{
  "target_id": "epscontractextpower",
  "proof_status": "defective",
  "summary": "The displayed coordinate estimates support the intended exterior-power contraction lemma for true singular values ordered decreasingly. However, the supplied Cartan conventions identify diagonal entries with squared singular values and also specify the opposite ordering. Read with those conventions, the decomposition and the prescribed attracting point are defective. This is a convention-level defect, not a failure of the exterior-power estimates. The argument needs neither the external BG proposition nor the restriction epsilon<1/4. Only direct finite-dimensional calculations are used here; no later result is relied upon, and model knowledge itself is not intrinsically date-filtered.",
  "used_dependencies": [
    "metricproperty1exterior"
  ],
  "missing_context": [
    "The lemma should specify that k is an integer with 1<=k<n; otherwise a_{k+1} and the asserted hyperplane need not be defined.",
    "A consistent Cartan convention is required: the target's decreasing diagonal convention conflicts with both the ascending chamber and the eigenvalue prescription in the supplied setup.",
    "The exterior-power metric notation must distinguish the outer exterior product in bigwedge^2(W), W=bigwedge^k(R^n), from exterior multiplication in bigwedge(R^n). The basis-induced metric description suggests the former."
  ],
  "findings": [
    {
      "kind": "error",
      "source_quote": "The matrix $a_g$ is uniquely determined by $g$: it is the diagonal matrix with the eigenvalues of $g^t g$ on the diagonal, ordered from highest to lowest.",
      "explanation": "In an orthogonal\u2013diagonal\u2013orthogonal decomposition, the positive diagonal entries are the square roots of these eigenvalues. Orthogonal multiplication preserves singular values, so the stated prescription generally cannot satisfy the displayed decomposition. The target's diagonal reduction instead requires the actual singular values. The analogous parenthetical in dependency gap is also incorrect and cannot establish the required convention."
    },
    {
      "kind": "error",
      "source_quote": "$$A=\\{{\\mbox{diag}}(e^{j_1},\\dots,e^{j_n})\\; |\\; j_i\\in\\mathbb R, j_i\\leq j_{i+1},\\sum_i j_i=0\\}.$$",
      "explanation": "This chamber orders diagonal entries increasingly, whereas the target proof and its attracting point require decreasing order. This matters to the precise conclusion: with gamma=diag(1/100,1,100), k=1 and epsilon=1/8, the true singular-value ratio is 1/100<=epsilon^2. Using the stated ascending chamber with both orthogonal factors equal to the identity gives attracting point [e_1] and hyperplane span(e_2,e_3). The point [e_1+e_3] lies outside the epsilon-neighborhood of that hyperplane, but its image is almost [e_3], not within epsilon of [e_1]. Thus the precise formula fails under the supplied ascending convention."
    },
    {
      "kind": "context",
      "source_quote": "since $||\\gamma v\\wedge e_1\\wedge e_2\\wedge\\cdots\\wedge e_k||\\leq a_1\\cdots a_{k-1}a_{k+1}||v||$, and $||\\gamma v||\\geq a_1\\cdots a_{k}|v_{1,2,\\dots,k}|$.",
      "explanation": "These bounds are correct when a_1>=...>=a_n>0 are actual singular values and the outer wedge is taken in the exterior square of W=bigwedge^k(R^n). Every nonleading basis vector of W has diagonal multiplier at most a_1...a_{k-1}a_{k+1}; the leading coordinate gives the denominator bound. Together with the supplied point-to-hyperplane formula, these prove the displayed distance estimate. For 0<epsilon<1, the nonzero leading coordinate also makes the numerator bound strict, so an open epsilon-ball causes no boundary problem. Epsilon=1 maps the sole possible boundary direction to the attracting point; epsilon>1 has empty outside region."
    },
    {
      "kind": "map_correction",
      "source_quote": "Since $K$ acts by isometries on $\\mathbb P(\\bigwedge^k(\\mathbb R^n))$, we may reduce to the diagonal case,",
      "explanation": "Exterior-power orthogonal invariance is an actual proof input missing from the listed dependency edges. The references to gap and BG describe an analogous proof, rather than supply a necessary theorem invocation. Their external verification is unnecessary for checking these coordinate estimates."
    }
  ],
  "dependency_proposals": [
    {
      "label": "metricproperty1exterior",
      "role": "proof_input",
      "source_quote": "d([v],[\\ker f])=\\frac{|f(v)|}{||f||\\cdot ||v||}",
      "explanation": "Used to convert separation from the coordinate hyperplane into the lower bound on the leading coordinate."
    },
    {
      "label": "exterior_K_isometry",
      "role": "proof_input",
      "source_quote": "The group $K=\\textrm{SO}_n(\\mathbb R)$ acts isometrically on $\\mathbb P(\\bigwedge^k(\\mathbb R^n))$ with the metric $d$.",
      "explanation": "Justifies transporting the diagonal estimate and its attracting and repelling data through the two orthogonal factors."
    },
    {
      "label": "KAK",
      "role": "conditional",
      "source_quote": "\\gamma=a_{\\gamma}=\\textrm{diag}(a_1,\\dots,a_n).",
      "explanation": "The proof requires this diagonal matrix to contain actual singular values in decreasing order. That requirement conflicts with the supplied governing definitions."
    },
    {
      "label": "gap",
      "role": "background",
      "source_quote": "The proof of this is a straightforward generalization of the proof of Theorem~\\ref{gap}, or Proposition~3.1 in \\cite{BG}.",
      "explanation": "A proof analogy, not a necessary premise: the target supplies its own estimates. In particular, gap's epsilon<1/4 hypothesis need not be imported."
    },
    {
      "label": "external:BG",
      "role": "background",
      "source_quote": "\\newblock {\\em Journal of Algebra}, 261(2):448--467, 2003.",
      "explanation": "The supplied bibliographic year precedes the 2015-07-28 cutoff, but the exact proposition and version have not been inspected. No external BG result is imported to close this obligation."
    }
  ],
  "_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": "f5b194192680fa8d95d5af6e26bf273641f735341f0ef43a346ebfda7dd4177c",
    "original_assessment_sha256": "f6c173e6f9b8aceae4d446baa1e09eeed97920bbdeeff45d91c54ff2ea22d812",
    "note": "Original final assessment fields retained; embedded roadmap and private provider/run metadata omitted. No raw reasoning stream is exported."
  }
}