{
  "case_id": "case-6fec05a0316d951924727739",
  "target_id": "eq:cKzBnd",
  "original_proof_status": "defective",
  "statement_status": "locally_supported",
  "statement_argument": "New repair (self-contained): Under the manuscript's conventions in (2.15)-(2.16) and Section 5.2, a primitive form in a class K is of the shape f(m,n) = A m^2 + 2 B m n + C n^2 with gcd(A,B,C) = 1 and discriminant 4(B^2 - A C) = -4 a^2 (with a != 0, so A C - B^2 = a^2 > 0 and f is positive definite for A > 0). Suppose f in K represents a positive integer z, say f(m,n) = z. Let w = gcd(m,n) and z_1 = z / w^2, so z_1 divides z and (m/w, n/w) is a primitive integer vector. Extending (m/w, n/w)^t to the first column of a matrix M in SL(2,Z) subset GL(2,Z) and applying the transformation law (5.20), f is equivalent to a form f_1(x,y) = z_1 x^2 + 2 B x y + C y^2. Because det(M) = +-1, the invariant A C - B^2 is preserved, yielding z_1 C - B^2 = a^2 (or, if 'determinant -4a^2' were instead taken literally to mean B^2 - A C = -4a^2, then z_1 C - B^2 = 4a^2). Applying a unipotent substitution (x,y) -> (x + k y, y) preserves z_1 and shifts B by k z_1, so we may choose a representative with B in [0, z_1). Since C = (B^2 + a^2)/z_1 is then uniquely determined by z_1 and B, the number of inequivalent classes K representing z_1 primitively is at most #{B (mod z_1) : B^2 == -a^2 (mod z_1)} (or #{B (mod z_1) : B^2 == -4a^2 (mod z_1)} under B^2 - A C = -4a^2). Write d = a^2 (respectively d = 4a^2). By the Chinese Remainder Theorem, the number of solutions B (mod z_1) to B^2 == -d (mod z_1) is the product over prime powers p^e || z_1 of #{B (mod p^e) : B^2 == -d (mod p^e)}. Fix p^e || z_1 and let p^f || sqrt(d), so d = u^2 p^{2f} with gcd(u,p) = 1. If 2f >= e, the congruence is B^2 == 0 (mod p^e), which holds if and only if p^{ceil(e/2)} | B, giving p^{floor(e/2)} <= p^{e/2} = min(p^{e/2}, p^f) solutions modulo p^e. If 2f < e, any solution satisfies p^f | B; writing B = p^f B_0 with B_0 (mod p^{e-f}) gives B_0^2 == -u^2 (mod p^{e-2f}). Since gcd(u,p) = 1, setting B_1 == u^{-1} B_0 (mod p^{e-2f}) gives B_1^2 == -1 (mod p^{e-2f}), which has at most 2 solutions modulo p^{e-2f} (at most 2 for odd p by Hensel's lemma, and at most 1 for p = 2 since e - 2f >= 1). Each solution modulo p^{e-2f} lifts to p^f residues for B_0 (mod p^{e-f}), hence p^f residues for B (mod p^e), giving at most 2 p^f = 2 min(p^{e/2}, p^f) local solutions. Multiplying over p^e || z_1 yields at most 2^{omega(z_1)} gcd(z_1, d)^{1/2} <= 2^{omega(z)} gcd(z, 4a^2)^{1/2} <<_{epsilon} z^{epsilon} gcd(z, 4a^2)^{1/2} classes for each z_1 | z. Summing over the O_{epsilon}(z^{epsilon}) divisors z_1 of z establishes (5.21).",
  "summary": "The displayed proof of Lemma (5.21) [eq:cKzBnd] is defective because it writes the determinant relation for the transformed form as -4a^2 = z_1 C - B^2, which both conflicts with the manuscript's normalization A C - B^2 = a^2 from (2.16) and (5.20) and reduces modulo z_1 to B^2 == +4a^2 (mod z_1) rather than the negative-square congruence B^2 == -4a^2 (mod z_1) displayed in (5.22) [eq:BtoZ1]. In addition, the proof states that the number of inequivalent forms is equal to (rather than bounded above by) the number of congruence solutions. However, the lemma statement itself is locally supported: replacing -4a^2 = z_1 C - B^2 by the manuscript's invariant z_1 C - B^2 = a^2 (or by B^2 - z_1 C = -4a^2) and replacing equality of counts with an upper bound immediately yields the negative-square congruence and establishes (5.21) and its downstream use in (5.25) [eq:eqeq].",
  "decisions": [
    {
      "finding_id": "finding-a2343127af3149f8584e",
      "outcome": "confirmed_error",
      "severity": "substantive",
      "reasoning": "The displayed equation -4a^2 = z_1 C - B^2 has the wrong sign (and an extraneous factor of 4 relative to the paper's discriminant formula 4(B^2 - AC) = -4a^2 in (2.16)). As written, reducing -4a^2 = z_1 C - B^2 modulo z_1 gives B^2 == +4a^2 (mod z_1), not the negative-square congruence B^2 == -4a^2 (mod z_1) in (5.22) [eq:BtoZ1]. Consequently, the reduction to B_1^2 == -1 (mod p^{e-2f}) does not follow from the displayed equation. Using the paper's actual invariant z_1 C - B^2 = a^2 (or B^2 - z_1 C = -4a^2) restores the negative sign in the congruence and repairs the proof.",
      "hypothesis_check": "In the full manuscript, (2.15)-(2.16) define f(x,y) = A x^2 + 2 B x y + C y^2 with discriminant 4(B^2 - AC) = -4a^2, so AC - B^2 = a^2 with a >= T/100 > 0 by (3.3). Under GL(2,Z) equivalence (5.20), z_1 C - B^2 = AC - B^2 = a^2 > 0, so -4a^2 = z_1 C - B^2 is impossible for these forms.",
      "affected_claims": [
        "eq:cKzBnd",
        "eq:eqeq"
      ],
      "revised_status": "not_checked",
      "revised_reasoning": "No later edition was supplied for comparison.",
      "source_quote": "-4a^{2}=z_{1}C-B^{2}",
      "source_span": {
        "char_start": 88785,
        "char_end": 88805,
        "match_mode": "exact"
      },
      "revised_quote": "",
      "revised_span": null
    },
    {
      "finding_id": "finding-5b404ca3d9608b9030c8",
      "outcome": "packet_context",
      "severity": "none",
      "reasoning": "The missing convention requested by the reviewer is supplied in the full manuscript prior to the local excerpt: Section 2.2, equations (2.15)-(2.16), defines f(x,y) = A x^2 + 2 B x y + C y^2 and its discriminant Delta = 4(B^2 - AC) = -4a^2, which is equivalent to AC - B^2 = a^2. Section 5.2 (proof of (5.13)) states the primitivity convention gcd(A,B,C) = 1, and (5.20) defines equivalence via GL(2,Z). In Lemma (5.21), 'determinant -4a^2' refers to this discriminant -4a^2 (as in the preceding Lemma (5.19) and following Proposition (5.25)). The resulting sign and factor-of-4 slip in the target proof's equation -4a^2 = z_1 C - B^2 is accounted for under finding-a2343127af3149f8584e.",
      "hypothesis_check": "Checked against (2.15), (2.16), (5.13), (5.19), (5.20), and (5.25) in the full reviewed source.",
      "affected_claims": [
        "eq:cKzBnd"
      ],
      "revised_status": "not_checked",
      "revised_reasoning": "No later edition was supplied for comparison.",
      "source_quote": "$AC-B^{2}=a^{2}$",
      "source_span": {
        "char_start": 87213,
        "char_end": 87229,
        "match_mode": "exact"
      },
      "revised_quote": "",
      "revised_span": null
    },
    {
      "finding_id": "finding-529308878a6ce74dc1df",
      "outcome": "confirmed_error",
      "severity": "cosmetic",
      "reasoning": "Every equivalence class K of primitive forms representing z_1 primitively contains at least one representative f_1(m,n) = z_1 m^2 + 2 B m n + C n^2 with B in [0, z_1), so the number of such inequivalent classes is bounded above by the number of solutions B (mod z_1) to the congruence. However, it is not equal to that count in general, both because distinct residues B (mod z_1) can yield GL(2,Z)-equivalent forms (for instance B and -B mod z_1 via reflection) and because not every congruence solution need produce a primitive form. Because the target (5.21) is an upper bound and the rest of the proof only bounds the congruence count from above, replacing 'is equal to' with 'is at most' is a cosmetic correction.",
      "hypothesis_check": "Under the GL(2,Z) equivalence defined in (5.20) and primitivity gcd(A,B,C) = 1, unipotent reduction modulo z_1 gives a surjection from a subset of congruence roots onto equivalence classes representing z_1 primitively, justifying an upper bound.",
      "affected_claims": [
        "eq:cKzBnd"
      ],
      "revised_status": "not_checked",
      "revised_reasoning": "No later edition was supplied for comparison.",
      "source_quote": "So the number of inequivalent  such $f_{1}$ is equal to",
      "source_span": {
        "char_start": 88962,
        "char_end": 89017,
        "match_mode": "exact"
      },
      "revised_quote": "",
      "revised_span": null
    },
    {
      "finding_id": "finding-564e1b7bc4a878de48e2",
      "outcome": "unsupported_objection",
      "severity": "none",
      "reasoning": "As the finding itself notes, conditional on the displayed congruence in (5.22), the local root count and the equality of cardinalities in (5.22) are mathematically valid: writing 2a = u p^f with gcd(u,p) = 1, multiplication by the unit u modulo p^e is a bijection between solutions of B^2 == -p^{2f} (mod p^e) and solutions of B^2 == -4a^2 (mod p^e), and the bound 2 min(p^{e/2}, p^f) holds for all primes p (including p = 2). Omitting the routine mention of the unit factor u is standard and not a proof error.",
      "hypothesis_check": "In the manuscript, a >= T/100 > 0 by (3.3), so 2a != 0 and the prime-power factorization p^f || 2a (or p^f || a) is well-defined.",
      "affected_claims": [],
      "revised_status": "not_checked",
      "revised_reasoning": "No later edition was supplied for comparison.",
      "source_quote": "If $2f\\ge e$, then the number of local solutions is at most $p^{e/2}$.",
      "source_span": {
        "char_start": 89178,
        "char_end": 89248,
        "match_mode": "exact"
      },
      "revised_quote": "",
      "revised_span": null
    },
    {
      "finding_id": "finding-184b511ee6718d625fe5",
      "outcome": "unsupported_objection",
      "severity": "none",
      "reasoning": "This finding is a verification note confirming that the reduction to primitive representations of z_1 = z/w^2 and the final divisor/epsilon bounds are valid. Note that in the application (Proposition (5.25)), z = f(m,-n) for a positive-definite primitive form f with A, Casymp T and AC - B^2 = a^2 > 0, and if m = n = 0 then z = 0 contributes only a single term to the sum over m, n < M.",
      "hypothesis_check": "Checked against the hypotheses of Lemma (5.21) ('For a fixed large integer z') and its application in Proposition (5.25).",
      "affected_claims": [],
      "revised_status": "not_checked",
      "revised_reasoning": "No later edition was supplied for comparison.",
      "source_quote": "The number of divisors $z_{1}$ of $z$ is $\\ll_{\\vep} z^{\\vep}$, completing the proof.",
      "source_span": {
        "char_start": 89629,
        "char_end": 89714,
        "match_mode": "exact"
      },
      "revised_quote": "",
      "revised_span": null
    }
  ],
  "chronology": {
    "review_policy": {
      "schema": 1,
      "cutoff_date": "2012-05-24",
      "basis": "journal_received",
      "evidence": "https://link.springer.com/article/10.1007/s00222-013-0475-y : Received 24 May 2012; publisher.html retained locally",
      "status": "recorded",
      "rule": "external-results-available-by-cutoff"
    },
    "note": "Recorded cutoff date is 2012-05-24 (journal received date). All steps in the verification and self-contained repair of eq:cKzBnd use only classical elementary number theory (unimodular completion of primitive vectors, GL(2,Z) invariance of binary quadratic form discriminants, Chinese Remainder Theorem, Hensel's lemma, and elementary divisor bounds) available long before the cutoff. While later knowledge cannot be removed from model weights, no post-cutoff external results are used.",
    "external_results": []
  },
  "origin": "independent_model_arbitration_not_formal_verification",
  "_meta": {
    "pipeline": "proof_arbitration",
    "model": "gemini-4-argon",
    "reasoning_effort": "high",
    "provider": "vertex",
    "batch_job_id": "projects/292312164194/locations/global/batchPredictionJobs/771129973195931648",
    "custom_id": "arbitration-2cdbac8b64168dbcda810092",
    "source_sha256": "d49f6fddb192e57eed39cbe5c62abf4da917100b8a970f47a2a1519415601e46",
    "usage": {
      "candidatesTokenCount": 3514,
      "candidatesTokensDetails": [
        {
          "modality": "TEXT",
          "tokenCount": 3514
        }
      ],
      "promptTokenCount": 69609,
      "promptTokensDetails": [
        {
          "modality": "TEXT",
          "tokenCount": 69609
        }
      ],
      "thoughtsTokenCount": 14985,
      "totalTokenCount": 88108,
      "trafficType": "ON_DEMAND"
    },
    "max_output_tokens": 32768,
    "prompt_profile": "proof_arbitration-v2",
    "prompt_profile_sha256": "fae29551415fcc5477020a7419ce2118906aabc98d7518bc7aeaff0e41c61974",
    "review_policy": {
      "schema": 1,
      "cutoff_date": "2012-05-24",
      "basis": "journal_received",
      "evidence": "https://link.springer.com/article/10.1007/s00222-013-0475-y : Received 24 May 2012; publisher.html retained locally",
      "status": "recorded",
      "rule": "external-results-available-by-cutoff"
    },
    "chronology_prompted": true
  }
}