# BGS: false intermediate assertions and a checked local spectral repair

**20 October 2026 · Acta Mathematica 207 (2011), 255–290 · arXiv 0912.5021v1.**

**The supplied proof contains substantive defects. A replacement for the forward expansion-to-spectral-gap comparison is now checked under explicit inputs; the complete proof chain remains unresolved in this review.** We found no counterexample to the intended main spectral-gap or qualitative affine-sieve conclusions. This campaign also does not establish that those results are unproved elsewhere.

The useful distinction is between the printed local assertions, the intended theorems, and the particular replacements that have actually been supplied. Several intermediate statements are false as written. Some defects have checked local corrections, while other proof inputs and quantitative transitions remain open.

## Read the sources and argument first

The primary independent panel input was the [36-page arXiv v1 PDF](../paper-arxiv-v1.pdf) and its [native source](../manuscript-expanded.tex), dated 26 December 2009. We separately inspected consequential passages in the [Acta typesetting](../journal-tsinghua-mirror.pdf). The journal records receipt on 29 December 2009 and revision on 6 March 2011. These editions are not identical; the complete proofs have not been collated line by line.

The neutral reading map records the proposed path from expansion to spectral and counting results, then to sieve estimates. Its arrows identify source-supported proof inputs or separately typed references. They are not a correctness certificate. Node labels explicitly distinguish native IDs from corresponding Acta numbering.

External proof inputs are constrained to the received-date cutoff, **29 December 2009**. A newly supplied self-contained repair is separately attributed. Later literature is retrospective context, not evidence that the original argument was complete.

## What failed, what was repaired, and what remains open

| Passage in Acta | Finding | Current consequence |
|---|---|---|
| §2, (2.1), p. 259 | Global orthogonality to the ground state does not imply pointwise orthogonality to the invariant fiber. The printed expansion assertion has an explicit counterexample. | The original inference is false. The new repair separates the invariant component. |
| §2, p. 260 | The proof handles a fixed positive scalar remainder and then exactly zero, omitting arbitrarily small positive remainders. | A compact-region, weighted Poincaré argument now supplies a uniform replacement for the forward comparison. |
| §3, pp. 264–265 | The unit ground state on a degree-$d$ cover needs normalization $d^{-1/2}$, rather than the printed $d^{-1}$. The lattice-point error and its uniformity require separate verification. | The normalization is locally corrected. The complete congruence-counting proof is not certified. |
| §10, pp. 281–282 | The resolvent residue has the wrong sign, and the smoothed counting coefficient loses its dependence on the kernel width. | The literal formulas are false. Local corrected formulations are described; the full inversion/counting chain remains open. |
| §13, (13.1), p. 287 | Rational irreducibility is used as though it implied geometric irreducibility. Two elementary examples contradict the pointwise local-density formula. | The formula is false. An earlier BGS source explicitly describes a Chebotarev route to the averaged sieve condition. |
| §13, pp. 286–288 | The abstract sieve theorem omits a level-scale hypothesis; a composite-modulus bound loses a prime-divisor factor; the allowable exponent is reciprocated. | These statements need correction. A complete corrected sieve argument has not been assembled here. |
| §12 to Theorem 1.5 | The claimed quantitative unsmoothing does not follow by the usual sandwich argument from the displayed estimates. | Open quantitative deduction; correcting the smoothing coefficient alone does not settle it. |

The [detailed source comparison](comparison-and-repair.md) gives the arguments, scope and limitations. The [independent arbitration analyses](independent-analyses.md) preserve the returned assessments and distinguish original rejected deliveries from the separately source-validated copies.

## The checked spectral repair

The [complete replacement argument](spectral-gap-repair.md) proves the forward expansion-to-gap comparison with constants independent of the covering degree and representation dimension. It assumes a fixed base surface with a positive isolated ground state, a scalar gap above it, and uniform expansion of the actual finite-quotient representations.

Write $F=h e+F_\perp$, where $e$ is the invariant unit vector and $F_\perp(z)\perp e$ pointwise. The scalar gap handles $h$. For the other component, divide by the ground state and use the ground-state transform. Small excess energy forces mass into a fixed ball; a weighted Poincaré inequality on a fixed connected compact region containing that ball and its generator translates then produces a nonzero nearly invariant vector. Expansion excludes it. All constants are independent of fiber dimension.

This bypasses the tube argument and covers the omitted small-positive-energy regime. It repairs the **forward comparison**, not the entire equivalence, the arithmetic expansion input, its exceptional-modulus range, uniform lattice-point counting or the transfer-operator and sieve machinery. The public graph keeps those scopes separate.

The construction came from a user-supplied Web GPT Pro argument and was written out and checked in the coordinating audit, with a separate agent subsequently checking its mathematical scope. The external model identity was not verified. This is an attributed mathematical repair, not a formal certificate or a retroactive claim that the original panel had already supplied it.

## Explicit arithmetic counterexamples

For $\Lambda=\mathrm{SL}_2(\mathbb Z)$, the polynomial $f=a^2+b^2$ is irreducible in the specified ring $\mathbb Q[a,b,c,d]/(ad-bc-1)$ and satisfies $f(I)=1$. Yet its zero density at an odd prime is

$$
\beta(p)=\begin{cases}
0,&p\equiv3\pmod4,\\
2/(p+1),&p\equiv1\pmod4,
\end{cases}
$$

contradicting $1/p+O(p^{-3/2})$. Sol independently found the same phenomenon using $f=a^2+1$, whose split-prime density is $2p/(p^2-1)$ and whose inert-prime density is zero. The [exact checker](exact-arithmetic-checks.py) and [results](exact-arithmetic-results.json) enumerate all determinant-one matrices for ten odd primes. The all-prime proofs and coordinate-ring arguments are in the detailed comparison.

The earlier BGS *Inventiones* paper, published online on 26 November 2009, discusses on p. 635 the modification needed when rational factors split geometrically: use a splitting field and quantitative Chebotarev for the averaged prime sum. We checked that passage and its pre-cutoff date. It is a documented earlier repair route, not a verification of the complete 86-page paper or every step of the corrected affine-sieve application.

## Edition corrections and rejected objections

The published Lemma 7.2 permits a separate mass majorant and §8 uses it. That fixes the preprint's mismatch between a mass bound and actual total variation. The journal also corrects the preprint's unsliced Fubini display. These are not unchanged errors in Acta.

Full-source arbitration resolves several packet-context objections, including omitted definitions and some coefficient/exponent bookkeeping. Proposition 12.1's conductor summation survives the local check conditional on its inputs; its inherited constant-mode and smoothing issues remain separate. Neither a returned reviewer objection nor absence of a passage from a bounded packet is automatically adopted as a manuscript defect.

## Present-day literature and the historical proof

Later work supplies concrete spectral theorems with stated hypotheses. Oh–Winter treats convex cocompact congruence covers at good square-free moduli; Soares's Theorem 1.10 treats all moduli coprime to a fixed integer in its convex cocompact setting. We inspected those statements, not their complete proofs or every imported finite-group result. They must not be silently substituted into the 2009 argument.

Our conclusion concerns the audited proof and specified repairs. It neither certifies all headline formulations from those later citations nor asserts that the intended main results lack proofs in the literature. See the detailed comparison for the exact source links and remaining obligations.

## Coverage, failures and cost

There were **39 native generation attempts**: four whole-paper, 29 local/narrative, and six arbitration attempts. One whole-paper Sol report and 25 local Sol reports passed their delivery validators. Opus, Fable and Gemini's whole-paper attempts exhausted output allowances and supplied no usable complete reviews. Four local Sol deliveries failed validation.

All six arbitration responses contained complete mathematical analyses but originally failed metadata/schema checks. Separately logged metadata-normalized copies pass the unchanged strict source validator; the original statuses and mathematical text are preserved. They are not six originally valid deliveries, and source matching is not a correctness certificate.

Total conservative usage-based accounting is **$9.141407**, including failures, against the **$25** authorization. This is not invoice reconciliation. Publication makes no new referee calls.

The [coverage and cost report](coverage-and-cost.md), attempt ledger, selected assessments, arithmetic checker/results and provenance records are included in the curated review download. The neutral reading packet remains separate. Uninspected references, unresolved chronology, missing proof coverage and remaining deductions are recorded rather than inferred to be correct.