BGS: comparison with the supplied Web audit and a checked local repair

20 October 2026. Reviewed sources: frozen arXiv 0912.5021v1 and separately frozen Acta Math. 207 (2011), 255–290. External-proof cutoff: 29 December 2009. This public supplement records source checks and an attributed local repair. Preparing the comparison and publication used no additional paid inference calls.

What the new evidence changes

The supplied Web GPT Pro audit is substantially useful. Its invariant-component/ground-state argument can be completed into a uniform replacement for the forward expansion-to-spectral-gap comparison. Its rational-versus-geometric irreducibility counterexample and smoothing-factor objection also check out. The earlier Inventiones source really does describe the Chebotarev modification, and was published online before the cutoff.

These findings support a more precise disposition than a single blanket label. The printed §2 argument is defective, but a checked replacement for its forward implication is now available under explicitly stated base-spectral and quotient-expansion inputs. The pointwise arithmetic formula is false, but an earlier cited source identifies the correct averaged route. Several other analytic and quantitative obligations remain open. This comparison does not certify the complete Acta argument, nor does it establish that a headline theorem is false.

The separately supplied Web model identity and conversation history were not authenticated. Its arguments are evaluated as supplied mathematical evidence. They are not counted as another returned native panel assessment.

The OpenAI batch has returned

Sol's whole-paper review was collected on 20 October, 01:27 UTC. Its input was the frozen arXiv v1 text, not the Acta PDF or the other referees' findings. It independently identifies the global/fiberwise orthogonality mismatch, the omitted small-positive-remainder regime, the ground-state normalization error, resolvent-sign and smoothing errors, rational/geometric irreducibility, composite-modulus factors and the sieve-level reciprocal.

Some preprint objections are corrected in Acta. In particular, the separate mass-majorant formulation in published Lemma 7.2 and the corrected Fubini display must not be counted as unchanged journal errors. Sol's overall assessment is therefore not adopted wholesale as a journal-version verdict.

All four whole-paper attempts are now collected: one usable Sol report, and three failed/incomplete reports. The local campaign has 25 accepted reports from 29 Sol attempts. All six arbitration mathematical responses were returned, with separately logged metadata-normalized/source-validated copies; their original invalid-assessment statuses remain unchanged. Usage-based conservative accounting is $9.141407 against the authorized $25, including failed attempts. This is not invoice reconciliation.

Findings compared

A checked replacement for the §2 forward implication

The complete argument, including explicit constants, is in spectral-gap-repair.md. Its hypotheses are a fixed base surface with an isolated positive ground state and a positive scalar gap, and uniform expansion on the non-invariant quotient representations.

First split the section as $F=h e+F_\perp$ with $F_\perp(z)\perp e$ pointwise. The base scalar gap controls $h$. For the other component, divide by the base ground state and use the ground-state transform. Small excess energy forces definite mass in one fixed ball. A weighted Poincaré inequality on one fixed compact connected region containing the ball and its generator translates produces a nearly invariant vector of nonzero norm. Expansion contradicts that conclusion. Summing the scalar Poincaré inequality over components preserves its constant, so increasing fiber dimension does not spoil uniformity.

The written repair replaces the tube construction entirely and handles arbitrarily small positive energy. It proves the forward comparison conditional on the stated inputs. It does not inspect the arithmetic expansion theorem, establish its exact exceptional-modulus range, prove the reverse comparison, or supply the later counting and transfer-operator arguments. This is a coordinating mathematical check of a new self-contained repair, not an assertion that the original proof was already complete or a new formal certificate.

The arithmetic counterexample is valid in the stated ring

Let

In $A/(f)$, the exact identity

shows that $a$ is a unit. Putting $i=b/a$ gives $i^2=-1$, $b=ia$ and $d=a^{-1}+ic$, and these substitutions provide mutually inverse maps with

This is a domain, so $(f)$ is a proper prime ideal and $f$ is irreducible. Also $f(I)=1$, satisfying weak primitivity. For $\Lambda=\mathrm{SL}_2(\mathbb Z)$, direct finite-field counting gives

For split primes there are $2(p-1)$ permitted nonzero first rows and $p$ completions per row. For inert primes there is no permitted first row. Thus the proposed $1/p+O(p^{-3/2})$ formula is false. Infinitely many primes congruent to 3 modulo 4 already suffice for the contradiction; no prime-density theorem is required for this counterexample.

Sol's separate example is $f=a^2+1$. Here $A/(f)\cong\mathbb Q(i)[b,c]$, again a domain. Its local density is 0 at inert odd primes and $2p/(p^2-1)$ at split primes, likewise contradicting (13.1). The exact checker enumerates every determinant-one matrix for ten odd primes through 31 and verifies both formulas; its JSON results preserve the counts. The all-prime arguments are the algebra and row counts above, not finite enumeration alone.

For square-free $m$ composed of split primes, CRT gives

Every factor is at least $8/5$, so this ratio is unbounded as the number of prime factors grows. The usual replacement $C_f^{\omega(m)}m^2\ll_{f,\eta}m^{2+\eta}$ requires corresponding slack in the sieve level. It does not by itself verify the complete sieve application.

The cited arithmetic remedy predates the cutoff

The inspected earlier paper is Affine linear sieve, expanders, and sum-product, Invent. Math. 179 (2010), 559–644. Its title page records publication online on 26 November 2009. On printed p. 635 it explicitly describes removing the absolute-irreducibility assumption by modifying the local formulas over a splitting field and using quantitative Chebotarev for the prime sum. Its earlier setup imposes absolute irreducibility for the simpler pointwise formula.

This is a real, historically admissible repair route, not an appeal to an unwritten later argument. The present check verifies that source passage, not the entire 86-page paper or the quantitative Chebotarev input. The averaged sieve dimension uses the number of distinct rational irreducible factors; a repeated-factor convention must also be made explicit.

At good primes, the natural leading coefficient counts Frobenius-fixed geometric components. A Galois average counts component orbits, so rational factors determine the averaged coefficient. Transferring this to the weighted prime sum uses quantitative Chebotarev and partial summation. These are explicit external proof obligations if one wants a fully audited replacement of the Acta sieve deduction.

Source: earlier BGS paper, printed pp. 574–575 and 635. The downloaded PDF and its hash are retained in this follow-up's acquisition record.

The smoothing multiplier cannot be dropped

For an allowed nonnegative even nontrivial bump $K$, normalized to integral 1, put $k_\gamma(t)=\gamma^{-1}K(t/\gamma)$. If $N(a)\sim c e^{\delta a}$, then for each fixed positive $\gamma$,

Uniformity of the asymptotic over the bounded integration interval follows directly from its definition. Evenness gives the strict inequality by $\cosh(\delta\gamma u)>1$ off zero, and

One need not assume a separate counting theorem to see the inconsistency of a common positive leading coefficient for every fixed smoothing width. Monotonicity gives

If the printed proposition held for all sufficiently small fixed $\gamma$ with the same positive coefficient $C_0$, let $a\to\infty$ and then $\gamma\downarrow0$. The sandwich forces $N(a)\sim C_0e^{\delta a}$. Applying the integral calculation at any fixed positive width then forces coefficient $C_0M_\gamma$, a contradiction. The discrepancy is of main exponential order for fixed width and cannot lie in the printed exponentially smaller remainder.

Thus the factor $e^{-\delta t}$ in (10.5) cannot simply disappear in Proposition 10.2. Retaining $M_\gamma$ or recording an appropriate shrinking-width error is necessary. This corrects one local formulation; the other residue, normalization and inversion issues remain separate.

The Lax–Phillips quotation: precise limit of this check

Acta pp. 264–265 print $O(R^{5/6}e^{R/2})$. The inspected Jakobson–Naud paper, p. 4, attributes to Lax–Phillips the remainder $O(R^{5/6}e^{(\delta+1)R/3})$. These are different exponents, and the latter is still smaller than the main exponent precisely when $\delta>1/2$.

The original 1982 article was not retrieved; publisher access returned 403. Accordingly this comparison records a source-backed quotation discrepancy and an unverified stronger input, not a counterexample to the stronger bound or a reproduction of the original Lax–Phillips proof. The congruence-uniform dependence and secondary spectral terms also remain unchecked. Jakobson–Naud is used here as retrospective diagnostic evidence, not as an imported pre-cutoff proof input.

Source: Jakobson–Naud, On the critical line of convex co-compact hyperbolic surfaces, p. 4.

A further quantitative issue in Sol's returned report

Sol flags the claimed immediate unsmoothing from Proposition 12.1 to Theorem 1.5. This is worth keeping separate from the Web audit's repair of the smoothing coefficient.

Write $b=\log(1/\gamma)$. The printed relative conductor error contains a factor of the form

The usual monotonicity sandwich also displaces the main term by order $\gamma=e^{-b}$. To obtain the stated relative error of order $e^{-c'a/\log a}$ by this route would require $b$ of order $a/\log a$, but then the first expression need not decay: the positive term has order $a/\log a$ and the negative term only order $\log a$. Taking $b$ of order a sufficiently small multiple of $\sqrt a$ instead gives a weaker decaying scale from the displayed bounds. This is a failure of that direct deduction, not a counterexample to the stronger unsmoothed theorem. It remains open in this review, and the local coefficient correction does not settle it.

Later literature and the supplied final verdict

There are concrete later results, rather than merely expressions of confidence. Oh–Winter's Theorem 1.1 assumes a convex cocompact group and gives uniform mixing for square-free moduli coprime to a fixed integer. Their Theorems 1.3–1.4 give corresponding resonance conclusions, and their introduction discusses the small-critical-exponent applications. Soares's Theorem 1.10 gives no new resonances in a fixed half-plane for non-elementary convex cocompact congruence covers at all moduli coprime to a fixed integer.

Inspected sources: Oh–Winter, arXiv:1410.4401, and Soares, arXiv:2101.05757v6, Theorem 1.10. These are later than the historical cutoff and are retained as retrospective context, never justification of the 2009 proof. We inspected the relevant statements and hypotheses, not all their proofs or imported finite-group machinery.

The supplied conclusion that the principal results are established in the literature may be defended by a separate theorem-by-theorem mapping to those later results and admissible inputs. It does not follow merely from repairing the few passages audited here. The present defensible statement is: a checked local spectral-comparison repair is now supplied, the arithmetic error has a documented earlier averaged remedy, and later theorem statements cover significant spectral regimes; the complete original argument and every headline formulation have not been certified by this campaign. Nor has this campaign established that those intended main results are unproved in the literature.

What remains, and provenance

Open obligations include exact imported expansion/flattening versions and proofs; exceptional-modulus and torsion/coding reductions; the reverse comparison; uniform lattice-point counting; transfer-operator/determinant identifications; quantitative unsmoothing; and the corrected complete sieve argument. No claim that a new repair is historically the first correct proof is made.

The follow-up preserves the earlier adjudication, all native responses, original failure statuses and original source snapshots. New sources have hashes and page-separated extraction records. The new mathematical argument is separately attributed; no original response was rewritten, no provider job was resubmitted, and no new referee-generation calls were made.