Analytic number theory

On the Local-Global Conjecture for Integral Apollonian Gaskets

Jean Bourgain and Alex Kontorovich · Appendix by Péter P. Varjú

64 pages46 claims indexed49 referencesManuscript 2013-05-14

Coverage, failures and API accounting

Final collected campaign, 19 October 2026. 84 attempts; 70 usable assessments; no pending requests. Recorded API accounting is $11.377502, within the $25 authorization. No new inference was run for publication. This sums terminal attempt accounting once, including failed deliveries; it is not an invoice reconciliation. Historical reports contain cumulative prior accounting and earlier reservations which must not be added again.

Stage Attempts Usable Accounted USD
Local claim/narrative assessment 73 66 $5.579903
Initial arbitration 4 2 $1.090650
Technical follow-up arbitration 7 2 $4.706949
Total 84 70 $11.377502

Actual model coverage

Model Attempts Usable Accounted USD
gpt-6.1-sol 66 64 $3.201094
claude-opus-5-5 3 0 $1.151704
claude-fable-5-1 1 0 $0.927345
gemini-4-argon 12 5 $3.122896
gpt-6-astra 2 1 $2.974462

Opus and Fable exhausted their allotted output/reasoning allowance without a usable visible report. They supplied no mathematical verdict. Other rejected deliveries failed assessment validation, including source-evidence checks. Available rejected visible text is included for transparency and explicitly labelled; it is not promoted to an accepted assessment.

The first panel assigned 41 claim targets and 25 narrative packets covering the manuscript body. A repeated attempt is not another independent mathematical target. Task assignment does not establish full verification: graph completeness is unverified, external proofs were not comprehensively inspected, and text-only packets did not include figure pixels. The 238-node, 436-edge roadmap indexes 49 external references. Bibliography coverage is not theorem verification.

Independent technical arbitration

Target Astra max Gemini high
lem:spec3 Accepted: substantive proof defects; statement unresolved Invalid assessment
lem:spec0 Invalid assessment Invalid assessment
lem:GGq Not assigned Invalid assessment
lem:spec1 Not assigned Invalid assessment
eq:cKzBnd Not assigned Accepted: local proof correction; statement locally supported

The independent briefs used the frozen full manuscript and original accepted referee findings. Coordinating computations and repair leads were not inserted as author responses. No author response was submitted. Independence of calls is not a guarantee against shared model errors or prior exposure to this published paper.

Chronology and edition limits

External proof inputs are constrained to results available by 24 May 2012, the recorded received date. The reviewed edition itself is arXiv v2 dated 14 May 2013. Model weights cannot be rolled back; the policy constrains cited evidence and does not certify absence of later influence. The journal typesetting was not inspected. Newly supplied self-contained local repairs are attributed as review work, not historical proof inputs.

Machine-readable attempt ledger · Coverage and costs JSON · Complete reports index.

20 October literature and repair follow-up

The dated reconciliation and elementary lifting completion used no additional paid campaign API calls. Historical attempted/completed counts and usage above are unchanged; they do not count the new local completion as another delivered referee report.

Source record

Additional supporting archive items are identified in the original report. View the manuscript hashes and repository snapshot.