{
  "case_id": "case-7bcd03f1ff7867735a8684d3",
  "target_id": "lem:spec3",
  "original_proof_status": "defective",
  "statement_status": "unresolved",
  "statement_argument": "",
  "summary": "The vector-stabilizer and residue-fiber counts have incorrect powers of the modulus, invalidating both displayed spectral-error balances. The lifting argument also does not control coefficient heights after its coordinate change. The separate prime-avoidance objection is unsupported: the existential integral solution can be normalized primitively. The full source supplies the Descartes setting and the T\u2013N relation missing from the earlier report. These conclusions concern the displayed proof of lem:spec3, not a disproof of the lemma or a verdict on the entire paper. No author response or later edition is supplied.",
  "decisions": [
    {
      "finding_id": "finding-1e6b6475f3e54a2cc69d",
      "outcome": "confirmed_error",
      "severity": "substantive",
      "reasoning": "At a split prime p>3, the Descartes form is equivalent to a determinant form. Its nonzero null vectors correspond to rank-one 2\u00d72 matrices, numbering (p\u00b2\u22121)\u00b2/(p\u22121)=(p\u22121)(p+1)\u00b2. Left/right SL2 actions are transitive on them: write a rank-one matrix as uv\u1d57 and move its two nonzero factors separately. The requisite spin image can also be checked from the supplied generators rather than importing Fuchs's theorem: the appendix's first two generators give all elementary matrices over F_p, and its displayed identity with a=1 produces the lower unipotent L(\u22123i/2). Powers and conjugation then give all elementary matrices over F_p[i], which at split primes is SL2(F_p)\u00d7SL2(F_p). Thus the vector-stabilizer index is of order p\u00b3. The p\u00b2 scale describes projective null lines, not the stabilizer actually defined. This also directly contradicts the deduction from eq:Fuc as stated.",
      "hypothesis_check": "The source defines F explicitly, places v0 on F=0, and fixes a primitive integral gasket. Its integral unimodular reflection action implies v0 is primitive, hence nonzero modulo every prime. The subgroup fixes v0 itself, not its projective line. Good prime moduli are included in the lemma's range.",
      "affected_claims": [
        "lem:spec3"
      ],
      "revised_status": "not_checked",
      "revised_reasoning": "No separately labelled later edition was supplied.",
      "source_quote": "[\\G:\\tilde\\G(q)]\\asymp q^{2},",
      "source_span": {
        "char_start": 62829,
        "char_end": 62858,
        "match_mode": "exact"
      },
      "revised_quote": "",
      "revised_span": null
    },
    {
      "finding_id": "finding-e20889d0f893f373cb31",
      "outcome": "confirmed_error",
      "severity": "substantive",
      "reasoning": "An orbit of order p\u00b3 partitioned among p curvature residues necessarily has a fiber of order at least p\u00b2, so the asserted uniform O(p) number of admissible vector cosets is impossible. The appropriate uniform upper bound is O(p\u00b2): after fixing the first coordinate and two further coordinates, the Descartes equation is a monic quadratic in the last coordinate. Writing \u0394=\u03b4\u2212\u03980 and granting the source's coset-counting estimate, the resulting bound is T1^\u03b4(T2^\u03b4/p+p\u00b2T2^\u03980). Obtaining T^\u03b4/p requires p\u00b3\u2272T2^\u0394, not merely the advertised p\u00b2<T2^\u0394. Moreover, at q0\u224dT1^(\u0394/2), the corresponding error q0\u00b2T1^\u03980 is of order T1^\u03b4, providing no power saving. These are substantive failures of the displayed estimates, not harmless changes of notation.",
      "hypothesis_check": "The residue r is unrestricted, so a uniform fiber estimate must cover the large fibers. Prime moduli already expose the failure; no all-moduli approximation theorem is needed for this diagnostic calculation. The analytic counting inputs are granted conditionally, not independently certified.",
      "affected_claims": [
        "lem:spec3"
      ],
      "revised_status": "not_checked",
      "revised_reasoning": "No separately labelled later edition was supplied.",
      "source_quote": "T_{1}^{\\gd}\\\nq\\\n\\left(\n\\frac1{q^{2}}\\\nT_{2}^{\\gd}\n+\nT_{2}^{\\gT_{0}}\n\\right)",
      "source_span": {
        "char_start": 63283,
        "char_end": 63358,
        "match_mode": "exact"
      },
      "revised_quote": "",
      "revised_span": null
    },
    {
      "finding_id": "finding-b50410999141a2199ffd",
      "outcome": "confirmed_error",
      "severity": "substantive",
      "reasoning": "The existence of an integral unimodular coordinate change is not the problem: primitivity permits sending v to a vector with first coordinate 1. The missing implication concerns heights. If the new coordinates are Uv, the row defining the equation becomes e1\u1d57\u03b3U\u207b\u00b9. Its coefficients can therefore have size O(T1\u2016U\u207b\u00b9\u2016), rather than O(T1). The subsequent assertion that the polynomials have height at most log T1 does not follow unless this change is quantitatively controlled. No such bound is supplied in the lifting argument or elsewhere in the reviewed source. This confirms a proof gap, not impossibility of a suitable controlled normalization.",
      "hypothesis_check": "Here v=\u03b32v0 is indeed primitive and Descartes-null, but q may be composite and U may depend on \u03b32 and q. Neither hypothesis supplies the missing height control. The referee's example (2,3,0,0) is not Descartes-null and is not itself a counterexample under the manuscript's hypotheses. Harmless constant factors from the norm convention are distinct from the uncontrolled coordinate change.",
      "affected_claims": [
        "lem:spec3"
      ],
      "revised_status": "not_checked",
      "revised_reasoning": "No separately labelled later edition was supplied.",
      "source_quote": "Since $v$ is a primitive vector, after a linear change of variables we may assume that $(v_{1},q)=1$.",
      "source_span": {
        "char_start": 65464,
        "char_end": 65565,
        "match_mode": "exact"
      },
      "revised_quote": "",
      "revised_span": null
    },
    {
      "finding_id": "finding-fbc15a63a1f198d2135d",
      "outcome": "unsupported_objection",
      "severity": "none",
      "reasoning": "The Claim already implies existence of a primitive integral solution. If S is nonempty and d is the gcd of the coordinates of v*, any \u03b3\u2208S gives z*=e1\u1d57\u03b3v*, hence d divides z*. Dividing both v* and z* by d preserves every equality and makes v* nonzero modulo every prime. If S is empty, its estimate is trivial. Thus denominator size creates no prime-avoidance obstruction. Nor does nonnullness introduce a separate dimensional obstruction: in the determinant/Hermitian finite-field models, nonzero fixed-norm vector orbits have order p\u00b3, while a prescribed first coordinate gives O(p\u00b2) possibilities uniformly. The printed q0 estimate nevertheless remains defective for the incorrect counts and error balance already addressed in finding-e20889d0f893f373cb31.",
      "hypothesis_check": "The normalization uses the integral equality in the Claim and an integral matrix row, not an unproved height estimate. The Claim does not require v* to be null. The conclusions here reject additional normalization and arbitrary-vector obstructions; they do not validate the source's q0-saving.",
      "affected_claims": [],
      "revised_status": "not_checked",
      "revised_reasoning": "No separately labelled later edition was supplied.",
      "source_quote": "Let $q_{0}$ be a         \nprime\n of size $\\asymp T_{1}^{(\\gd-\\gT_{0})/2}$, say, such that $v_{*}\\not\\equiv0(\\mod q_{0})$; then",
      "source_span": {
        "char_start": 64649,
        "char_end": 64775,
        "match_mode": "whitespace_and_outer_quotes"
      },
      "revised_quote": "",
      "revised_span": null
    },
    {
      "finding_id": "finding-fd0b1e0a60459ec7f586",
      "outcome": "packet_context",
      "severity": "none",
      "reasoning": "This concluding contradiction is correct conditional on the displayed integer identity, modular evaluation, and height bound: q divides positive d, whereas d\u226aT1^(10^28) and q\u2265T1^(10^29). Also 8^(4\u00b72\u00b3\u22121)=8^31<10^28. A consistent affine-linear system over Q has a rational solution by elimination, and clearing denominators preserves the nonzero normalized vector. Dropping the family restriction is valid because the summands are nonnegative and the source allows factorization multiplicities. Finally, eq:TXNis supplies T=N^(1/100); together with T=T1^(1+C), it gives q0\u224dN^c with c=\u0394/[200(1+C)]>0. Hence a hypothetical q0\u207b\u00b9 saving would imply a q\u207bc saving for q<N.",
      "hypothesis_check": "The required T\u2013N relation is present in the full reviewed source. These elementary implications neither verify the cited effective Bezout theorem nor cure the missing coordinate-height control or the invalid spectral estimates.",
      "affected_claims": [],
      "revised_status": "not_checked",
      "revised_reasoning": "No separately labelled later edition was supplied.",
      "source_quote": "But then since $\\fd\\ge1$, we in fact have $\\fd\\ge q$, which is incompatible with \\eqref{eq:fdIs} and \\eqref{eq:qToT1}.",
      "source_span": {
        "char_start": 67836,
        "char_end": 67954,
        "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": "The recorded cutoff is 2012-05-24, based on the supplied journal-received record. No external theorem or later proof is imported to decide these findings; the decisive checks are direct algebra and conditional error calculations. The bibliography identifies Fuchs's thesis as 2010, so its date is not wholly absent, but its theorem text remains uninspected. The cited external counting and effective Bezout results are likewise not certified here. Model knowledge may contain later developments; it is not treated as evidence of historical availability or correctness.",
    "external_results": []
  },
  "origin": "independent_model_arbitration_not_formal_verification",
  "_meta": {
    "pipeline": "proof_arbitration",
    "model": "gpt-6-astra",
    "reasoning_effort": "max",
    "provider": "openai",
    "batch_job_id": "batch_6ad4ad7742988190940597f330f69815",
    "custom_id": "arbitration-bb33f63f970879a7529db71f",
    "source_sha256": "e74f3827f49eeb36169c162faaa0c94255f88e26b32a0452d942762df6049031",
    "usage": {
      "input_tokens": 74609,
      "input_tokens_details": {
        "cache_write_tokens": 74606,
        "cached_tokens": 0
      },
      "output_tokens": 22422,
      "output_tokens_details": {
        "reasoning_tokens": 20164
      },
      "total_tokens": 97031
    },
    "max_output_tokens": 65536,
    "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
  }
}