Preliminary audit—repair verification ongoing

Fuchs–Rivin: source-adjudicated progress report

21 October 2026. Several local statements are false; the main generic-thinness theorem has not been refuted or completely repaired by this campaign.

Scope and editions

The native review used arXiv 1506.01735v1 (4 June 2015), with the documented received-date cutoff of 28 July 2015. The published IMRN article has a substantially revised Section 2. Its complete text was not available to the native panel; the publisher's indexed excerpts do not constitute a complete journal audit. In particular, native Lemma 2.2 is journal Lemma 2.5. A separate author PDF and the journal excerpts are frozen as distinct sources.

Sol returned the full-paper assessment and all thirty-two local/prose packets. Gemini returned three local assessments and two accepted arbitrations. Three Opus attempts hit their output caps. The external Astra review supplied by the user was excluded from the initial native prompts; its proposed repairs are not independently verified merely because the panel repeats them.

False numerical bound

Theorem 3.4 is false even after interpreting its displayed ratio of divergent limits as the intended limit of normalized measures. In dimension three put L=log X and use j₁=L−x, j₂=L−x−y, j₃=−2L+2x+y. The chamber is x,y≥0 and 3x+2y≤3L. The rescaled positive Cartan density converges to ¼exp(−6x−3y)sinh y and is dominated by that same integrable expression. Thus the limiting gap y has probability density 8exp(−3y)sinh y and tail

At T=2log η this is $2\eta^{-4}-\eta^{-8}$, strictly larger than the stated η⁻⁴ for every η>4. At η=5 the exact probability is 1249/390625, while the proposed bound is 1/625. The integral identity and normalization were checked symbolically; the dominated-convergence argument just given supplies the limiting-measure justification. This refutes a particular quantitative assertion, not the headline thinness result.

False product lemma

Lemma 5.2 fails for A=[[13,8],[8,5]] and B=[[13,−8],[−8,5]], the cubes of symmetric positive definite integral determinant-one matrices. Both complete singular spectra are {9+4√5,9−4√5}. All four forward/inverse operator norms equal λ=9+4√5. The self overlaps equal one and the squared cross overlap is 1/5>1/9, satisfying ε=1/3.

For the reduced word A⁻¹, however, its action on the specified expanding unit vector u₁(A) has norm λ⁻¹<λ/3, contradicting the claimed lower bound. Powers give arbitrarily large singular gaps. This exact counterexample also exposes the incorrect inverse-singular-vector identification in the proof. A correct cone argument must allow an initial vector appropriate to the last letter and control all permitted generator/inverse transitions uniformly in word length.

False reduced-length estimate

Fact 5.3 is refuted under the intended uniform independent four-letter random-product model. Away from the identity, the reduced length has outward probability 3/4 and inward probability 1/4, hence drift 1/2. The accepted Gemini arbitration supplies a direct refutation without relying on a quoted limit theorem: putting q=√3/2 gives P(L_t=0)≤q^t, so E[L_n]≤n/2+1/(2(1−q)). Markov's inequality implies limsup P(L_n≥3n/4)≤2/3, contradicting the printed limit one.

The usual almost-sure speed is 1/2. To see this, the summable return probabilities imply only finitely many returns almost surely. Subtracting the conditional drifts produces a martingale with bounded increments, whose value divided by n tends to zero by Azuma's inequality and Borel–Cantelli. The finitely many identity times contribute o(n); therefore L_n/n→1/2. This is an argument about uniform unreduced input words; a different sampling law needs a different statement.

Generic freeness and the main theorem

The second accepted Gemini arbitration finds the supplied proof of Proposition 3.5 defective, with its statement unresolved. It confirms the impossible odd-dimensional bound X^(n²−2log X) for an integral over an expanding region containing a fixed positive-measure subset, and identifies missing angular/ping-pong justifications. Cosmetic reversed inequalities and notation are kept separate from those substantive omissions.

The external Astra reconstruction proposes the corrected symmetrized-ball exponent floor(n²/2), partial-flag sector counting, and a uniform cone argument for the Lyapunov conclusion. These are plausible routes requiring substantial checks. They are not yet a completed campaign repair. In particular, one must verify the counting theorem for the actual sampling regions, all inverse-generator incidences, the order of the two limiting parameters, and Zariski density in symmetrized rather than ordinary balls.

The later Bulinski–Ostafe–Shparlinski paper (arXiv 2304.10980, published 2025) explicitly disproves the published fixed-separation lemma and supplies an independent quantitative SL₂ freeness argument. This is retrospective corroboration, not a proof input available at the 2015 cutoff. It does not on its own repair the higher-dimensional argument or every other conclusion.

The new journal trace-counting proof also deserves its own edition-specific audit. The proposed supporting bound r₂(N)≪log N fails, since r₂(65^m)=4(m+1)². That observation alone does not refute the journal lemma's stated counting bound. The incomplete journal text is a real scope limitation, not something to fill in from the arXiv edition.

Evidence and decision graph

The earlier executable exact checker and results record the matrix counterexample, integral identities, and finite drift calculation. The source-bound decision records and decorated graph at the top of this panel page distinguish refuted local statements from defective proofs of statements whose truth remains unresolved. Dependencies are explicitly audited; missing edges remain a limitation of the partial graph. None of the three false local assertions is used to infer that the main theorem is false.

Cost and remaining work

All paid work is terminal: 41 attempts, 38 usable outputs (36 initial assessments and two accepted arbitrations), and three capped/incomplete attempts. Recorded usage-based cost ceiling is $4.0776285 against $10 authorized, including failures; it is not invoice reconciliation. This adjudication incurred no additional API usage.

The next step is a substantive correction audit of the volume and angular-counting replacements, followed by the uniform cone/Lyapunov argument and complete journal Section 2 comparison if that edition becomes available. The current evidence supports publishing precise local defects with their scope; it does not support calling the main generic-thinness theorem false or its proposed replacement proof fully checked.

Reports and reproducibility

Independent model outputs and attempt coverage · Coverage and costs · Exact-check instructions. All native calls are terminal. Publication does not imply that the proposed corrections have been fully verified.