{
  "target_id": "wellrounded1",
  "proof_status": "defective",
  "summary": "The supplied proof contains a false wavefront containment and an invalid passage from polynomial volume asymptotics to the required inequalities. The stated odd-dimensional volume degree also fails a direct check at n=3. The spectral-norm estimates themselves are sound under the intended interpretation of the neighborhoods. These defects concern the supplied argument, not a general verdict that large symmetrized balls cannot be well-rounded.",
  "used_dependencies": [
    "external:GO",
    "modified",
    "wellrounded"
  ],
  "missing_context": [
    "The actual volume calculation in Section modified, including its exact odd-dimensional exponent and any normalization of p.",
    "The exact statement and hypotheses of Gorodnik\u2013Oh Theorem 2.1 in the cited journal version, especially its regularity restrictions and neighborhood factors.",
    "An explicit parameter range or indexing convention excluding zero-volume initial regions, together with the required uniform quantifier argument.",
    "No external theorem text was inspected or imported to close this obligation. Later-trained model knowledge cannot certify historical availability; the assessment uses the supplied source and direct checks."
  ],
  "findings": [
    {
      "kind": "error",
      "source_quote": "$$\\mathcal N_{1,\\epsilon}\\gamma\\subset k_1U_1aVk_2=k_1U_1Vak_2$$",
      "explanation": "This containment cannot hold as stated. For T>=1, take gamma=I with k_1=a=k_2=I. It would require an open identity neighborhood to lie in U_1V, hence in KA. Matrices in KA have mutually orthogonal columns, whereas I+tE_{12} lies in SL_n(R), approaches I, and has nonorthogonal columns for t!=0. Thus the displayed assertion is false independently of the unavailable external theorem text."
    },
    {
      "kind": "context",
      "source_quote": "The last containment follows from the submultiplicativity of the spectral norm.",
      "explanation": "The subsequent norm estimates are valid for the stated n>2, provided a neighborhood 'of radius r' means {g: ||g||<r}. Indeed, determinant one gives ||g^{-1}||<=||g||^{n-1}. This justifies both inverse-norm bounds; the inward calculation requires the already stated restriction 0<epsilon<1. The meaning of radius should nevertheless be explicit: an ordinary ball defined by ||g-I||<r would not supply the asserted bounds."
    },
    {
      "kind": "error",
      "source_quote": "$m(KC_T'K)\\sim p(T)$ where $p$ is a polynomial of degree $n^2/2$ if $n$ is even and $n(n-1)/2$ if $n$ is odd.",
      "explanation": "The odd-dimensional degree fails already for n=3. Using the Cartan density displayed elsewhere in the supplied excerpts, consider logarithmic diagonal coordinates (log T-u,v,-log T+u-v), with 2<u<3 and 0<v<1. For large T this fixed-area region satisfies the defining ordered bounds. The product of the three positive-root sinh factors is bounded below by a positive constant times T^4: its exponential exponent is 2(j_1-j_3)=4 log T-4u+2v. Consequently the volume is at least cT^4, incompatible with asymptotic equivalence to a polynomial of the asserted degree 3. This is a direct check of the supplied premise, not an imported asymptotic theorem."
    },
    {
      "kind": "error",
      "source_quote": "For any $0<\\epsilon'<1$ one can find $\\epsilon_1>0$ such that $(1-\\epsilon') =p(\\frac{1}{1+\\epsilon_1}$)",
      "explanation": "Even conditional on a positive polynomial asymptotic of degree d, the relevant volume ratio tends to (1+epsilon_1)^d, not to an evaluation of the unnormalized polynomial p. Existence of the displayed solution is not implied by its degree. The analogous equation for epsilon_2 has the same problem. Moreover, equality at the limiting threshold supplies no strict margin for the strict well-roundedness inequalities, and asymptotics alone establish no assertion for every index."
    },
    {
      "kind": "error",
      "source_quote": "Now, given that $\\{KC_T'K\\}_{T\\geq 1}$ is a well-rounded sequence of sets",
      "explanation": "The literal T>=1 formulation conflicts with Definition wellroundeddefn, which requires strict inequalities for every index. At T=1, C_1'={I}, so KC_1'K=K has Haar measure zero. For any open identity neighborhood U, UK contains U and has positive measure. Thus the first defining inequality fails for 0<epsilon'<1. The surrounding text permits an eventual sequence for applications, but that qualification is not part of this displayed conclusion."
    },
    {
      "kind": "map_correction",
      "source_quote": "Recall from Section~\\ref{modified}",
      "explanation": "The modified edge points to a section, not to an identified volume-asymptotic result. Its supplied excerpt introduces the sampling model and a freeness proposition; it does not supply the asserted volume calculation. Likewise, external:GO is a citation obligation, not a verified justification of the displayed containment."
    }
  ],
  "dependency_proposals": [
    {
      "label": "external:GO",
      "role": "conditional",
      "source_quote": "By the strong wavefront lemma (Theorem~2.1 in \\cite{GO})",
      "explanation": "Proposed external input only. Historical record: available_from=2007 at year resolution; date_evidence is the supplied Duke Math. J. bibliography entry; date_basis=bibliographic publication year; version=cited journal article, 139(3):483--525. This predates the 2015-07-28 cutoff, but the exact theorem and hypotheses remain uninspected. No imported version of that theorem is accepted here."
    },
    {
      "label": "modified",
      "role": "conditional",
      "source_quote": "Recall from Section~\\ref{modified} that $m(KC_T'K)\\sim p(T)$",
      "explanation": "The proof requires a precise volume-asymptotic input. The attached section opening does not establish it, and the stated odd-dimensional degree fails the direct n=3 check."
    },
    {
      "label": "wellrounded",
      "role": "proof_input",
      "source_quote": "for all $i$.",
      "explanation": "This is the governing definitional quantifier, not an independently proved antecedent. It makes the missing initial-index qualification and the asymptotic-to-uniform transition material."
    }
  ],
  "_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": "2eee386bd66386200c8d44248a22de2844d1527fe7002590d898b69524a4e07a",
    "original_assessment_sha256": "f9acfc260cf2156dd82bb6bfaf82b52407e6d234d14b63373aca3a585b18a5c5",
    "note": "Original final assessment fields retained; embedded roadmap and private provider/run metadata omitted. No raw reasoning stream is exported."
  }
}