{
  "target_id": "lem:spec3",
  "proof_status": "defective",
  "summary": "The supplied proof contains an incorrect stabilizer-index estimate: conditional on the stated full orthogonal-group reduction, the orbit of a primitive Descartes vector has order p^3, not p^2, at good primes. The accompanying residue-fiber estimate and both spectral-error balances therefore fail as written. There is also an uncontrolled change of variables in the lifting argument. Several elementary transitions are valid, but they do not close these gaps. This is a local proof assessment, not a disproof of the lemma.",
  "used_dependencies": [
    "eq:1",
    "eq:2",
    "eq:3",
    "eq:Fuc",
    "eq:G1G",
    "eq:TT1T2",
    "eq:cCis",
    "eq:fFdef",
    "eq:fdIs",
    "eq:lemRHS",
    "eq:qToT1",
    "eq:wStar"
  ],
  "missing_context": [
    "Explicit definitions of F, v0, the matrix norm and \u03b4; confirmation that v0 is primitive and F(v0)=0.",
    "A valid uniform congruence orbit/fiber estimate and spectral-error calculation covering every q<N and every residue r.",
    "Height-controlled coordinates for the lifting argument and justified normalization of its integral solution.",
    "Exact external theorem statements and hypotheses supporting orbit counting and the effective arithmetic Bezout bound.",
    "A dated source/version record for FuchsThesis. Its availability by the 2012-05-24 cutoff is not established here."
  ],
  "findings": [
    {
      "kind": "error",
      "source_quote": "[\\G:\\tilde\\G(q)]\\asymp q^{2}",
      "explanation": "This contradicts the proposed full-group reduction in the standard Descartes setting. Direct finite-field check: at a good split prime p, the four-variable Descartes form is equivalent to the determinant on 2\u00d72 matrices. Nonzero null vectors are rank-one matrices, numbering (p\u00b2\u22121)\u00b2/(p\u22121)=(p\u22121)(p+1)\u00b2. Left/right SL(2) actions preserve the determinant and act transitively on these matrices. Thus, conditional on eq:Fuc and primitive null v0, the vector-stabilizer index is asymptotic to p\u00b3, not p\u00b2. The distinction between a vector and its projective line is essential."
    },
    {
      "kind": "error",
      "source_quote": "T_{1}^{\\gd}\\\nq\\\n\\left(\n\\frac1{q^{2}}\\\nT_{2}^{\\gd}\n+\nT_{2}^{\\gT_{0}}\n\\right)",
      "explanation": "The factor q also undercounts admissible vector-orbit cosets: an orbit of order p\u00b3 distributed among p curvature residues has average fiber of order p\u00b2. Consequently the stated uniform O(p) fiber bound cannot hold. Using vector cosets instead produces a spectral error of order T1^\u03b4 p\u00b2 T2^\u03980. To dominate this by T^\u03b4/p requires p\u00b3\u2264T2^(\u03b4\u2212\u03980), which is not ensured by the advertised range p<T2^((\u03b4\u2212\u03980)/2). The same problem affects the final q0 argument."
    },
    {
      "kind": "unverified",
      "source_quote": "Since $v$ is a primitive vector, after a linear change of variables we may assume that $(v_{1},q)=1$.",
      "explanation": "Primitivity does not ensure that an existing coordinate is a unit modulo composite q; for example (2,3,0,0) is primitive but has no unit coordinate modulo 6. An integral coordinate change can address this, but its coefficients and the transformed row coefficients must be controlled. No such control is supplied, whereas the later height bound depends on it."
    },
    {
      "kind": "unverified",
      "source_quote": "such that $v_{*}\\not\\equiv0(\\mod q_{0})$",
      "explanation": "An unrestricted denominator-cleared solution need not admit a prime of the prescribed size avoiding its coordinate gcd. A primitive normalization, with integral z*, needs justification. Moreover, Case 1 was formulated for v0, while its reuse here requires a uniform finite-field orbit/fiber estimate for every nonzero reduction of v*. Neither prerequisite is supplied."
    },
    {
      "kind": "context",
      "source_quote": "which is incompatible with \\eqref{eq:fdIs} and \\eqref{eq:qToT1}",
      "explanation": "This contradiction is valid conditional on the height estimate and modular normalization: q divides positive d, while d\u226aT1^(10^28) and q\u2265T1^(10^29), for sufficiently large T1. The numerical exponent check also passes: 8^31<10^28. Dropping the family restriction, rational solvability of a consistent rational linear system, and clearing denominators are valid. Converting a hypothetical q0-saving into a q-saving additionally uses T=N^(1/100), not merely T=T1T2."
    },
    {
      "kind": "map_correction",
      "source_quote": "bound the left hand side of \\eqref{eq:spec3} by",
      "explanation": "eq:spec3 is the target conclusion, not an independent proof input. Equations 1, 2, 3, fdIs, qToT1 and wStar are internal proof stages. The map also omits the essential scale relation eq:TXNis and the other parts of lem:GGq needed for composite moduli and primes 2 and 3."
    }
  ],
  "dependency_proposals": [
    {
      "label": "eq:TXNis",
      "role": "proof_input",
      "source_quote": "T=N^{1/100}",
      "explanation": "Needed to convert a T1-power saving into a uniform q-power saving using q<N."
    },
    {
      "label": "lem:GGq",
      "role": "conditional",
      "source_quote": "the powers of $2$ stabilize at $8$",
      "explanation": "The complete quotient lemma, not eq:Fuc alone, is needed for all moduli. Its cited external source and chronology remain unverified."
    },
    {
      "label": "external:LaxPhillips1982",
      "role": "conditional",
      "source_quote": "46:280--350, 1982.",
      "explanation": "Required result: # {\u03b3\u2208\u0393: ||\u03b3||<Y}\u226aY^\u03b4 for this \u0393 and norm. available_from=1982 (bibliographic year); date_evidence=supplied journal citation; date_basis=journal publication; version=1982 JFA article. Exact theorem and geometric hypotheses were not inspected."
    },
    {
      "label": "external:Hermann1926",
      "role": "conditional",
      "source_quote": "{\\em Math. Ann.}, 95(1):736--788, 1926.",
      "explanation": "Required result: effective integer Bezout bounds for an inconsistent affine-linear system in four variables. available_from=1926; date_evidence=supplied citation; date_basis=journal publication; version=1926 Math. Ann. article. Exact applicable theorem remains unverified."
    },
    {
      "label": "external:MasserWustholz1983",
      "role": "conditional",
      "source_quote": "\\cite[Theorem IV]{MasserWustholz1983}",
      "explanation": "Required theorem: the stated d-bound for integer affine-linear polynomials of height \u2264log T1 with empty complex zero set. available_from=1983; date_evidence=supplied Invent. Math. bibliography; date_basis=journal publication; version=72(3):407\u2013464. Theorem IV and its hypotheses were not inspected."
    },
    {
      "label": "external:BourgainGamburd2009",
      "role": "background",
      "source_quote": "see a related argument",
      "explanation": "An analogy, not a supplied theorem establishing this lifting claim."
    },
    {
      "label": "external:BerensteinYger1991",
      "role": "excluded",
      "source_quote": "Much better bounds are known",
      "explanation": "Optional comparison, not used by the argument. No external theorem has been independently verified or imported here. Model knowledge cannot itself certify historical availability; no later result is used to close an obligation."
    }
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