{
  "target_id": "freesln",
  "proof_status": "defective",
  "summary": "The supplied candidate proof does not establish generic freeness in the symmetrized-ball model. Its odd-dimensional integral estimate is false, and its sector-counting region does not match that sampling model. The exterior-power contraction calculation is locally valid when a_i are genuine singular values, but the quantitative separation and counting inputs remain unverified. These are defects or outstanding obligations in the supplied derivation, not a disproof of Proposition freesln. No later results are used; possible later knowledge in model weights is not treated as evidence.",
  "used_dependencies": [
    "external:BG"
  ],
  "missing_context": [
    "A valid odd-dimensional large-gap integral argument and the lattice-counting transfer, including the statements and applicable hypotheses of emtheorem, wellrounded1, and gapwellrounded.",
    "A quantitative separation estimate for all required signed attracting-point/repelling-hyperplane pairs, and an explicit two-scale limiting argument.",
    "External BG and GO theorem texts; EM and GO version/date evidence establishing availability by 2015-07-28. These external texts have not been inspected.",
    "The earlier n=2 argument, if freesln is intended to cover all n>=2 rather than this section's n>2 scope. Single-element probabilities in the lemmas also require interpreting mu'_X through its marginal."
  ],
  "findings": [
    {
      "kind": "error",
      "source_quote": "\\int_{R\\cap R_{T_1,T_2}} f(j_1,\\dots,j_{n-1})dj_{n-1}\\cdots dj_1\\ll X^{n^2-2\\log X}",
      "explanation": "In the odd case, R_{T_1,T_2} is defined by the desired large-gap inequalities, not their failure. The displayed estimate is false. For n=3, put L=log X and restrict to j_1 in [L-2,L-1], j_2 in [-1/4,1/4], and j_3=-j_1-j_2. For large X this fixed-area region lies in R and satisfies both fixed gap thresholds. The three positive-root sinh factors have product comparable to X^4 there. Thus the integral is at least cX^4, whereas the asserted upper bound tends to zero. Consequently the supplied proof of middletwo does not establish its odd-dimensional case."
    },
    {
      "kind": "error",
      "source_quote": "it is the diagonal matrix with the eigenvalues of $g^t g$ on the diagonal, ordered from highest to lowest.",
      "explanation": "For g=k_ga_gk'_g with orthogonal factors, the diagonal entries of a_g are the positive square roots of the eigenvalues of g^t g. The stated eigenvalue convention is incompatible with KAK and changes the squared gap thresholds. Also, the earlier definition of A uses ascending order, whereas later arguments use descending order. The exterior-power contraction calculation is correct under the genuine, descending singular-value convention, but not under the definitions literally supplied."
    },
    {
      "kind": "error",
      "source_quote": "Let $S_{\\Omega, X}:=\\{\\kappa a\\kappa' \\;|\\; (\\kappa,\\kappa')\\in\\Omega, \\, ||a||<X, ||a^{-1}||>1/X\\}$.",
      "explanation": "This is not the symmetrized-ball sector. For determinant-one matrices, ||a^{-1}|| is at least 1, so its stated lower bound is automatic for X>1. The model instead requires the upper bound ||a^{-1}||<X. Therefore the subsequent inclusion into a union of sets imposing both norm upper bounds is not justified, and its normalized count does not establish prox for the stated sampling model."
    },
    {
      "kind": "unverified",
      "source_quote": "$\\mu(S_i')\\leq \\epsilon'$ since the dimension of the orbit under $K$ of any point in the hyperplane $H_{I}$, where $I$ is identity, is at least $1$.",
      "explanation": "Orbit dimension alone does not provide this uniform measure bound for hyperplanes meeting a shrinking ball. A quantitative incidence/tube estimate is required. Additionally, the final ping-pong step needs simultaneous separation for each generator and its inverse, and for all cross-generator signed point\u2013hyperplane pairs. The supplied prose does not explicitly establish these estimates or the order of limits: contraction at fixed epsilon, separation at a larger scale, followed by that scale tending to zero. GO sector counting and its boundary/well-roundedness hypotheses also remain unverified."
    },
    {
      "kind": "error",
      "source_quote": "\\lim_{X\\rightarrow\\infty}\\, \\mu_X(\\{(g_1,g_2)\\in G^2\\, | \\, \\Gamma(g_1,g_2) \\mbox{ is free}\\})=1",
      "explanation": "The concluding display uses the ordinary-ball measure mu_X, while the proposition concerns mu'_X. The preceding argument, even conditionally on its inputs, addresses the modified model only. This must be treated as a sampling-notation error, not as an established ordinary-ball freeness result."
    },
    {
      "kind": "map_correction",
      "source_quote": "Since free subgroups of $\\textrm{SL}_n(\\mathbb Z)$ are infinite index in $\\textrm{SL}_n(\\mathbb Z)$, and since the generic finitely generated subgroup of $\\textrm{SL}_n(\\mathbb Z)$ is Zariski dense in $\\SLn(\\mathbb C)$ by \\cite{R2}, Theorem~\\ref{thinsln} follows.",
      "explanation": "The source presents freesln as an input toward thinsln, not conversely. The supplied thinsln-to-freesln edge is therefore not a proof-input edge. Thinness alone would not imply freeness. The supplied governing section contains candidate proof prose despite the missing proof anchor; that anchor flag is only a mapping limitation."
    }
  ],
  "dependency_proposals": [
    {
      "label": "thinsln",
      "role": "excluded",
      "source_quote": "As we note below, this is almost Theorem~\\ref{thinsln}.",
      "explanation": "A downstream conclusion, not a premise establishing generic freeness."
    },
    {
      "label": "middletwo",
      "role": "proof_input",
      "source_quote": "Lemma~\\ref{epscontractextpower} together with Lemma~\\ref{middletwo} immediately imply the following:",
      "explanation": "The proposed source of generic contraction. Its supplied proof contains the false odd-dimensional estimate identified above."
    },
    {
      "label": "prox",
      "role": "proof_input",
      "source_quote": "Putting this together with  Lemma~\\ref{middletwo} and Lemma~\\ref{prox}",
      "explanation": "The proposed separation input, not established by the supplied region definition and measure argument."
    },
    {
      "label": "external:BG",
      "role": "conditional",
      "source_quote": "\\newblock {\\em Journal of Algebra}, 261(2):448--467, 2003.",
      "explanation": "Required input: the ping-pong freeness criterion for the stated (r,epsilon)-very proximal pair with r>2epsilon and cross separation, applied on the exterior-power projective space. available_from: 2003, bibliographic year only; date_evidence: supplied bibliography; date_basis: journal publication; version: cited Journal of Algebra article. Availability predates the cutoff, but the exact theorem and hypotheses have not been checked."
    },
    {
      "label": "external:GO",
      "role": "conditional",
      "source_quote": "To prove Lemma~\\ref{prox}, we use Theorem~1.6 from \\cite{GO}",
      "explanation": "Required input: the sector lattice-counting theorem for these symmetrized regions and angular sets, with every regularity hypothesis checked. available_from: unknown; date_evidence: absent; date_basis: unresolved; version: unknown. It is not admitted as an established pre-cutoff input."
    }
  ],
  "_public_provenance": {
    "origin": "returned_native_final_assessment_not_campaign_verdict",
    "model": "gpt-6.1-sol",
    "effort": "high",
    "provider": "openai",
    "manuscript_sha256": "8adb9681e2cd5e4bdcb292162a32796b00ae7b757a5a900fbd580d0bc939e34e",
    "review_input_sha256": "970f56278bb06a579fcb21dadd2f5154f664cc0d486286bb7f76bc5b4790594d",
    "original_assessment_sha256": "60b7d372d71956041d6fb7a14bb298d1aa28c0c8b2198dce2b330a387c2a0db4",
    "note": "Original final assessment fields retained; embedded roadmap and private provider/run metadata omitted. No raw reasoning stream is exported."
  }
}