[ { "id": "quasi-riemann-source-0", "label_origin": "existing_source_audit_annotation_not_independent_gold", "reference_role": "proof_input", "source_lines": [ 16453, 16461 ], "edge": [ "prop:parameter-order", "thm:main" ], "state": { "consumer": { "id": "thm:main", "label": "Theorem 1.1", "title": "" }, "prerequisite": { "id": "prop:parameter-order", "label": "Proposition 20.3", "title": "Order of choices" }, "source_passage": "\\end{proof}\n\nProposition~\\ref{prop:parameter-order} and\nEquation~\\eqref{eq:shared-principal-nonvanishing} verify the hypotheses of\nProposition~\\ref{lem:continuation-criterion} with\n\\(\\sigma_0=7/8\\), \\(J_\\eta=J_{\\mathrm{II},\\eta}\\), and\n\\(f_\\eta=f_{\\mathrm{II},\\eta}\\). This contradicts \\(\\beta_*>7/8\\);\nhence \\(\\beta_*\\le7/8\\). Proposition~\\ref{prop:hecke-dirichlet-transfer}\nat \\(\\sigma_0=7/8\\) extends the primitive Hecke conclusion to all", "context_limit": "Only this source span is supplied; absent context may justify abstention." }, "paper_slug": "quasi-riemann", "source_path": "paper.tex", "source_sha256": "42a5ee0febca59fd1def55cfd6c6808c322ef4237d7726303ea52f711deac6a3" }, { "id": "quasi-riemann-source-1", "label_origin": "existing_source_audit_annotation_not_independent_gold", "reference_role": "background", "source_lines": [ 144, 152 ], "edge": [ "external:BaierBansal", "manuscript" ], "state": { "consumer": { "id": "manuscript", "label": "The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane Re(s)>7/8", "title": null }, "prerequisite": { "id": "external:BaierBansal", "label": "[1] BaierBansal", "title": "S. Baier and A. Bansal, The large sieve with power moduli for $\\mathbb Z[i]$, International Journal of Number Theory 14 (2018), no. 10, 2737--2756. \\href{https://doi.org/10.1142/S1793042118501658}{doi:10.1142/S1793042118501658}; \\href{https://arxiv.org/abs/1802.08964}{arXiv:1802.08964}." }, "source_passage": "fixed ray class \\cite[Theorem 1.1]{TZ}.\n\nThe planar additive large sieve used in the first stage is classical;\ncompare Huxley's multivariable and number-field inequality \\cite{Huxley}\nand the Poisson proof in \\cite[Section 3, Theorem 3]{BaierBansal}.\nWe include a direct Eisenstein-lattice proof to record the normalization\nneeded for the reduced-fraction expansion. David, de Faveri, Dunn, and\nStucky combine Patterson's coefficients, the cubic large sieve, and\nmollified moments to prove nonvanishing at $s=1/2$ for a positive", "context_limit": "Only this source span is supplied; absent context may justify abstention." }, "paper_slug": "quasi-riemann", "source_path": "paper.tex", "source_sha256": "42a5ee0febca59fd1def55cfd6c6808c322ef4237d7726303ea52f711deac6a3" }, { "id": "quasi-riemann-source-2", "label_origin": "existing_source_audit_annotation_not_independent_gold", "reference_role": "explicitly_excluded", "source_lines": [ 122, 130 ], "edge": [ "external:OpenAICubicFirstMoment", "manuscript" ], "state": { "consumer": { "id": "manuscript", "label": "The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane Re(s)>7/8", "title": null }, "prerequisite": { "id": "external:OpenAICubicFirstMoment", "label": "[23] OpenAICubicFirstMoment", "title": "The unconditional first-moment asymptotic in the independent OpenAI manuscript, not the unconditional cusp expansions of Dunn and Radziwill" }, "source_passage": "the automorphic setting \\cite{Kubota,Patterson}. We use the unconditional\nexplicit cusp expansions recorded by Dunn and Radziwi\\l\\l{}\n\\cite[Section 5 and Appendix A]{DR}; their GRH-conditional prime asymptotic\nis not an input. The corresponding first-moment asymptotic is proved\nunconditionally in an independent manuscript\n\\cite[Theorem~1.1]{OpenAICubicFirstMoment}; that result is also not used\nhere. Proposition~\\ref{prop:completed-reflection} derives the\nreflection used here while retaining the characters' zero-on-nonunit\nrestrictions.", "context_limit": "Only this source span is supplied; absent context may justify abstention." }, "paper_slug": "quasi-riemann", "source_path": "paper.tex", "source_sha256": "42a5ee0febca59fd1def55cfd6c6808c322ef4237d7726303ea52f711deac6a3" }, { "id": "quasi-riemann-source-3", "label_origin": "existing_source_audit_annotation_not_independent_gold", "reference_role": "proof_input", "source_lines": [ 16453, 16462 ], "edge": [ "lem:continuation-criterion", "thm:main" ], "state": { "consumer": { "id": "thm:main", "label": "Theorem 1.1", "title": "" }, "prerequisite": { "id": "lem:continuation-criterion", "label": "Proposition 2.1", "title": "Continuation from a common signal" }, "source_passage": "\\end{proof}\n\nProposition~\\ref{prop:parameter-order} and\nEquation~\\eqref{eq:shared-principal-nonvanishing} verify the hypotheses of\nProposition~\\ref{lem:continuation-criterion} with\n\\(\\sigma_0=7/8\\), \\(J_\\eta=J_{\\mathrm{II},\\eta}\\), and\n\\(f_\\eta=f_{\\mathrm{II},\\eta}\\). This contradicts \\(\\beta_*>7/8\\);\nhence \\(\\beta_*\\le7/8\\). Proposition~\\ref{prop:hecke-dirichlet-transfer}\nat \\(\\sigma_0=7/8\\) extends the primitive Hecke conclusion to all\nfinite-order Hecke and Dirichlet \\(L\\)-functions, with the principal pole", "context_limit": "Only this source span is supplied; absent context may justify abstention." }, "paper_slug": "quasi-riemann", "source_path": "paper.tex", "source_sha256": "42a5ee0febca59fd1def55cfd6c6808c322ef4237d7726303ea52f711deac6a3" }, { "id": "quasi-riemann-source-4", "label_origin": "existing_source_audit_annotation_not_independent_gold", "reference_role": "background", "source_lines": [ 75, 83 ], "edge": [ "external:BCSS", "manuscript" ], "state": { "consumer": { "id": "manuscript", "label": "The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane Re(s)>7/8", "title": null }, "prerequisite": { "id": "external:BCSS", "label": "[3] BCSS", "title": "K. Billington, M. Cheng, J. Schettler, and A. I. Suriajaya, The average number of Goldbach representations and zero-free regions of the Riemann zeta-function, International Journal of Number Theory 21 (2025), no. 2, 289--316. \\href{https://doi.org/10.1142/S1793042125500150}{doi:10.1142/S1793042125500150}; \\href{https://arxiv.org/abs/2306.09102}{arXiv:2306.09102}." }, "source_passage": "$\\Re s>\\sigma_0$.\n\nFor $\\zeta$ alone, the existence of a fixed $\\sigma_0<1$ for which\n$\\zeta(s)\\ne0$ in $\\Re s>\\sigma_0$ is called the quasi-Riemann\nhypothesis \\cite[Section 1]{BCSS}. It asks for one gap valid at every\nheight, not merely nonvanishing on $\\Re s=1$. The corresponding uniform\nformulation over all Dirichlet characters appears in\n\\cite[p.~288, Equation (1.4) and the following paragraph]{FG}.\n", "context_limit": "Only this source span is supplied; absent context may justify abstention." }, "paper_slug": "quasi-riemann", "source_path": "paper.tex", "source_sha256": "42a5ee0febca59fd1def55cfd6c6808c322ef4237d7726303ea52f711deac6a3" }, { "id": "quasi-riemann-source-5", "label_origin": "existing_source_annotation_with_result_scoping_not_human_gold", "reference_role": "explicitly_excluded", "source_lines": [ 118, 130 ], "edge": [ "external:DR", "manuscript" ], "state": { "consumer": { "id": "manuscript", "label": "The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane Re(s)>7/8", "title": null }, "prerequisite": { "id": "external:DR:conditional-prime-asymptotic", "title": "The GRH-conditional prime asymptotic of Dunn and Radziwill, distinguished from their unconditional cusp expansions" }, "source_passage": "Riemann hypothesis or its generalized versions, which place nontrivial\nzeros on $\\Re s=1/2$. The Riemann hypothesis remains open \\cite{Clay}.\n\nKubota's metaplectic theory and Patterson's cubic theta series provide\nthe automorphic setting \\cite{Kubota,Patterson}. We use the unconditional\nexplicit cusp expansions recorded by Dunn and Radziwi\\l\\l{}\n\\cite[Section 5 and Appendix A]{DR}; their GRH-conditional prime asymptotic\nis not an input. The corresponding first-moment asymptotic is proved\nunconditionally in an independent manuscript\n\\cite[Theorem~1.1]{OpenAICubicFirstMoment}; that result is also not used\nhere. Proposition~\\ref{prop:completed-reflection} derives the\nreflection used here while retaining the characters' zero-on-nonunit\nrestrictions.", "context_limit": "Only this source span is supplied; absent context may justify abstention." }, "paper_slug": "quasi-riemann", "source_path": "paper.tex", "source_sha256": "42a5ee0febca59fd1def55cfd6c6808c322ef4237d7726303ea52f711deac6a3" }, { "id": "quasi-riemann-source-6", "label_origin": "existing_source_audit_annotation_not_independent_gold", "reference_role": "proof_input", "source_lines": [ 16457, 16466 ], "edge": [ "prop:hecke-dirichlet-transfer", "thm:main" ], "state": { "consumer": { "id": "thm:main", "label": "Theorem 1.1", "title": "" }, "prerequisite": { "id": "prop:hecke-dirichlet-transfer", "label": "Proposition 11.3", "title": "Quadratic transfer" }, "source_passage": "Proposition~\\ref{lem:continuation-criterion} with\n\\(\\sigma_0=7/8\\), \\(J_\\eta=J_{\\mathrm{II},\\eta}\\), and\n\\(f_\\eta=f_{\\mathrm{II},\\eta}\\). This contradicts \\(\\beta_*>7/8\\);\nhence \\(\\beta_*\\le7/8\\). Proposition~\\ref{prop:hecke-dirichlet-transfer}\nat \\(\\sigma_0=7/8\\) extends the primitive Hecke conclusion to all\nfinite-order Hecke and Dirichlet \\(L\\)-functions, with the principal pole\nallowed, and proves Theorem~\\ref{thm:main}.\n\\begin{thebibliography}{99}\n\n\\bibitem{BaierBansal}", "context_limit": "Only this source span is supplied; absent context may justify abstention." }, "paper_slug": "quasi-riemann", "source_path": "paper.tex", "source_sha256": "42a5ee0febca59fd1def55cfd6c6808c322ef4237d7726303ea52f711deac6a3" }, { "id": "quasi-riemann-source-7", "label_origin": "existing_source_audit_annotation_not_independent_gold", "reference_role": "background", "source_lines": [ 115, 123 ], "edge": [ "external:Clay", "manuscript" ], "state": { "consumer": { "id": "manuscript", "label": "The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane Re(s)>7/8", "title": null }, "prerequisite": { "id": "external:Clay", "label": "[5] Clay", "title": "Clay Mathematics Institute, Riemann Hypothesis, Millennium Prize Problems, \\url{https://www.claymath.org/millennium/riemann-hypothesis/}." }, "source_passage": "nontrivial zeros of $\\zeta$, namely its zeros in $0<\\Re s<1$, is at most\n$7/8$. The boundary $\\Re s=7/8$ is not included, while any exceptional\nreal zero in $(7/8,1)$ is excluded. The theorem does not establish the\nRiemann hypothesis or its generalized versions, which place nontrivial\nzeros on $\\Re s=1/2$. The Riemann hypothesis remains open \\cite{Clay}.\n\nKubota's metaplectic theory and Patterson's cubic theta series provide\nthe automorphic setting \\cite{Kubota,Patterson}. We use the unconditional\nexplicit cusp expansions recorded by Dunn and Radziwi\\l\\l{}", "context_limit": "Only this source span is supplied; absent context may justify abstention." }, "paper_slug": "quasi-riemann", "source_path": "paper.tex", "source_sha256": "42a5ee0febca59fd1def55cfd6c6808c322ef4237d7726303ea52f711deac6a3" }, { "id": "noncoordinate-polynomial-role-1", "paper_slug": "noncoordinate-polynomial", "state": { "consumer": "Theorem 1.1: the zero-fibre quotient is a three-variable polynomial ring", "prerequisite": "Lemma 2.1: the cusp lifting algebra is isomorphic to B[T] under x\u00b2+y\u00b3=hs and (h,y)=B", "source_passage": "Form the algebra in Lemma~\\ref{lem:lift} over this $B$, using lower-case\n$u,v,w$ for its three added variables. Its presentation is\n\\[\n C=B[u,v,w]/\\bigl(u^3+hv-x,\\,-u^2+hw-y,\\,\n S(h,u,v,w)-s\\bigr).\n\\]\nIn the polynomial ring on $h,x,y,s,u,v,w$, the last three equations eliminate\n$x,y,s$. The relation $x^2+y^3-sh$ then becomes the zero polynomial by\n\\eqref{eq:cusp}, and the other relation in \\eqref{eq:B} becomes exactly $F$.\nThus $C\\cong R/(F)$. Lemma~\\ref{lem:lift} and \\eqref{eq:Bplane} now give\n\\begin{equation}\n R/(F)\\cong C\\cong B[T]\\cong\\CC[X,Y,T]. \\label{eq:fiber}" }, "reference_role": "proof_input", "label_origin": "researcher_source_annotation_not_human_gold", "source_lines": [ 292, 303 ], "full_role_evidence_lines": [ 288, 303 ], "source_path": "manuscript-expanded.tex", "source_sha256": "60993c0b524291b7f707e42eec89ec7e910c22760d81f1d3af007c229228c13c", "researcher_annotation": "The proof constructs the lifting algebra, identifies it with the zero-fibre quotient, and applies the lemma.", "human_gold": false }, { "id": "noncoordinate-polynomial-role-2", "paper_slug": "noncoordinate-polynomial", "state": { "consumer": "Corollary 1.2: counterexamples after adjoining variables", "prerequisite": "Theorem 1.1: affine-three-space zero fibre and an ambient critical point for the explicit polynomial F", "source_passage": "\\begin{proof}[Proof of Corollary~\\ref{cor:dimension}]\nFor $n>4$, use the same $F$ in $R[z_1,\\ldots,z_{n-4}]$. Equation~\\eqref{eq:fiber}\nshows that the quotient is a polynomial ring in $n-1$ variables.\nAll derivatives of $F$ in the added variables vanish, and its original four\nderivatives vanish at $P$. Thus the extended polynomial has a critical point\nat $(P,0,\\ldots,0)$, and the same Jacobian argument rules out a coordinate.\nThe case $n=4$ is Theorem~\\ref{thm:main}." }, "reference_role": "proof_input", "label_origin": "researcher_source_annotation_not_human_gold", "source_lines": [ 332, 338 ], "full_role_evidence_lines": [ 332, 338 ], "source_path": "manuscript-expanded.tex", "source_sha256": "60993c0b524291b7f707e42eec89ec7e910c22760d81f1d3af007c229228c13c", "researcher_annotation": "The corollary uses the original quotient and critical point, and expressly cites the theorem for n=4.", "human_gold": false }, { "id": "noncoordinate-polynomial-role-3", "paper_slug": "noncoordinate-polynomial", "state": { "consumer": "Theorem 1.1: identifying the zero-fibre quotient with C[X,Y,T]", "prerequisite": "The auxiliary base algebra B is isomorphic to C[X,Y] through X=x+s\u00b3 and Y=y\u2212s\u00b2", "source_passage": "We first identify the base algebra to which Lemma~\\ref{lem:lift} applies.\nAfter obtaining the coordinates on the zero fibre, we use the same polynomial\nidentities in the ambient ring to locate a critical point on another fibre." }, "reference_role": "proof_input", "label_origin": "researcher_source_annotation_not_human_gold", "source_lines": [ 250, 252 ], "full_role_evidence_lines": [ 269, 303 ], "source_path": "manuscript-expanded.tex", "source_sha256": "60993c0b524291b7f707e42eec89ec7e910c22760d81f1d3af007c229228c13c", "researcher_annotation": "The opening announces a base-algebra identification without supplying its precise isomorphism. Later equations Bplane and fiber explicitly apply it.", "human_gold": false, "short_reference_role": "insufficient_context" }, { "id": "noncoordinate-polynomial-role-4", "paper_slug": "noncoordinate-polynomial", "state": { "consumer": "Theorem 1.1: a noncoordinate polynomial in four complex variables", "prerequisite": "The Abhyankar\u2013Moh\u2013Suzuki theorem answers the embedding/coordinate question affirmatively in ambient dimension two", "source_passage": "For $n=2$, the question is answered affirmatively by the\nAbhyankar--Moh--Suzuki Theorem~\\cite{AM,Suzuki}.\nThe higher-dimensional problem belongs to a family of questions about\npolynomial maps with affine-space fibres. Sathaye~\\cite[pp.~1--2]{Sathaye}\ndistinguished the coordinate, or epimorphism, question from the weaker question\nwhether an affine-space zero fibre forces all fibres to be affine space.\nA coordinate polynomial defines a projection after an automorphism, so its\nfibres are all isomorphic to $\\Aff^{n-1}_{\\CC}$; an isomorphism of the zero fibre\nalone supplies no such global trivialization." }, "reference_role": "background", "label_origin": "researcher_source_annotation_not_human_gold", "source_lines": [ 47, 55 ], "full_role_evidence_lines": [ 47, 55 ], "source_path": "manuscript-expanded.tex", "source_sha256": "60993c0b524291b7f707e42eec89ec7e910c22760d81f1d3af007c229228c13c", "researcher_annotation": "The two-dimensional theorem supplies historical contrast, not a step in the four-dimensional construction.", "human_gold": false }, { "id": "noncoordinate-polynomial-role-5", "paper_slug": "noncoordinate-polynomial", "state": { "consumer": "Theorem 1.1: the explicit four-variable construction", "prerequisite": "Ghosh Theorem 1.4 gives an affirmative characteristic-zero embedding theorem for the displayed special linear equation with restricted coefficients", "source_passage": "Ghosh~\\cite[Theorem~1.4]{Ghosh} proves an affirmative embedding theorem in\ncharacteristic zero for\n$a(x_m)b(x_1,\\ldots,x_{m-1})y+f(z,t)+x_m$, where the displayed variables are\nindependent, $b\\ne0$, and $a$ has at least two distinct roots over the algebraic\nclosure.\nThese results illustrate the role of restrictions on the defining equation;\nan abstract polynomial presentation of the quotient does not itself provide\nan automorphism of the ambient polynomial ring." }, "reference_role": "background", "label_origin": "researcher_source_annotation_not_human_gold", "source_lines": [ 72, 79 ], "full_role_evidence_lines": [ 72, 79 ], "source_path": "manuscript-expanded.tex", "source_sha256": "60993c0b524291b7f707e42eec89ec7e910c22760d81f1d3af007c229228c13c", "researcher_annotation": "The paragraph describes prior affirmative results with additional hypotheses; it does not apply them to the constructed polynomial.", "human_gold": false }, { "id": "noncoordinate-polynomial-role-6", "paper_slug": "noncoordinate-polynomial", "state": { "consumer": "Theorem 1.1: the explicit four-variable construction", "prerequisite": "Kaliman\u2013V\u00e9n\u00e9reau\u2013Zaidenberg Theorem 4.2(a) straightens affine-three-space zero fibres of a(x)y+b(x,z,t), with coefficient depending on one variable", "source_passage": "In four variables, Kaliman, V\\'en\\'ereau and Zaidenberg~\\cite[Theorem~4.2(a)]{KVZ}\nand Maubach~\\cite[Corollaries~4.2--4.3]{Maubach} established affirmative results\nfor equations $a(x)y+b(x,z,t)$ with $a\\ne0$: an affine-three-space zero fibre\ncan be straightened by an ambient automorphism. Here $x,y,z,t$ are independent\nvariables, and the coefficient of $y$ depends on only one of them.\nFurther algebraic treatments appear in work of El~Kahoui, Essamaoui and\nOuali~\\cite{EEO} and Ghosh, Gupta and Pal~\\cite{GGPthreefold}." }, "reference_role": "background", "label_origin": "researcher_source_annotation_not_human_gold", "source_lines": [ 62, 68 ], "full_role_evidence_lines": [ 62, 79 ], "source_path": "manuscript-expanded.tex", "source_sha256": "60993c0b524291b7f707e42eec89ec7e910c22760d81f1d3af007c229228c13c", "researcher_annotation": "This is an earlier restricted-equation result in the introductory comparison, not an imported proof step.", "human_gold": false }, { "id": "noncoordinate-polynomial-role-7", "paper_slug": "noncoordinate-polynomial", "state": { "consumer": "Theorem 1.1: proving the zero-fibre isomorphism for the displayed cusp polynomials", "prerequisite": "Maubach Theorem 3.2 case (3) triangular monomial kernel calculation produces the displayed invariant polynomials", "source_passage": "They arise in the triangular monomial kernel calculation of\nMaubach~\\cite[proof of Theorem~3.2, case~(3)]{MaubachMonomial}." }, "reference_role": "background", "label_origin": "researcher_source_annotation_not_human_gold", "source_lines": [ 143, 144 ], "full_role_evidence_lines": [ 140, 145 ], "source_path": "manuscript-expanded.tex", "source_sha256": "60993c0b524291b7f707e42eec89ec7e910c22760d81f1d3af007c229228c13c", "researcher_annotation": "The isolated attribution does not establish whether the cited kernel theorem is applied. The next sentence says the written proof uses the cusp relation directly through the local lifting lemma.", "human_gold": false, "short_reference_role": "insufficient_context" }, { "id": "noncoordinate-polynomial-role-8", "paper_slug": "noncoordinate-polynomial", "state": { "consumer": "Theorem 1.1: F is not an ambient coordinate", "prerequisite": "A polynomial automorphism has invertible Jacobian at every point, by the chain rule applied to its polynomial inverse", "source_passage": "We now use the ambient critical-point obstruction emphasized by\nShpilrain and Yu~\\cite[\\S1]{SY}.\nIf a polynomial automorphism had $F$ as its first component, its Jacobian\nmatrix at $P$ would have a zero first row. But the chain rule applied to the\nautomorphism and its polynomial inverse makes that matrix invertible.\nThis contradiction proves that $F$ is not a coordinate." }, "reference_role": "proof_input", "label_origin": "researcher_source_annotation_not_human_gold", "source_lines": [ 324, 329 ], "full_role_evidence_lines": [ 318, 329 ], "source_path": "manuscript-expanded.tex", "source_sha256": "60993c0b524291b7f707e42eec89ec7e910c22760d81f1d3af007c229228c13c", "researcher_annotation": "The proof uses invertibility to contradict the zero first Jacobian row at the critical point. The obstruction is explained locally.", "human_gold": false }, { "id": "foulkes-howe-stabilization-role-1", "paper_slug": "foulkes-howe-stabilization", "state": { "consumer": "Proposition 3: vanishing of the symmetric multilinear form T", "prerequisite": "Lemma 2: directional differentiation from U toward Y is injective on a homogeneous space with U-degree greater than Y-degree", "source_passage": "Immediately before the $i$-th operator, the degrees in $t$ and $y_i$ are\n\\[\n p=b-i+1,\\qquad q=r,\\qquad p-q=m-i+1\\ge1.\n\\]\nHere $y_i$ already has degree $r$ in $H_x$, but no earlier operator has\nincreased its degree. Lemma~\\ref{lem:shift} makes every operator injective,\nand hence $H_x=0$." }, "reference_role": "proof_input", "label_origin": "researcher_source_annotation_not_human_gold", "source_lines": [ 372, 378 ], "full_role_evidence_lines": [ 350, 378 ], "source_path": "manuscript-expanded.tex", "source_sha256": "89b46306d4bc756ff5fc261deaf3075ce1afa37087a313bab7abaf1569b8af21", "researcher_annotation": "The proof checks the degree inequality before each derivative and applies the local injectivity lemma.", "human_gold": false }, { "id": "foulkes-howe-stabilization-role-2", "paper_slug": "foulkes-howe-stabilization", "state": { "consumer": "Theorem 1: surjectivity of the canonical map for b\u2265a(a\u22121)", "prerequisite": "Proposition 3: symmetric forms with diagonal vanishing on products vanish identically", "source_passage": "\\begin{proof}[Proof of Theorem~\\ref{thm:main}]\nFor $a\\ge2$ and $b\\ge a(a-1)$, Proposition~\\ref{prop:vanishing} supplies\nthe vanishing assertion in the criterion of Section~\\ref{sec:map}.\nThat criterion gives the surjectivity of $\\mu_{a,b,V}$.\n\\end{proof}" }, "reference_role": "proof_input", "label_origin": "researcher_source_annotation_not_human_gold", "source_lines": [ 432, 436 ], "full_role_evidence_lines": [ 432, 436 ], "source_path": "manuscript-expanded.tex", "source_sha256": "89b46306d4bc756ff5fc261deaf3075ce1afa37087a313bab7abaf1569b8af21", "researcher_annotation": "The named theorem proof invokes the proposition and the diagonal criterion.", "human_gold": false }, { "id": "foulkes-howe-stabilization-role-3", "paper_slug": "foulkes-howe-stabilization", "state": { "consumer": "Corollary 4: existence of a GL(V)-equivariant embedding in the opposite direction", "prerequisite": "Theorem 1: the specified canonical map is surjective for b\u2265a(a\u22121)", "source_passage": "\\begin{proof}\nAveraging and multiplication commute with the action of $\\mathrm{GL}(V)$,\nso the surjection $\\mu_{a,b,V}$ of Theorem~\\ref{thm:main} is equivariant.\nFinite-dimensional polynomial representations over $\\C$ are completely\nreducible \\cite[Chapter~I, Appendix~A, Sections~7--8]{Macdonald}.\nChoose an invariant complement $M$ to its kernel. The restriction\n$\\mu_{a,b,V}|_M$ is an equivariant isomorphism onto\n$\\Sym^a(\\Sym^b V)$. Its inverse followed by the inclusion of $M$ in\n$\\Sym^b(\\Sym^a V)$ is the required embedding. This construction asserts\nexistence and does not identify the embedding with a prescribed canonical map." }, "reference_role": "proof_input", "label_origin": "researcher_source_annotation_not_human_gold", "source_lines": [ 450, 459 ], "full_role_evidence_lines": [ 450, 459 ], "source_path": "manuscript-expanded.tex", "source_sha256": "89b46306d4bc756ff5fc261deaf3075ce1afa37087a313bab7abaf1569b8af21", "researcher_annotation": "The corollary splits the theorem\u2019s equivariant surjection.", "human_gold": false }, { "id": "foulkes-howe-stabilization-role-4", "paper_slug": "foulkes-howe-stabilization", "state": { "consumer": "Corollary 4: splitting the surjection equivariantly", "prerequisite": "Finite-dimensional polynomial representations over C are completely reducible, as cited in Macdonald Chapter I Appendix A \u00a7\u00a77\u20138", "source_passage": "Finite-dimensional polynomial representations over $\\C$ are completely\nreducible \\cite[Chapter~I, Appendix~A, Sections~7--8]{Macdonald}.\nChoose an invariant complement $M$ to its kernel. The restriction\n$\\mu_{a,b,V}|_M$ is an equivariant isomorphism onto\n$\\Sym^a(\\Sym^b V)$. Its inverse followed by the inclusion of $M$ in\n$\\Sym^b(\\Sym^a V)$ is the required embedding. This construction asserts\nexistence and does not identify the embedding with a prescribed canonical map." }, "reference_role": "proof_input", "label_origin": "researcher_source_annotation_not_human_gold", "source_lines": [ 453, 459 ], "full_role_evidence_lines": [ 450, 459 ], "source_path": "manuscript-expanded.tex", "source_sha256": "89b46306d4bc756ff5fc261deaf3075ce1afa37087a313bab7abaf1569b8af21", "researcher_annotation": "Complete reducibility supplies the invariant complement used to build the embedding.", "human_gold": false }, { "id": "foulkes-howe-stabilization-role-5", "paper_slug": "foulkes-howe-stabilization", "state": { "consumer": "Theorem 1: the dimension-independent sufficient stabilization degree a(a\u22121)", "prerequisite": "Raicu\u2013Sam\u2013Weyman Theorem 3.9 bounds regularity of the Segre ring and modules of covariants, including normalization", "source_passage": "highest nonzero degree, or zero if $R=A$. Raicu, Sam, and Weyman bound the\nregularity of the Segre ring and modules of covariants, including the\nnormalization \\cite[Theorem~3.9]{RSW}. Their theorem does not compute the\ntop degree of the separate quotient $R/A$ that controls stabilization.\nTheorem~\\ref{thm:main} gives the sufficient bound $a(a-1)$ in every dimension.\nIt also answers positively the request for a polynomial bound in\n\\cite[Problem~7.19]{Landsberg}: in that source's notation $d=b$, $n=a$,\nand $w=\\dim V$, the bound is $d\\ge n(n-1)$, including when $w=n$.\nThe present argument does not determine the least stabilization degree\nin general; the bound is not asserted to be sharp." }, "reference_role": "background", "label_origin": "researcher_source_annotation_not_human_gold", "source_lines": [ 119, 128 ], "full_role_evidence_lines": [ 107, 128 ], "source_path": "manuscript-expanded.tex", "source_sha256": "89b46306d4bc756ff5fc261deaf3075ce1afa37087a313bab7abaf1569b8af21", "researcher_annotation": "The introduction distinguishes normalization regularity from the quotient top degree controlling this stabilization problem. The written proof proceeds via its local vanishing proposition.", "human_gold": false }, { "id": "foulkes-howe-stabilization-role-6", "paper_slug": "foulkes-howe-stabilization", "state": { "consumer": "Theorem 1: the uniform sufficient stabilization bound", "prerequisite": "Brion\u2019s eventual surjectivity and subsequent effective bound involving both a and dimension", "source_passage": "Brion established eventual surjectivity \\cite{Brion1993} and subsequently\ngave an effective bound involving both $a$ and the dimension\n\\cite[Theorem~3.3]{Brion}. In the normalization picture, eventual\nsurjectivity says that the graded quotient $R/A$ has only finitely many\nnonzero components. The least stabilization degree is one more than its\nhighest nonzero degree, or zero if $R=A$. Raicu, Sam, and Weyman bound the\nregularity of the Segre ring and modules of covariants, including the\nnormalization \\cite[Theorem~3.9]{RSW}. Their theorem does not compute the\ntop degree of the separate quotient $R/A$ that controls stabilization.\nTheorem~\\ref{thm:main} gives the sufficient bound $a(a-1)$ in every dimension.\nIt also answers positively the request for a polynomial bound in\n\\cite[Problem~7.19]{Landsberg}: in that source's notation $d=b$, $n=a$,\nand $w=\\dim V$, the bound is $d\\ge n(n-1)$, including when $w=n$.\nThe present argument does not determine the least stabilization degree\nin general; the bound is not asserted to be sharp." }, "reference_role": "background", "label_origin": "researcher_source_annotation_not_human_gold", "source_lines": [ 114, 128 ], "full_role_evidence_lines": [ 107, 128 ], "source_path": "manuscript-expanded.tex", "source_sha256": "89b46306d4bc756ff5fc261deaf3075ce1afa37087a313bab7abaf1569b8af21", "researcher_annotation": "Earlier stabilization results provide a comparison; the local proof does not import their bounds.", "human_gold": false }, { "id": "foulkes-howe-stabilization-role-7", "paper_slug": "foulkes-howe-stabilization", "state": { "consumer": "Lemma 2: directional-derivative injectivity with arbitrary additional variables", "prerequisite": "The positive-weight injectivity criterion in sl\u2082 weight strings, compared with Fulton\u2013Harris \u00a711.1 Claim 11.4 and Eq. (11.5)", "source_passage": "The criterion below is the positive-weight injectivity in the usual\n$\\mathfrak{sl}_2$ weight strings; compare\n\\cite[Section~11.1, Claim~11.4 and equation~(11.5)]{FultonHarris}." }, "reference_role": "background", "label_origin": "researcher_source_annotation_not_human_gold", "source_lines": [ 252, 254 ], "full_role_evidence_lines": [ 252, 290 ], "source_path": "manuscript-expanded.tex", "source_sha256": "89b46306d4bc756ff5fc261deaf3075ce1afa37087a313bab7abaf1569b8af21", "researcher_annotation": "The excerpt compares the criterion with a classical result but leaves its use as a premise unclear. The following text announces and supplies a direct inner-product proof.", "human_gold": false, "short_reference_role": "insufficient_context" }, { "id": "foulkes-howe-stabilization-role-8", "paper_slug": "foulkes-howe-stabilization", "state": { "consumer": "Lemma 2: the norm identity proving derivative injectivity", "prerequisite": "The Fischer/apolar Hermitian inner product makes distinct monomials orthogonal with squared norm \u03b1! and coordinate differentiation adjoint to multiplication", "source_passage": "\\begin{proof}\nGive the polynomial ring the positive definite Fischer, or apolar,\nHermitian inner product\n\\cite[Section~2, equation~(8) and Proposition~3]{AldazRender}, in which\ndistinct monomials are orthogonal and\n\\[\n \\|X^\\alpha\\|^2=\\alpha!:=\\prod_k\\alpha_k!;\n\\]\nhere $X$ lists all coordinates, including the additional variables.\nDifferentiation by a coordinate is adjoint to multiplication by that\ncoordinate. Thus, for $D=\\D{Y}{U}$ and $E=\\D{U}{Y}$, one has $D^*=E$.\nA direct calculation gives\n\\[\n ED-DE=N_U-N_Y,\n \\qquad\n N_U=\\sum_k U_k\\partial_{U_k},\\quad\n N_Y=\\sum_k Y_k\\partial_{Y_k}.\n\\]\nConsequently, for a polynomial $h$ of the indicated bidegree,\n\\begin{equation}\\label{eq:norm}\n \\|Dh\\|^2-\\|Eh\\|^2=(p-q)\\|h\\|^2." }, "reference_role": "proof_input", "label_origin": "researcher_source_annotation_not_human_gold", "source_lines": [ 267, 287 ], "full_role_evidence_lines": [ 267, 290 ], "source_path": "manuscript-expanded.tex", "source_sha256": "89b46306d4bc756ff5fc261deaf3075ce1afa37087a313bab7abaf1569b8af21", "researcher_annotation": "The proof equips the ring with this inner product and uses adjointness plus the commutator to obtain the displayed norm identity.", "human_gold": false } ]