{ "target_id": "lem:spec0", "proof_status": "defective", "summary": "The supplied sketch does not establish the claimed uniformity in gamma_1: its smoothing inserts a perturbation between gamma_1 and gamma, where conjugation can amplify it without bound. The translated bump also lacks the asserted uniform Sobolev control. This is a defect in the supplied proof, not a counterexample to the lemma. Spectral and external inputs remain conditional.", "used_dependencies": [ "eq:RepDecomp", "eq:decayMtrx", "eq:gl0Qgl0", "eq:specGap", "external:BourgainKontorovichSarnak2010", "external:LaxPhillips1982" ], "missing_context": [ "Definitions of the SO_F norm, its relation to the SL2(C) Frobenius norm, and the relevant Haar-ball growth.", "The complete Sobolev norm definition and a valid uniform estimate for the actual translated vectors.", "Ground-state simplicity, normalized lifting, and spectral decomposition for every intermediate subgroup.", "The exact Lax\u2013Phillips counting theorem and hypotheses needed for the shrinking shell estimate.", "For the right-sided assertion, the norm's inversion symmetry or another explicit reduction.", "Dated external theorem texts available by 2012-05-24. Model knowledge, including possible later knowledge, is not evidence of historical availability." ], "findings": [ { "kind": "error", "source_quote": "both $f(\\g_{1}g^{-1}\\g h)$ and $f(\\g_{1}\\g)$ are $1$, so their difference vanishes.", "explanation": "This does not follow uniformly in gamma_1. Writing k=gamma_1 gamma gives gamma_1 g^{-1} gamma h=(gamma_1 g^{-1} gamma_1^{-1})kh. Small g need not give a uniformly small conjugated perturbation. Thus the asserted inner/outer shell tests with width 10 eta are unjustified. For gamma=gamma_1^{-1}, the unperturbed product is the identity, while the perturbed product contains precisely this uncontrolled conjugation. Membership gamma_1 in Gamma permits reindexing the full-group sum, but does not remove the amplification." }, { "kind": "error", "source_quote": "The functions $\\psi$ are bump functions in six real dimensions, so can be chosen \nto have second-order Sobolev norms bounded by $\\ll \\eta^{-5}$.", "explanation": "The eta^{-5} scaling is appropriate for an unshifted six-dimensional L2 bump with two derivatives. The matrix coefficient, however, also uses Psi_{q,gamma_1}=R(gamma_1)Psi_q. Right-action derivatives of this vector involve Ad(gamma_1^{-1}); no uniform bound for its required Sobolev norm is supplied. Consequently the eta^{-10} product bound, with a constant independent of gamma_1, does not follow." }, { "kind": "context", "source_quote": "Of course the projection onto the base representation is just $[\\G:\\tilde\\G(q)]^{-1}$ times the same projection at level one", "explanation": "This requires a simple ground state whose representation at every intermediate level is exactly the normalized lift from level one. Equality of lowest eigenvalues alone does not establish that assertion. Conditional on this lift statement, the index scaling is natural. The spectral-gap exclusion passes to intermediate subgroups through the covering-space inclusion; their normality is unnecessary." }, { "kind": "unverified", "source_quote": "apply \\cite{LaxPhillips1982}, bounding the count by", "explanation": "The supplied bibliography does not verify the claimed shell estimate, its hypotheses, or its norm normalization. The section switches from SL2(C) to SO_F; transferring the supplied matrix-coefficient decay and integrating it over the stated norm ball also requires the spin-cover norm comparison and Haar-volume growth. These prerequisites are not supplied." }, { "kind": "map_correction", "source_quote": "After proving the lemma below, we will use the argument as a template for the more complicated statement.", "explanation": "lem:spec1 is downstream, not a premise. The family definition eq:fFdef and narrative:105898 also belong to subsequent applications. eq:lemLHS and eq:lemRHS are parts of this target, not independent inputs. Despite proof_anchor_missing=true, an actual sketch is present in source_slice; the anchor flag is a mapping limitation, not an additional proof defect." }, { "kind": "context", "source_quote": "Optimizing $\\eta$ and renaming $\\gT_{0}<\\gd$", "explanation": "Conditional on eq:NqErr with uniform constants, choosing eta=Y^{-epsilon/11} gives Theta_0=delta-epsilon/11. This elementary final step is correct; it does not resolve the preceding uniformity errors." } ], "dependency_proposals": [ { "label": "lem:spec1", "role": "excluded", "source_quote": "a slightly more complicated result will be needed later", "explanation": "A later application of this proof template, not an input." }, { "label": "eq:fFdef", "role": "excluded", "source_quote": "Next we exploit the previous lemma and the product structure of the family $\\fF$", "explanation": "Belongs to the following argument, outside this lemma's proof." }, { "label": "narrative:105898", "role": "excluded", "source_quote": "Such a bound is needed at several places in \\S\\ref{sec:QX}.", "explanation": "A downstream-use reference, not a proof dependency." }, { "label": "eq:lemLHS", "role": "excluded", "source_quote": "Denote the left hand side \\eqref{eq:lemLHS} by $\\cN_{q}$", "explanation": "Target notation rather than an independent premise." }, { "label": "eq:lemRHS", "role": "excluded", "source_quote": "the first term of \\eqref{eq:lemRHS}", "explanation": "Self-reference to the claimed conclusion." }, { "label": "external:BourgainKontorovichSarnak2010", "role": "conditional", "source_quote": "{\\em GAFA}, 20(5):1175--1200, 2010.", "explanation": "available_from: 2010, bibliographic year only; date_evidence: supplied bibliography; date_basis: journal citation; version: cited GAFA article. Needed results are Lemma 3.7's matrix-coefficient identity and Proposition 4.18's comparison argument. Exact statements and hypotheses are uninspected, so applicability is unresolved." }, { "label": "external:LaxPhillips1982", "role": "conditional", "source_quote": "{\\em Journal of Functional Analysis}, 46:280--350, 1982.", "explanation": "available_from: 1982, bibliographic year only; date_evidence: supplied bibliography; date_basis: journal citation; version: cited JFA article. The exact theorem yielding O(eta Y^delta+Y^{delta-epsilon}), including geometric and norm hypotheses, remains unidentified and unverified." }, { "label": "external:KontorovichOh2011", "role": "conditional", "source_quote": "\\cite[Prop. 5.3]{KontorovichOh2011}", "explanation": "available_from: unresolved; date_evidence: citation key only; date_basis: unresolved; version: unknown. The supplied statement requires Theta>1, exclusion of weakly contained complementary series with s>Theta, and smooth vectors. Obtain the dated Proposition 5.3 containing this precise decay and Sobolev formulation; the citation key alone does not establish availability." } ], "_meta": { "pipeline": "claim_review", "model": "gpt-6.1-sol", "reasoning_effort": "high", "provider": "openai", "batch_job_id": "batch_6ad3ffda23ec8190bb2333acdad49cf6", "custom_id": "claim-a9bff874fecc46248f2002d4", "source_sha256": "eb1e65c8c38518070b6d06f1a27bc0ec61db913ce26c1bab66340f4410f44c2a", "usage": { "input_tokens": 17068, "input_tokens_details": { "cache_write_tokens": 17065, "cached_tokens": 0 }, "output_tokens": 5425, "output_tokens_details": { "reasoning_tokens": 3624 }, "total_tokens": 22493 }, "max_output_tokens": 24576, "prompt_profile": "claim_review-v2", "prompt_profile_sha256": "4aa7aa2ef9a18c790385b9e0e0f7ad6c44ff664be253c2d9f790a1469765c3a4", "review_policy": { "schema": 1, "cutoff_date": "2012-05-24", "basis": "journal_received", "evidence": "https://link.springer.com/article/10.1007/s00222-013-0475-y : Received 24 May 2012; publisher.html retained locally", "status": "recorded", "rule": "external-results-available-by-cutoff" }, "chronology_prompted": true } }