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

Reports and complete public package

The public package has two downloads so a reader can assess the argument before opening any verdict:

Both are required for the complete archive. The panel synthesis is the current source-adjudicated account. The original model reports below retain their historical wording; successful delivery does not mean every objection was accepted. The technical evidence explains accepted defects, rejected objections and local completions.

Dated literature and repair update

The 20 October reconciliation documents the missed 2022 literature connection and the checked lifting repair. Its exact checker and source record are now included in the panel download; the original reports below retain their historical wording.

Historical reports and completed arbitrations

Every accepted assessment

There are 66 local assessments and four arbitration assessments. Files retain visible mathematical explanations and source quotations; private provider envelopes and hidden reasoning are not exported.

Attempt Target Model Assessment
local-001 eq:cIQ3def gpt-6.1-sol JSON
local-002 eq:cKzBnd gpt-6.1-sol JSON
local-003 eq:cRfuIs gpt-6.1-sol JSON
local-004 eq:cSfbnd1 gpt-6.1-sol JSON
local-005 eq:decayMtrx gpt-6.1-sol JSON
local-006 eq:fincK gpt-6.1-sol JSON
local-007 eq:gls gpt-6.1-sol JSON
local-008 eq:modBndaEqAp gpt-6.1-sol JSON
local-009 eq:modBndaNeqAp gpt-6.1-sol JSON
local-010 eq:vgIs gpt-6.1-sol JSON
local-011 lem:GGq gpt-6.1-sol JSON
local-012 lem:kEllD gpt-6.1-sol JSON
local-014 lem:spec0 gpt-6.1-sol JSON
local-015 lem:spec3 gpt-6.1-sol JSON
local-017 lm_local gpt-6.1-sol JSON
local-019 lm_transference gpt-6.1-sol JSON
local-020 narrative:103198 gpt-6.1-sol JSON
local-021 narrative:104579 gpt-6.1-sol JSON
local-022 narrative:105898 gpt-6.1-sol JSON
local-023 narrative:109314 gpt-6.1-sol JSON
local-024 narrative:110408 gpt-6.1-sol JSON
local-025 narrative:112352 gpt-6.1-sol JSON
local-026 narrative:114852 gpt-6.1-sol JSON
local-027 narrative:115416 gpt-6.1-sol JSON
local-028 narrative:118709 gpt-6.1-sol JSON
local-029 narrative:119956 gpt-6.1-sol JSON
local-030 narrative:122162 gpt-6.1-sol JSON
local-031 narrative:123431 gpt-6.1-sol JSON
local-032 narrative:123918 gpt-6.1-sol JSON
local-033 narrative:13050 gpt-6.1-sol JSON
local-034 narrative:144695 gpt-6.1-sol JSON
local-035 narrative:47002 gpt-6.1-sol JSON
local-036 narrative:55546 gpt-6.1-sol JSON
local-037 narrative:68030 gpt-6.1-sol JSON
local-038 narrative:71322 gpt-6.1-sol JSON
local-040 narrative:86590 gpt-6.1-sol JSON
local-041 narrative:92045 gpt-6.1-sol JSON
local-042 narrative:93454 gpt-6.1-sol JSON
local-043 narrative:98161 gpt-6.1-sol JSON
local-044 narrative:99181 gpt-6.1-sol JSON
local-045 th_sg gpt-6.1-sol JSON
local-046 th_sg gemini-4-argon JSON
local-047 thm:CircMeth gpt-6.1-sol JSON
local-048 thm:IQM gpt-6.1-sol JSON
local-049 thm:IQX gpt-6.1-sol JSON
local-050 thm:IQX gemini-4-argon JSON
local-051 thm:Vin gpt-6.1-sol JSON
local-052 thm:cMNtoMNU gpt-6.1-sol JSON
local-053 thm:specGap gpt-6.1-sol JSON
local-054 eq:cI3bnd gpt-6.1-sol JSON
local-055 eq:cI42bnd gpt-6.1-sol JSON
local-056 eq:cIQ12bnd gpt-6.1-sol JSON
local-057 eq:cIQ1def gpt-6.1-sol JSON
local-058 eq:hatRfuBnd gpt-6.1-sol JSON
local-059 eq:nmUpToM gpt-6.1-sol JSON
local-060 lem:spec1 gpt-6.1-sol JSON
local-061 lem:spec2 gpt-6.1-sol JSON
local-062 lm_generation gpt-6.1-sol JSON
local-064 thm:RNU gpt-6.1-sol JSON
local-065 eq:cIQ11bnd gpt-6.1-sol JSON
local-066 eq:cIQ2bnd gpt-6.1-sol JSON
local-067 eq:eqeq gpt-6.1-sol JSON
local-068 thm:IQ0 gpt-6.1-sol JSON
local-069 thm:Main gpt-6.1-sol JSON
local-071 thm:cMNis gpt-6.1-sol JSON
local-073 eq:cI41bnd gpt-6.1-sol JSON
initial-1-01 thm:Main gemini-4-argon JSON
initial-4-01 eq:cIQ12bnd gemini-4-argon JSON
astra-technical-02 lem:spec3 gpt-6-astra JSON
gemini-technical-01 eq:cKzBnd gemini-4-argon JSON

Rejected or incomplete deliveries

These 14 attempts are failures of delivery/validation, not additional mathematical votes. Nonempty visible rejected responses are available for inspection and have not been source-adjudicated as accepted reports.

Attempt Target Model Delivery Visible rejected response
local-013 lem:line:1307 gpt-6.1-sol invalid_assessment Text
local-016 lem:spec3 claude-opus-5-5 provider_failed_or_incomplete No visible assessment returned
local-018 lm_local claude-fable-5-1 provider_failed_or_incomplete No visible assessment returned
local-039 narrative:73512 gpt-6.1-sol invalid_assessment Text
local-063 lm_generation claude-opus-5-5 provider_failed_or_incomplete No visible assessment returned
local-070 thm:Main gemini-4-argon invalid_assessment Text
local-072 thm:cMNis claude-opus-5-5 provider_failed_or_incomplete No visible assessment returned
initial-2-01 thm:RNU gemini-4-argon invalid_assessment Text
initial-3-01 eq:cIQ11bnd gemini-4-argon invalid_assessment Text
astra-technical-01 lem:spec0 gpt-6-astra invalid_assessment Text
gemini-technical-02 lem:GGq gemini-4-argon invalid_assessment Text
gemini-technical-03 lem:spec0 gemini-4-argon invalid_assessment Text
gemini-technical-04 lem:spec1 gemini-4-argon invalid_assessment Text
gemini-technical-05 lem:spec3 gemini-4-argon invalid_assessment Text

Reproduction and provenance

The review ZIP includes all 66 local assignment briefs, all eleven arbitration briefs, the exact arithmetic scripts and frozen outputs, per-attempt accounting, and historical reports. The reading ZIP includes the original arXiv source tarball and the frozen expanded source. Public files are selected by an explicit allowlist; runtime credentials, provider transport envelopes, hidden reasoning and private operational ledgers are excluded.

The original evidence manifest identifies cached external references by hash and URL; their full texts are not part of these two downloads. See the neutral roadmap for the full bibliography and the technical report for the exact references used.

Publication file inventory · Attempt ledger · Coverage and cost details.

Source record

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