Bourgain–Kontorovich: review and source adjudication

Originally published 19 October 2026; updated 20 October 2026. This audit finds substantive defects in the supplied proof of the residue-distribution lemma lem:spec3, together with additional source-level defects described below. The density-one theorem remains unresolved by this audit. We have no counterexample to that theorem and no checked repair of its complete proof.

The reviewed text is arXiv:1205.4416v2, 14 May 2013, 64 pages, including Péter P. Varjú's appendix. The 2014 Inventiones journal typesetting was not inspected; its identity with this snapshot is not assumed. External proof inputs must have been available by the journal received date, 24 May 2012. New self-contained local completions are separately attributed. These are AI-assisted source checks and model assessments, not a human referee verdict or a formal proof certificate.

Dated update: a published correction and a checked local repair

The coordinate-height defect was explicitly documented by Kontorovich–Zhang in 2022. Our original literature search missed that connection. Their later paper supplies a corrective mechanism; the lifting Claim in this four-coordinate argument now also has a checked elementary determinant proof. The lifting step is locally repaired; the other orbit-counting and analytic objections remain separate. The complete density-one proof is still unresolved in this audit.

Read the objection-by-objection reconciliation, full local proof and remaining obligations. This update cites the public paper and preserves the original 2012 received-date cutoff; the later correction is not retroactive justification of the reviewed text.

Start with the evidence

Read the technical evidence and reproduction instructions. The decorated graph links each recorded decision to its explanation. Red concerns the specified supplied proof or local assertion; it does not declare every descendant theorem false. Unadjudicated claims stay yellow, and incomplete dependency extraction does not establish independence.

What the strongest finding says

In lem:spec3, the manuscript fixes a vector modulo a prime but uses the scale for its projective line. At split primes the nonzero null-vector orbit has order $p^3$, while the printed index is of order $p^2$. A fixed-coordinate fiber has scale $p^2$, rather than the printed $O(p)$. These powers enter both spectral-error balances; replacing them changes the required modulus cutoffs. Astra's accepted arbitration gives an all-prime generator argument, so this conclusion does not depend on trusting Fuchs' strong-approximation proof or extrapolating a few finite computations.

The same arbitration identifies missing height control after the coordinate change. The dated reconciliation records the 2022 acknowledgment and the separately checked replacement for the lifting Claim. It also rejects a separate prime-avoidance objection and resolves several apparent omissions using context in the full manuscript. Those rebuttals are included in the complete report; the original referee's objections are not all adopted.

Local repairs and remaining work

The quadratic-form counting slip has a local repair under the source's own normalization $AC-B^2=a^2$. The dyadic summation and small-exponent adjustments are likewise local completions, conditional on the upstream analytic and arithmetic estimates. None supplies the missing global proof. Proposed changes to smoothing placement, modulus cutoffs or subgroup conventions remain repair leads until their hypotheses, quantitative losses and downstream applications are checked.

An earlier arbitration treated missing explicit constants as a substantive effectivity objection. The current synthesis keeps effectivity unverified: absence of a numerical value does not prove a theorem ineffective. Earlier reports remain available as historical model assessments and are superseded by this dated synthesis where they differ.

Complete public package

Use the separate reading packet for the frozen PDF, expanded and original native sources, source hashes and neutral internal/external roadmap. Use the curated review package for this synthesis, decorated decision graph, all 70 accepted assessment records, the available visible rejected responses, historical reports, assignment briefs, exact scripts/results and coverage/cost records.

Reports and package index · Coverage and costs · Source comparison with Bourgain–Fuchs.

The first panel delivered 66 usable local assessments from 73 attempts; four of eleven arbitration attempts were usable. Opus and Fable returned no usable assessment in this campaign. Delivery/schema acceptance is not mathematical endorsement. No author response was submitted to these arbitrations. The review makes no claim of author agreement with its findings.