Epoch says AI solved second FrontierMath open problem with Fable 5

• AI filled a concrete 2-adic Galois gap: EpochAIResearch's announcement says the result gives a presentation for the absolute Galois group of the field of 2-adic numbers, the second solved item in FrontierMath: Open Problems.
• The result sits below the hype tier: EpochAIResearch's follow-up calls it the first solved problem in the benchmark's "Solid Result" category, with one consulted mathematician saying it would likely be publishable.
• It was a two-model milestone: EpochAIResearch's thread says David Roe first elicited the solution with Fable 5, and David Turturean also elicited it with GPT-5.5 Pro.
• Epoch built Open Problems around checkable research math, and EpochAIResearch's site link points to the benchmark page, where Epoch says the problem set will expand next week.

The Epoch problem page exposes the exact evaluation prompt, including the implicit relation tau^sigma = tau^2 and a modulus of 85667662080 for profinite exponents. The Open Problems about page says the pilot includes only research problems whose proposed solutions can be verified programmatically, with verifier access sold for a fee. In EpochAIResearch's quote card, Roe says Jannsen and Wingberg wrote down the odd-prime case in 1982, while the new p=2 proof counts finite quotients because the odd-prime embedding-problem route breaks at 2.

The deliverable was a presentation of Gal(Qbar_2 / Q_2) as a profinite group. The Epoch problem page frames that as a missing p=2 counterpart to the known explicit presentation of Gal(Qbar_p / Q_p) for p > 2.

The full prompt fixed the output shape:

• Variables: comma-separated generator names.
• Distinguished generators: the first two, conventionally sigma and tau.
• Implicit relation: tau^sigma = tau^2, omitted from the submitted relations section.
• Later generators: contained inside the 2-core, the intersection of all 2-Sylow subgroups.
• Sections: variables, optional definitions, and relations.
• Profinite constants: reductions modulo 85667662080 = 2^8 * 3^2 * 5 * 7 * 11 * 13 * 17 * 19 * 23.

Roe's quote, carried by EpochAIResearch's quote card, says the proof's individual techniques were unsurprising, but the assembly was intricate and the finite-quotient strategy should extend to all 2-adic fields.

Epoch placed the solve in the first occupied slot of its Solid Result category. The category graphic in EpochAIResearch's benchmark note gives the current solved counts:

• Moderately interesting: 1/4 solved.
• Solid Result: 1/5 solved.
• Major Advance: 0/3 solved.
• Breakthrough: 0/3 solved.

The same thread says one mathematician consulted before accepting the problem thought a solution "would certainly be publishable, probably in a pretty good journal." The problem survey lists 2 to 4 serious prior human attempts, a 1 to 3 month estimate for an expert human, and "a standard specialty journal" as the expected publication venue.

Epoch also cooled the victory-lap version. EpochAIResearch's caveat says the 1982 origin should not be read as proof of a major enigma, because the problem was "basically attention-bottlenecked" and Roe expected the answer to be messy.

Epoch said Roe first elicited the solution using Fable 5, while Turturean also elicited it using GPT-5.5 Pro, per EpochAIResearch's thread. Epoch's Claude Fable 5 model page lists Fable 5 as an Anthropic closed-weights model released on June 9, 2026.

Epoch also said the two created explanatory materials, including an interactive formal proof, per EpochAIResearch's thread. That detail matters because the claim reads like a benchmark milestone: prompt, model run, independent elicitation, and proof artifacts.

The community immediately focused on elicitation skill. doodlestein's reply said interpretation depends on how much skill was required to elicit the result.

Epoch's FrontierMath overview says the program has two parts: Tiers 1-4, a benchmark of unpublished expert-written problems, and Open Problems, a collection of unsolved research problems. The Open Problems pilot is the one at issue here.

The Open Problems about page says the pilot began with 14 problems and includes only problems whose proposed solutions can be verified programmatically. It also says verifier access is offered for a fee to help fund benchmark expansion.

The benchmark publishes several pieces engineers can inspect:

• problem significance write-ups,
• precise prompts for AI systems,
• AI attempt records,
• programmatic verifiers,
• mathematician survey metadata on familiarity, prior attempts, human effort, notability, and expected publishability.

In the reaction stream, BlackHC's reply put the hard part at verification and singled out Lean or similar formalisms.

AndrewLampinen argued that recent AI math progress fits a connectionist account where formal symbols are external tools rather than the foundation of intelligence. His essay uses human mathematical practice as the analogy: people write equations, use programs, and externalize symbol manipulation before internalizing useful approximations.

Denny Zhou pushed the workflow implication further: if proofs become cheap, mathematicians' work shifts toward asking deep questions and formulating bold conjectures. The counterpoint came fast, with dilipkay's reply arguing that it is hard to know in advance which questions are deep or bold, since serious mathematicians already aim at those problems.