[
  {
    "id": "quasi-riemann-retrieval-0",
    "label_origin": "post_hoc_context_omission_source_annotation_not_human_gold",
    "reference_role": "proof_input",
    "source_lines": [
      99,
      113
    ],
    "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": "by \\cite[Theorem 3.1]{TZ}. Such regions do not give uniform exclusion in\na fixed half-plane. Zero-density estimates address a different question:\nthey bound the number of zeros to the right of a given vertical line.\nFor example, Guth and Maynard's large-value estimates improve zero-density\nbounds for $\\zeta$ \\cite[Theorem 1.2]{GM}, but these zero-density bounds\ndo not exclude every zero.\n\n\\begin{theorem}\\label{thm:main}\nEvery finite-order Hecke $L$-function over\n$F=\\mathbb Q(\\sqrt{-3})$ has no zero in $\\Re s>7/8$.\nThe same holds for every Dirichlet $L$-function, including $\\zeta(s)$.\nA pole at $s=1$ for a principal character is allowed.\n\\end{theorem}\n\nIn particular, Theorem~\\ref{thm:main} resolves the quasi-Riemann",
      "context_limit": "Only this source span is supplied; absent context may justify abstention."
    },
    "paper_slug": "quasi-riemann",
    "source_path": "paper.tex",
    "source_sha256": "42a5ee0febca59fd1def55cfd6c6808c322ef4237d7726303ea52f711deac6a3",
    "short_reference_role": "insufficient_context",
    "context_selection_note": "Main claim excerpt supplied; prerequisite use/exclusion passage omitted. Expected excerpt abstention is separately annotated. This is a post-hoc follow-up, not heldout evidence."
  },
  {
    "id": "quasi-riemann-retrieval-1",
    "label_origin": "post_hoc_context_omission_source_annotation_not_human_gold",
    "reference_role": "background",
    "source_lines": [
      99,
      113
    ],
    "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": "by \\cite[Theorem 3.1]{TZ}. Such regions do not give uniform exclusion in\na fixed half-plane. Zero-density estimates address a different question:\nthey bound the number of zeros to the right of a given vertical line.\nFor example, Guth and Maynard's large-value estimates improve zero-density\nbounds for $\\zeta$ \\cite[Theorem 1.2]{GM}, but these zero-density bounds\ndo not exclude every zero.\n\n\\begin{theorem}\\label{thm:main}\nEvery finite-order Hecke $L$-function over\n$F=\\mathbb Q(\\sqrt{-3})$ has no zero in $\\Re s>7/8$.\nThe same holds for every Dirichlet $L$-function, including $\\zeta(s)$.\nA pole at $s=1$ for a principal character is allowed.\n\\end{theorem}\n\nIn particular, Theorem~\\ref{thm:main} resolves the quasi-Riemann",
      "context_limit": "Only this source span is supplied; absent context may justify abstention."
    },
    "paper_slug": "quasi-riemann",
    "source_path": "paper.tex",
    "source_sha256": "42a5ee0febca59fd1def55cfd6c6808c322ef4237d7726303ea52f711deac6a3",
    "short_reference_role": "insufficient_context",
    "context_selection_note": "Main claim excerpt supplied; prerequisite use/exclusion passage omitted. Expected excerpt abstention is separately annotated. This is a post-hoc follow-up, not heldout evidence."
  },
  {
    "id": "quasi-riemann-retrieval-2",
    "label_origin": "post_hoc_context_omission_source_annotation_not_human_gold",
    "reference_role": "explicitly_excluded",
    "source_lines": [
      99,
      113
    ],
    "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": "by \\cite[Theorem 3.1]{TZ}. Such regions do not give uniform exclusion in\na fixed half-plane. Zero-density estimates address a different question:\nthey bound the number of zeros to the right of a given vertical line.\nFor example, Guth and Maynard's large-value estimates improve zero-density\nbounds for $\\zeta$ \\cite[Theorem 1.2]{GM}, but these zero-density bounds\ndo not exclude every zero.\n\n\\begin{theorem}\\label{thm:main}\nEvery finite-order Hecke $L$-function over\n$F=\\mathbb Q(\\sqrt{-3})$ has no zero in $\\Re s>7/8$.\nThe same holds for every Dirichlet $L$-function, including $\\zeta(s)$.\nA pole at $s=1$ for a principal character is allowed.\n\\end{theorem}\n\nIn particular, Theorem~\\ref{thm:main} resolves the quasi-Riemann",
      "context_limit": "Only this source span is supplied; absent context may justify abstention."
    },
    "paper_slug": "quasi-riemann",
    "source_path": "paper.tex",
    "source_sha256": "42a5ee0febca59fd1def55cfd6c6808c322ef4237d7726303ea52f711deac6a3",
    "short_reference_role": "insufficient_context",
    "context_selection_note": "Main claim excerpt supplied; prerequisite use/exclusion passage omitted. Expected excerpt abstention is separately annotated. This is a post-hoc follow-up, not heldout evidence."
  },
  {
    "id": "quasi-riemann-retrieval-3",
    "label_origin": "post_hoc_context_omission_source_annotation_not_human_gold",
    "reference_role": "proof_input",
    "source_lines": [
      99,
      113
    ],
    "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": "by \\cite[Theorem 3.1]{TZ}. Such regions do not give uniform exclusion in\na fixed half-plane. Zero-density estimates address a different question:\nthey bound the number of zeros to the right of a given vertical line.\nFor example, Guth and Maynard's large-value estimates improve zero-density\nbounds for $\\zeta$ \\cite[Theorem 1.2]{GM}, but these zero-density bounds\ndo not exclude every zero.\n\n\\begin{theorem}\\label{thm:main}\nEvery finite-order Hecke $L$-function over\n$F=\\mathbb Q(\\sqrt{-3})$ has no zero in $\\Re s>7/8$.\nThe same holds for every Dirichlet $L$-function, including $\\zeta(s)$.\nA pole at $s=1$ for a principal character is allowed.\n\\end{theorem}\n\nIn particular, Theorem~\\ref{thm:main} resolves the quasi-Riemann",
      "context_limit": "Only this source span is supplied; absent context may justify abstention."
    },
    "paper_slug": "quasi-riemann",
    "source_path": "paper.tex",
    "source_sha256": "42a5ee0febca59fd1def55cfd6c6808c322ef4237d7726303ea52f711deac6a3",
    "short_reference_role": "insufficient_context",
    "context_selection_note": "Main claim excerpt supplied; prerequisite use/exclusion passage omitted. Expected excerpt abstention is separately annotated. This is a post-hoc follow-up, not heldout evidence."
  },
  {
    "id": "quasi-riemann-retrieval-4",
    "label_origin": "post_hoc_context_omission_source_annotation_not_human_gold",
    "reference_role": "background",
    "source_lines": [
      99,
      113
    ],
    "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": "by \\cite[Theorem 3.1]{TZ}. Such regions do not give uniform exclusion in\na fixed half-plane. Zero-density estimates address a different question:\nthey bound the number of zeros to the right of a given vertical line.\nFor example, Guth and Maynard's large-value estimates improve zero-density\nbounds for $\\zeta$ \\cite[Theorem 1.2]{GM}, but these zero-density bounds\ndo not exclude every zero.\n\n\\begin{theorem}\\label{thm:main}\nEvery finite-order Hecke $L$-function over\n$F=\\mathbb Q(\\sqrt{-3})$ has no zero in $\\Re s>7/8$.\nThe same holds for every Dirichlet $L$-function, including $\\zeta(s)$.\nA pole at $s=1$ for a principal character is allowed.\n\\end{theorem}\n\nIn particular, Theorem~\\ref{thm:main} resolves the quasi-Riemann",
      "context_limit": "Only this source span is supplied; absent context may justify abstention."
    },
    "paper_slug": "quasi-riemann",
    "source_path": "paper.tex",
    "source_sha256": "42a5ee0febca59fd1def55cfd6c6808c322ef4237d7726303ea52f711deac6a3",
    "short_reference_role": "insufficient_context",
    "context_selection_note": "Main claim excerpt supplied; prerequisite use/exclusion passage omitted. Expected excerpt abstention is separately annotated. This is a post-hoc follow-up, not heldout evidence."
  },
  {
    "id": "quasi-riemann-retrieval-5",
    "label_origin": "post_hoc_context_omission_source_annotation_not_human_gold",
    "reference_role": "explicitly_excluded",
    "source_lines": [
      99,
      113
    ],
    "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": "by \\cite[Theorem 3.1]{TZ}. Such regions do not give uniform exclusion in\na fixed half-plane. Zero-density estimates address a different question:\nthey bound the number of zeros to the right of a given vertical line.\nFor example, Guth and Maynard's large-value estimates improve zero-density\nbounds for $\\zeta$ \\cite[Theorem 1.2]{GM}, but these zero-density bounds\ndo not exclude every zero.\n\n\\begin{theorem}\\label{thm:main}\nEvery finite-order Hecke $L$-function over\n$F=\\mathbb Q(\\sqrt{-3})$ has no zero in $\\Re s>7/8$.\nThe same holds for every Dirichlet $L$-function, including $\\zeta(s)$.\nA pole at $s=1$ for a principal character is allowed.\n\\end{theorem}\n\nIn particular, Theorem~\\ref{thm:main} resolves the quasi-Riemann",
      "context_limit": "Only this source span is supplied; absent context may justify abstention."
    },
    "paper_slug": "quasi-riemann",
    "source_path": "paper.tex",
    "source_sha256": "42a5ee0febca59fd1def55cfd6c6808c322ef4237d7726303ea52f711deac6a3",
    "short_reference_role": "insufficient_context",
    "context_selection_note": "Main claim excerpt supplied; prerequisite use/exclusion passage omitted. Expected excerpt abstention is separately annotated. This is a post-hoc follow-up, not heldout evidence."
  },
  {
    "id": "quasi-riemann-retrieval-6",
    "label_origin": "post_hoc_context_omission_source_annotation_not_human_gold",
    "reference_role": "proof_input",
    "source_lines": [
      99,
      113
    ],
    "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": "by \\cite[Theorem 3.1]{TZ}. Such regions do not give uniform exclusion in\na fixed half-plane. Zero-density estimates address a different question:\nthey bound the number of zeros to the right of a given vertical line.\nFor example, Guth and Maynard's large-value estimates improve zero-density\nbounds for $\\zeta$ \\cite[Theorem 1.2]{GM}, but these zero-density bounds\ndo not exclude every zero.\n\n\\begin{theorem}\\label{thm:main}\nEvery finite-order Hecke $L$-function over\n$F=\\mathbb Q(\\sqrt{-3})$ has no zero in $\\Re s>7/8$.\nThe same holds for every Dirichlet $L$-function, including $\\zeta(s)$.\nA pole at $s=1$ for a principal character is allowed.\n\\end{theorem}\n\nIn particular, Theorem~\\ref{thm:main} resolves the quasi-Riemann",
      "context_limit": "Only this source span is supplied; absent context may justify abstention."
    },
    "paper_slug": "quasi-riemann",
    "source_path": "paper.tex",
    "source_sha256": "42a5ee0febca59fd1def55cfd6c6808c322ef4237d7726303ea52f711deac6a3",
    "short_reference_role": "insufficient_context",
    "context_selection_note": "Main claim excerpt supplied; prerequisite use/exclusion passage omitted. Expected excerpt abstention is separately annotated. This is a post-hoc follow-up, not heldout evidence."
  },
  {
    "id": "quasi-riemann-retrieval-7",
    "label_origin": "post_hoc_context_omission_source_annotation_not_human_gold",
    "reference_role": "background",
    "source_lines": [
      99,
      113
    ],
    "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": "by \\cite[Theorem 3.1]{TZ}. Such regions do not give uniform exclusion in\na fixed half-plane. Zero-density estimates address a different question:\nthey bound the number of zeros to the right of a given vertical line.\nFor example, Guth and Maynard's large-value estimates improve zero-density\nbounds for $\\zeta$ \\cite[Theorem 1.2]{GM}, but these zero-density bounds\ndo not exclude every zero.\n\n\\begin{theorem}\\label{thm:main}\nEvery finite-order Hecke $L$-function over\n$F=\\mathbb Q(\\sqrt{-3})$ has no zero in $\\Re s>7/8$.\nThe same holds for every Dirichlet $L$-function, including $\\zeta(s)$.\nA pole at $s=1$ for a principal character is allowed.\n\\end{theorem}\n\nIn particular, Theorem~\\ref{thm:main} resolves the quasi-Riemann",
      "context_limit": "Only this source span is supplied; absent context may justify abstention."
    },
    "paper_slug": "quasi-riemann",
    "source_path": "paper.tex",
    "source_sha256": "42a5ee0febca59fd1def55cfd6c6808c322ef4237d7726303ea52f711deac6a3",
    "short_reference_role": "insufficient_context",
    "context_selection_note": "Main claim excerpt supplied; prerequisite use/exclusion passage omitted. Expected excerpt abstention is separately annotated. This is a post-hoc follow-up, not heldout evidence."
  }
]