{
  "target_id": "gapwellrounded",
  "proof_status": "defective",
  "summary": "The supplied proof does not establish the stated well-roundedness lemma, even conditional on wellrounded1. It identifies the wrong gap faces, omits an additional face in odd dimension, and asserts an unjustified absolute Haar-volume bound where relative boundary control is needed. These are defects in the local argument, not a disproof of the lemma. The assessment uses elementary checks and supplied premises; no external result is imported to close the gaps.",
  "used_dependencies": [
    "wellrounded1",
    "polyR2",
    "middletwo",
    "weirdfirstinteven",
    "weirdfirstintodd"
  ],
  "missing_context": [
    "A noncircular continuous Haar-density argument for the correctly defined middle-gap regions, independent of lattice counting that invokes gapwellrounded.",
    "A uniform normalized Haar-volume estimate for neighborhoods of every additional K-saturated gap face, with n>2 and fixed beta>16.",
    "Control reconciling epsilon-dependent large-T estimates with the lemma's single-tail quantifier.",
    "If the cited EM equivalence of well-roundedness definitions is used, supply its exact hypotheses and a dated source/version available by 2015-07-28. External texts were not inspected; available_from, date_evidence, date_basis, and version remain unresolved.",
    "The frozen middletwo premise and wellrounded1 premise are not certified here. Model training knowledge cannot itself be chronologically filtered; no later theorem or revised proof has been used."
  ],
  "findings": [
    {
      "kind": "error",
      "source_quote": "the hyperplane $j_1=j_2+\\beta$",
      "explanation": "With j_i=log(alpha_i), a multiplicative gap beta becomes an additive gap log(beta), not beta. For even n=2k, the intended new face is j_k-j_{k+1}=log(beta); for odd n=2k+1, there are two faces, j_k-j_{k+1}=log(beta) and j_{k+1}-j_{k+2}=log(beta). The section assumes n>2 and fixed beta>16, so the asserted single top-gap face is not a valid description."
    },
    {
      "kind": "error",
      "source_quote": "the boundary of $KC_{T,\\beta}'K$ is the union of part of the boundary of $KC_T'K$ and a polygon $P_T'$ sitting inside the hyperplane $j_1=j_2+\\beta$.",
      "explanation": "This decomposition inherits the incorrect indices and logarithmic parameter, and misses the second new face for odd n. Moreover, polyR2 describes logarithmic Cartan coordinates, not the group region itself as a Euclidean polytope. Boundary estimates must concern the K-saturated faces and Haar measure, including the Cartan Jacobian."
    },
    {
      "kind": "error",
      "source_quote": "$m(U'P_T')<R\\epsilon/2$ for all $T>N'$",
      "explanation": "This is an absolute bound uniform in T, substantially stronger than the required normalized estimate. It fails for fixed identity neighborhoods on the relevant growing faces. For example, in n=3, put L=log(T), b=log(beta), and consider the face j_1-j_2=b with j_3=-L+O(1). Then j_1,j_2=L/2+O(1). A fixed-thickness logarithmic tube has Cartan density comparable to exp(3L), since the other two positive-root gaps are 3L/2+O(1). Its Haar volume therefore grows at least on the order of T^3. Separation from the dominant vertex does not imply bounded absolute tube volume."
    },
    {
      "kind": "unverified",
      "source_quote": "Now, by the proof of Lemma~\\ref{middletwo}, we have that for any $\\eta>0$, there is some $N\\in\\mathbb N$",
      "explanation": "The supplied middletwo statement concerns normalized lattice counting, whereas density1 concerns Haar volume. Its continuous-volume proof is not supplied here. The attached reference excerpt explicitly uses gapwellrounded when transferring volume ratios to lattice ratios, so the lattice conclusion cannot simply be fed back into this proof. A direct continuous-density argument could be an admissible input, but remains unverified. Also choose 0<eta<1 to obtain the required R>0."
    },
    {
      "kind": "unverified",
      "source_quote": "for all $T>\\max(N,N')$, as desired.",
      "explanation": "N' is chosen after epsilon, whereas the lemma asserts one tail T>N that is well-rounded under the supplied definition, which requires control for every index in that tail. The proof needs either uniform control of the remaining bounded T-range or an applicable equivalence with an asymptotic definition. Merely obtaining an epsilon-dependent tail does not establish the stated quantifiers. The final arithmetic also tacitly requires m(KC_T'K)>1, obtainable by enlarging a threshold if volume divergence is assumed."
    },
    {
      "kind": "map_correction",
      "source_quote": "\\begin{proof}",
      "explanation": "Despite proof_anchor_missing=true, the supplied target slice contains an explicit proof environment. That flag is a mapping limitation, not an additional mathematical defect. The middletwo edge should distinguish its unavailable continuous argument from its supplied counting statement; the exponential-integral edges provide background, not the missing uniform tube estimate."
    }
  ],
  "dependency_proposals": [
    {
      "label": "wellrounded1",
      "role": "conditional",
      "source_quote": "Then the sequence $\\{KC_{T}'K\\}$ of regions (whose volume tends to infinity as $T\\rightarrow\\infty$) is well-rounded.",
      "explanation": "This supplies outer-boundary control only, conditional on the lemma and the asserted boundary-definition equivalence."
    },
    {
      "label": "middletwo",
      "role": "conditional",
      "source_quote": "By the same argument as in the proof of Lemma~\\ref{middletwo}",
      "explanation": "Only a direct continuous-volume portion of that proof could support density and boundary estimates here. Its supplied counting conclusion is insufficient."
    },
    {
      "label": "polyR2",
      "role": "background",
      "source_quote": "e^{|j_1|}\\leq X",
      "explanation": "Identifies logarithmic radial coordinates and norm bounds; it does not identify the added middle-gap faces."
    },
    {
      "label": "weirdfirstinteven / weirdfirstintodd",
      "role": "background",
      "source_quote": "Again we expand this as a sum of integrals of exponentials.",
      "explanation": "These formulas are proposed asymptotic background, not a demonstrated neighborhood-volume estimate. No full verification of these referenced expansions is asserted."
    },
    {
      "label": "wellroundeddefn2",
      "role": "proof_input",
      "source_quote": "\\frac{m(U\\partial B_n)}{m(B_n)}<\\epsilon",
      "explanation": "This supplied criterion specifies the actual remaining obligation: normalized boundary-neighborhood control, uniformly over the claimed tail."
    }
  ],
  "_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": "6360c6c753f1e11abbc01e6146e738bcb2a6ccee60fcb831350da7e6634c20c4",
    "original_assessment_sha256": "e8a09629894900e0e88408b4a36bd4ba12a8ab9a1094589ad9bd084082cc2b42",
    "note": "Original final assessment fields retained; embedded roadmap and private provider/run metadata omitted. No raw reasoning stream is exported."
  }
}