{
  "target_id": "singleop",
  "proof_status": "defective",
  "summary": "The singular-value gap alone does not imply the asserted power asymptotic for arbitrary vectors. A direct two-dimensional counterexample satisfies an arbitrarily large gap but makes the proposed leading term negligible compared with the actual result. The meanings of the asymptotic symbols and the permitted dependence on k are also unspecified. This assessment concerns only the lemma, not downstream theorems. No external results are imported; model knowledge itself is not historically date-filtered.",
  "used_dependencies": [],
  "missing_context": [
    "A precise definition of the comparison and asymptotic symbols, including the limiting parameter and whether k is fixed or variable.",
    "Any intended restrictions on w and quantitative lower bounds on its leading projection and on the overlap of the leading left and right singular directions.",
    "If a weaker normalized additive approximation was intended rather than relative asymptotic equivalence, its exact statement and error estimate."
  ],
  "findings": [
    {
      "kind": "error",
      "source_quote": "Let $A \\in M^{n\\times n},$ with $\\lambda_1(A) \\gg \\lambda_2(A),$ and let $w$ be a vector. Then, $$A^k w \\sim \\langle w, v_1(A)\\rangle \\langle v_1(A), u_1(A)\\rangle^{k-1} \\lambda_1^k u_1(A).$$",
      "explanation": "Direct counterexample: let t>1, c=1/t, s=sqrt(1-t^(-2)), and A=diag(t,1)V, where V has rows (c,s) and (-s,c). This is precisely the stated SVD convention with U=I, singular values t and 1, u_1=e_1 and v_1=(c,s). Take the fixed vector w=e_1 and k=2. The displayed expression equals c^2 t^2 e_1=e_1. But direct multiplication gives A^2 e_1=(1-t+t^(-1))e_1-s(1+t^(-1))e_2, whose norm grows like t. Thus the gap tends to infinity, both relevant inner products are nonzero, and the claimed asymptotic fails even for fixed k=2. The example lies within the lemma's stated domain M^{n\\times n}."
    },
    {
      "kind": "unverified",
      "source_quote": "Simple computation.",
      "explanation": "The rank-one part R=lambda_1 u_1 v_1^t does satisfy the displayed power formula exactly. The missing step is controlling the remainder E=A-R, whose norm is lambda_2. Already for k=2, A^2=R^2+RE+ER+E^2. The hypothesis does not ensure that these additional terms are small relative to the displayed term, because that term also contains potentially small overlaps with w and between the singular directions. The counterexample shows that this omission is substantive, not merely an absent written calculation."
    },
    {
      "kind": "context",
      "source_quote": "$\\lambda_1(A) \\gg \\lambda_2(A),$",
      "explanation": "Neither the parameter governing this comparison nor the meaning of the conclusion's asymptotic equivalence is specified. In particular, the statement does not distinguish a fixed-power, increasing-gap limit from a large-k assertion, or provide uniformity in w and the singular-direction overlaps. The counterexample addresses the natural relative-asymptotic reading as the gap increases."
    }
  ],
  "dependency_proposals": [
    {
      "label": "Local SVD notation and singular-vector convention",
      "role": "proof_input",
      "source_quote": "The $\\lambda$'s are the \\emph{singular values} of $A.$ We will denote the first rows of $V$ and $U$ by $v_1(A)$ and $u_1(A),$ respectively.",
      "explanation": "This supplied local definition fixes the meanings of the symbols used in the lemma and counterexample. It is not an external theorem import or an established frozen-map dependency. No additional dependency edge would resolve the demonstrated failure under the stated hypotheses."
    }
  ],
  "_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": "cdf4f9db54a76a31032e01a109e090cf8807952d927e7322716ff1fcbfe858df",
    "original_assessment_sha256": "aeefb26b7710e02cc1def5ee42795386794bdb39bbc9daae98ab0c756965ea88",
    "note": "Original final assessment fields retained; embedded roadmap and private provider/run metadata omitted. No raw reasoning stream is exported."
  }
}