{
  "case_id": "case-757cd4d7b1e8cadb4b51fcb4",
  "target_id": "thm:Main",
  "original_proof_status": "defective",
  "statement_status": "unresolved",
  "statement_argument": "New repair (conditional on Theorem thm:RNU, eq:fFTbnd, eq:Uis, eq:TXNis, and Lemma 2.12 / eq:binQuad for the existence of a power-saving exponent): For a scale parameter M, define Z(M) to be the set of admissible n in (M/2, M) for which \\cR_M(n) = 0, and W(M) to be the set of admissible n in (M/2, M) satisfying eq:RNUbnd at scale M. By the M\u00f6bius-tail estimate and Theorem thm:RNU applied on (M/2, M), |Z(M) \\cap W(M)| \\ll_\\vep M^{1+\\vep-\\fu} and |Z(M) \\cap W(M)^c| \\le |W(M)^c| \\ll M^{1-\\eta_R}, where W(M)^c denotes the complement inside admissible n in (M/2, M). Taking \\vep = \\fu/2 and \\eta' = \\min(\\eta_R, \\fu/2) > 0 gives |Z(M)| \\ll M^{1-\\eta'}. Since \\cR_M(n) \\neq 0 implies n \\in \\sB by eq:cRNis and eq:binQuad, every admissible exception n \\in (M/2, M) \\setminus \\sB lies in Z(M). Summing over dyadic scales M = N/2^k for 0 \\le k \\le \\lfloor \\log_2 N \\rfloor + 1 yields #(\\{n < N : n \\in \\sA \\setminus \\sB\\}) \\ll \\sum_{k \\ge 0} (N/2^k)^{1-\\eta'} \\ll N^{1-\\eta'}. However, the full statement of Theorem thm:Main remains unresolved in this packet because (1) Theorem thm:Main additionally claims that \\eta > 0 is effectively computable, which is neither assumed in Theorem thm:RNU nor proved in the manuscript (the Appendix explicitly notes that explicit bounds are not pursued), and (2) upstream external inputs (such as Vinogradov 2013 / arXiv:1204.5498v1) are not supplied in full text for direct verification.",
  "summary": "In the displayed proof of Theorem thm:Main assuming Theorem thm:RNU, there are two substantive gaps against the stated target. First, Theorem thm:RNU establishes the singular-series lower bound and exceptional-set estimate O(N^{1-\\eta}) only on the dyadic interval \\frac{1}{2}N < n < N, whereas the proof of Theorem thm:Main defines Z and W over all admissible n < N. Because \\cR_N(n) is localized to scale TX^2 = N, \\cR_N(n) and \\cR_N^U(n) vanish for small n \\ll N, so |W^c| \\ll N^{1-\\eta} is literally false for W defined on all admissible n < N; a restriction to (N/2, N) followed by dyadic summation over scales is required but omitted. Second, Theorem thm:Main claims that \\eta > 0 is effectively computable, whereas Theorem thm:RNU only asserts existence of \\eta > 0 and the manuscript never proves effective computability. The M\u00f6bius-tail estimate and power-saving algebra on a single dyadic interval are valid in the full manuscript context. Regarding chronology, the full manuscript bibliography cites Vinogradov's preprint arXiv:1204.5498v1 (April 2012, prior to the 2012-05-24 cutoff), though the external preprint text is not supplied in the packet.",
  "decisions": [
    {
      "finding_id": "finding-94be3d9bf03ed68b4408",
      "outcome": "confirmed_error",
      "severity": "substantive",
      "reasoning": "In the full source preamble, \\foh is defined as \\frac{1}{2}. Theorem thm:RNU states that for \\frac{1}{2}N < n < N, \\fS(n) \\gg_\\vep N^{-\\vep} for n \\in \\sA and eq:RNUbnd holds for admissible n in (\\frac{1}{2}N, N) except on a set of cardinality \\ll N^{1-\\eta}. However, the proof of Theorem thm:Main defines Z and W over all admissible n < N ('let Z be the \"exceptional\" set of admissible n<N for which \\cR_N(n)=0. Futhermore, let W be the set of admissible n<N for which \\eqref{eq:RNUbnd} is satisfied'). Because \\cR_N(n) and \\cR_N^U(n) are localized via \\gU(2x/X)\\gU(y/X) and \\g \\in \\fF_T to values \\ff(2x,y) of order TX^2 = N, \\cR_N(n) and \\cR_N^U(n) vanish for small n \\ll N. Consequently, for W defined over all admissible n < N, Theorem thm:RNU does not imply |W^c| \\ll N^{1-\\eta} (and in fact |W^c| \\asymp N for a fixed N). The proof omits restricting Z and W to the dyadic interval \\frac{1}{2}N < n < N and summing over dyadic scales.",
      "hypothesis_check": "Checked against the full statement of Theorem thm:RNU and the definitions of \\cR_N (eq:cRNis) and \\cR_N^U (eq:cRNhatUis) in Section 3.1. Theorem thm:RNU explicitly restricts its hypotheses and conclusion to \\frac{1}{2}N < n < N.",
      "affected_claims": [
        "thm:Main"
      ],
      "revised_status": "not_checked",
      "revised_reasoning": "No later edition was supplied for comparison.",
      "source_quote": "Note also from Theorem \\ref{thm:RNU} that $|Z\\cap W^{c}|\n\\le |W^{c}|\\ll N^{1-\\eta}$.",
      "source_span": {
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        "char_end": 40897,
        "match_mode": "exact"
      },
      "revised_quote": "",
      "revised_span": null
    },
    {
      "finding_id": "finding-a0a39e9cc39cb243796d",
      "outcome": "packet_context",
      "severity": "none",
      "reasoning": "This finding is a context note confirming the elementary M\u00f6bius-tail estimate \\sum_{n<N} |\\cR_N(n) - \\cR_N^U(n)| \\ll T^\\gd X^2 / U. In the full manuscript (e.g. Section 3.1 and Lemma lem:spec3 in Section 5.1, eq:1), sums over \\fF_T are taken with the pair multiplicity (\\g_1, \\g_2) from eq:fFdef, and applying the Lax-Phillips ball bound to \\|\\g_1\\| < 2T_1 and \\|\\g_2\\| < 2T_2 with T = T_1 T_2 (eq:TT1T2) bounds the multiplicity-counted family by \\ll T_1^\\gd T_2^\\gd = T^\\gd. Since \\supp(\\gU) \\subset [1,2], 2x and y lie in [X, 2X], so for each u \\ge U there are O(X/u) choices for 2x and O(X/u) choices for y (and zero choices for u > 2X), yielding \\ll T^\\gd \\sum_{U \\le u \\le 2X} (X/u)^2 \\ll T^\\gd X^2 / U.",
      "hypothesis_check": "Consistent with the definitions of \\fF_T (eq:fFdef), T = T_1 T_2 (eq:TT1T2), eq:fFTbnd, and \\gU in Section 3.1.",
      "affected_claims": [
        "thm:Main"
      ],
      "revised_status": "not_checked",
      "revised_reasoning": "No later edition was supplied for comparison.",
      "source_quote": "T^{\\gd}            \n{X^{2}\\over U}",
      "source_span": {
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      "revised_quote": "",
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    },
    {
      "finding_id": "finding-5e7e13d1bcc21dc161e4",
      "outcome": "packet_context",
      "severity": "none",
      "reasoning": "This finding is a context note confirming that on the dyadic interval (N/2, N) covered by Theorem thm:RNU, the exponent arithmetic is valid: since U = N^\\fu with \\fu > 0 (eq:Uis) and TX^2 = N (eq:TXNis), choosing \\vep = \\fu/2 in N^{1-\\eta} + N^{1+\\vep}/U yields a power saving N^{1 - \\min(\\eta, \\fu/2)}. Moreover, the implication \\cR_N(n) \\neq 0 \\implies n \\in \\sB is established in Section 2.2 of the full manuscript (Lemma 2.12, eq:binQuad and eq:ffToInProd).",
      "hypothesis_check": "Checked against Lemma 2.12 (eq:binQuad), eq:ffToInProd, eq:TXNis, eq:Uis, and Theorem thm:RNU in the full manuscript.",
      "affected_claims": [
        "thm:Main"
      ],
      "revised_status": "not_checked",
      "revised_reasoning": "No later edition was supplied for comparison.",
      "source_quote": "which\n is a power savings since $\\vep>0$ is arbitrary.",
      "source_span": {
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        "char_end": 41108,
        "match_mode": "exact"
      },
      "revised_quote": "",
      "revised_span": null
    },
    {
      "finding_id": "finding-34cafcc1e4a10af2e9bc",
      "outcome": "confirmed_error",
      "severity": "substantive",
      "reasoning": "Theorem thm:Main explicitly states that the exponent \\eta > 0 in O(N^{1-\\eta}) is 'effectively computable'. However, in the reduction to Theorem thm:RNU, Theorem thm:RNU only states 'There exists an \\eta > 0', and the proof of Theorem thm:Main assuming Theorem thm:RNU provides no argument for effective computability. Inspecting the full manuscript confirms that effective computability is never mentioned or proved anywhere after the statement of Theorem thm:Main: Theorems thm:CircMeth, thm:specGap, and thm:Vin are all stated as existence results without effective bounds on the spectral gap \\vep(\\G) or counting exponent \\gT < \\gd, and the Appendix explicitly remarks 'we do not pursue explicit bounds'.",
      "hypothesis_check": "Checked against the full text of the manuscript, including Theorem thm:RNU, Theorem thm:CircMeth, Section 4 (Theorems thm:specGap and thm:Vin), Section 5 (Lemmata lem:spec0-lem:spec2), Section 6 (eq:Q0is, eq:fuIs), and the Appendix.",
      "affected_claims": [
        "thm:Main"
      ],
      "revised_status": "not_checked",
      "revised_reasoning": "No later edition was supplied for comparison.",
      "source_quote": "$O(N^{1-\\eta})$, where $\\eta>0$ is effectively computable.",
      "source_span": {
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        "char_end": 15775,
        "match_mode": "exact"
      },
      "revised_quote": "",
      "revised_span": null
    },
    {
      "finding_id": "finding-3f2d492685ad2dff96d1",
      "outcome": "unresolved",
      "severity": "none",
      "reasoning": "In the local reduction of Theorem thm:Main to Theorem thm:RNU and eq:fFTbnd, no external result is directly invoked in the displayed proof step (eq:fFTbnd is an internal display cited to Lax-Phillips 1982 or eq:Vin). Upstream, Theorem thm:Vin (eq:Vin) is deduced from \\cite[Thm 2.2]{Vinogradov2013}. Inspecting the full manuscript's bibliography (omitted from the reviewer's local section packet) resolves the citation metadata: \\bibitem[Vin13]{Vinogradov2013} cites 'Princeton University Thesis 2012, {\\tt arxiv:1204.5498v1}', which dates to April 2012, prior to the 2012-05-24 received cutoff. However, because the external text of arXiv:1204.5498v1 is not supplied in the review packet, Theorem 2.2 of Vinogradov2013 and its exact hypotheses cannot be directly inspected in the supplied material, leaving the external input verification unresolved.",
      "hypothesis_check": "Checked against Section 4.4 (Theorem thm:Vin) and the bibliography (\\bibitem[Vin13]{Vinogradov2013}) in the full reviewed source.",
      "affected_claims": [
        "thm:Main",
        "thm:Vin"
      ],
      "revised_status": "not_checked",
      "revised_reasoning": "No later edition was supplied for comparison.",
      "source_quote": "The following theorem follows easily              \nfrom \\cite[Thm 2.2]{Vinogradov2013}.",
      "source_span": {
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        "match_mode": "exact"
      },
      "revised_quote": "",
      "revised_span": null
    }
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      "status": "recorded",
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    "note": "The recorded review policy cutoff is 2012-05-24 (journal received date). In the full reviewed manuscript's bibliography, the citation \\cite{Vinogradov2013} (\\bibitem[Vin13]{Vinogradov2013}) explicitly records the preprint version 'Princeton University Thesis 2012, arxiv:1204.5498v1' in addition to 'IMRN, 2013'. The arXiv identifier 1204.5498v1 indicates April 2012 availability, prior to the 2012-05-24 cutoff. However, because the external text of arXiv:1204.5498v1 (and of Lax-Phillips 1982) is not supplied in the review packet, applicability and exact hypotheses of those external results cannot be verified directly from supplied text. No post-cutoff knowledge or later proof is used in this arbitration.",
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      {
        "reference": {
          "role": "proof_input",
          "imported_result": "Vinogradov [Vin13, Thm 2.2]: effective bisector counting in G = SL(2,C) for geometrically finite non-elementary Kleinian groups with \\delta > 1, used to deduce Theorem thm:Vin (eq:Vin) and cited as an alternative reference for eq:fFTbnd.",
          "available_from": "2012-04",
          "date_evidence": "Reviewed manuscript bibliography \\bibitem[Vin13]{Vinogradov2013}: 'I. Vinogradov. Effective bisector estimate with application to Apollonian circle packings. IMRN, 2013. Princeton University Thesis 2012, {\\tt arxiv:1204.5498v1}.'",
          "date_basis": "arXiv identifier YYMM prefix (1204 = April 2012) documented in the manuscript bibliography prior to the 2012-05-24 cutoff; external preprint text is not supplied in the review packet.",
          "version": "arXiv:1204.5498v1 / Princeton University Thesis 2012",
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        },
        "status": "admissible",
        "reason": "The documented version was available by the cutoff."
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        "reference": {
          "role": "proof_input",
          "imported_result": "Lax and Phillips [LP82]: asymptotic distribution of orbit points in hyperbolic space, cited for the ball counting bound #\\fF_T \\ll T^\\delta in eq:fFTbnd.",
          "available_from": "1982",
          "date_evidence": "Reviewed manuscript bibliography \\bibitem[LP82]{LaxPhillips1982}: 'P.D. Lax and R.S. Phillips. The asymptotic distribution of lattice points in Euclidean and non-Euclidean space. Journal of Functional Analysis, 46:280--350, 1982.'",
          "date_basis": "Published journal article dated 1982 in the manuscript bibliography, prior to the 2012-05-24 cutoff; external text not supplied in the review packet.",
          "version": "Journal of Functional Analysis 46 (1982), 280-350",
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        },
        "status": "admissible",
        "reason": "The documented version was available by the cutoff."
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