizontal joins tend to zero.'",
      "derivation": "Given the hypotheses on V (all (x d/dx)^j V rapidly decaying at 0 and \u221e), hat V is entire and rapidly decreasing on vertical strips by repeated integration by parts in log x; combined with the proved polynomial strip bound for T this suffices, but the cited equation itself is unseen.",
      "hypotheses_checked": [
        "V smooth with (x\u2202_x)^j V=O(x^{\u00b1C})",
        "T polynomial growth on strips (maximum-principle argument in proof)"
      ],
      "possible_repair": "Standard; integrate by parts B+2 times as done later in the proof for eq:reflection-test-seminorm.",
      "dependency_impact": "Validity of eq:reflection for a>1 only; no effect on local factors."
    }
  ],
  "unverified_obligations": [
    {
      "anchor": "DR (chidef), Pat1 \u00a72",
      "obligation": "Confirm theta(gw)=chi(g)theta(w) on Gamma_2 with chi=(c/a)_3 (unconjugated) and chi homomorphic with chi|_{SL_2(Z)}=1; the paper's kappa(L(3w))=1 and kappa(E)=1 depend on this."
    },
    {
      "anchor": "DR (cubiccoeff),(cdcond), Pat1 Thm 8.1",
      "obligation": "Confirm tau(\u00b1lambda^{-3}cd^3)=overline{g(1,c)}|d/c|3^{5/2} with g(1,c)=sum (x/c)_3 check-e(x/c) and with no coprimality required between c and d (shared primes)."
    },
    {
      "anchor": "DR Appendix table rows 19,10; (cubiccoeff1),(cubiccoeff2)",
      "obligation": "Confirm d_19(mu)=omega^2 tau_1(omega^2 mu)check-e(mu), d_10(mu)=omega tau_2(omega mu)check-e(mu) and hence support in lambda^{-4}O and |overline{d_j(-mu)}|\u226427\u00b73^{k/6}|b|; also |g(omega^k lambda^2,c)|=|c| for squarefree primary c including c=-2."
    },
    {
      "anchor": "paper.tex eq:pointwise-mellin-tail (not in packet)",
      "obligation": "Verify the rapid vertical decay of hat V used for the contour shift and for the V^sharp bounds."
    },
    {
      "anchor": "DR (cuberecip),(supprecip)",
      "obligation": "Cubic reciprocity (a/b)_3=(b/a)_3 for primary a,b and supplementary laws mod 9, used in all three kappa(G) cases and in the L-periodicity of phi."
    },
    {
      "anchor": "paper.tex Prop 5.1 statement, Psi_0 'extended by zero away from the E-units'",
      "obligation": "Confirm that x\u21a6Psi_0((x)) on primary x coprime to E is periodic modulo the conductor (ray-class periodicity), as assumed for L-periodicity of phi."
    }
  ],
  "assessment": "no_demonstrated_error_found"
}