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Benefit Ledger · provisional · Mathematics

Claude Fable 5 produces a presentation of Gal(Q̄₂/Q₂)

On 9 June 2026 David Roe elicited a Claude Fable 5 solution to Epoch’s open problem asking for a presentation of the absolute Galois group of the 2-adics. Epoch later accepted verifier output and noted a second, independent GPT-5.5 Pro solution. Epoch warns the verifier is short of a full proof on this problem.

Tier 2 NotableMethodology 0.1
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Current score

+0.33

3 base · Notable (tier 2 of 5, 3 pts)
× 0.7500 attribution · Critical contribution
× 0.5000 evidence · External expert evaluation
× 0.4500 realization · Complete public manuscript
× 0.6500 durability · Useful for many years
Event-level product before credit split: 0.33

An explicit Q₂ Galois presentation is notable (tier 2) until a complete proof is written. Fable 5 was first (0.75). Evidence is Epoch’s verifier plus a later writeup (0.40). Realization is a candidate presentation, not a textbook theorem (0.45). Follow-up ingest: independent source raised evidence; added a follow-up source. Follow-up ingest: added a follow-up source.

What happened

Presentations of Gal(Q̄_p/Q_p) are classical for odd p; p=2 was a documented gap. Epoch lists Fable 5 as first solver, with a later GPT-5.5 Pro writeup by David Turturean. Because Epoch says verifier evidence here is short of a full proof, the event is provisional Notable rather than Major, and Fable 5 keeps first-solver credit rather than splitting equally. Update (2026-06-26): Summary of METR's predeployment evaluation of GPT-5.6 Sol June 26, 2026 A summary of METR's independent, predeployment evaluation of GPT-5.6 Sol Read more. Summary of METR's predeployment evaluation of GPT-5.6 Sol June 26, 2026 A summary of METR's independent, predeployment evaluation of GPT-5.6 Sol Read more Update (2026-09-21): Advisory Group on Mathematics and Artificial Intelligence. OpenAI is working with an independent Advisory Group on Mathematics and Artificial Intelligence to guide the review and communication of emerging AI results.

Model attribution

+0.33

Claude

First system Epoch records as solving the Q₂ absolute Galois presentation problem, elicited by David Roe.

Epoch names Claude Fable 5 as first solved; a later GPT-5.5 Pro solution is independent confirmation, not a co-search on the original run.

Attribution 0.7500 · Credit share 100% · Anthropic

Claims

  • Epoch records a Claude Fable 5 solution, elicited 9 June 2026, to a presentation of the absolute Galois group of Q₂.

    outcome · supported

  • Epoch’s verifier constitutes a complete proof of the presentation.

    significance · disputed

Sources

primary sources

independent sources

Revision history

  • 13 Sep 2026 · 0.00 → 0.26

    Imported events under methodology 0.1.

  • 17 Sep 2026 · 0.26 → 0.33

    Ingest update: independent source raised evidence; added a follow-up source.

  • 21 Sep 2026 · 0.33 → 0.33

    Ingest update: added a follow-up source.

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