Epoch's FrontierMath Benchmark Cracks a 40-Year-Old Math Problem Using Fable 5

Epoch AI has announced that AI has produced an explicit presentation for the absolute Galois group of the field of 2-adic numbers, a problem that resisted mathematicians since the early 1980s. It is the second problem solved on FrontierMath: Open Problems, Epoch's benchmark of genuinely unsolved research mathematics, and the first in the "Solid Result" category, meaning it carries real weight in its subfield.

What was actually solved

The absolute Galois group of a field is a single algebraic object that encodes all of that field's extension theory at once: every way you can adjoin roots and build larger number systems. For p-adic fields, which are number systems built around a prime p that capture modular arithmetic in a continuous way, explicit presentations of this group were known for all odd primes since the early 1980s, thanks to work by Jannsen and Wingberg. The prime 2 was the stubborn exception.

No explicit presentation of the absolute Galois group of Q₂ had ever been written down: no concrete generators, no concrete relations. The new result fills that gap, giving an explicit presentation in an enriched, "marked" sense that extends Jannsen and Wingberg's framework for odd primes.

Concretely, the absolute Galois group of Q₂ is the profinite group generated by four marked elements σ, τ, x₀, x₁, subject to the tame relation τ^σ = τ², one explicit wild word relation, and the requirement that the closed normal subgroup generated by the wild generators x₀, x₁ be pro-2. A profinite group is a group constructed as a limit of finite groups, the natural algebraic structure for capturing symmetries of infinite field extensions.

Who found it and how

The problem was proposed by David Roe, a Research Scientist at MIT, who also built the automated verifier used to check submissions. The solution arrived from two independent paths:

• Roe elicited a solution using Fable 5 (Anthropic's Claude Fable 5).
• David Turturean independently found a solution using GPT-5.5 Pro.

Turturean spent much of early 2026 attempting the problem with the strongest models then available, but those submissions failed. In mid-June, he began a ChatGPT Pro conversation that produced a candidate presentation along with an informal proof, with the initial effort running over a 26-hour autonomous session.

Turturean directed much of the work by voice, using voice mode in Claude Code to operate the ChatGPT research harnesses and speech-to-text inside ChatGPT while developing a 60-page manuscript later used for autoformalization. The workflow is worth noting: a human researcher orchestrating AI tools across multiple platforms, not a single model running autonomously.

How the result was verified

Mathematical results found by AI need more than a plausible-looking proof. The team applied three layers of validation:

1. Automated verifier: Roe ran the finite-quotient verifier he built for Epoch AI, and the submission predicted the correct extension count for every test group. Specifically, it confirmed the predicted count of Galois extensions for all 5,402 finite test groups, which is strong evidence, though short of a proof on its own.
2. Lean 4 formalization: Two separately initiated Lean 4 formalizations verify the theorem. Roe's reduces to nine named interfaces to the classical literature; Turturean's reduces to seven external inputs after a documented later adaptation. Lean 4 is a proof assistant that forces every logical step to be machine-checkable.
3. Written manuscript: The team produced an interactive paper with the complete proof, though it has not yet been externally peer-reviewed.

The formalization process did more than confirm the result. Several standing hypotheses, normalization choices, and citations had to be made explicit before the proof became machine-checkable, and those refinements are now part of the manuscript. That kind of forced precision is itself a contribution to the literature.

The benchmark behind the problem

FrontierMath: Open Problems is a benchmark of open problems from research mathematics that professional mathematicians have tried and failed to solve. To allow evaluation at scale, it includes only problems for which proposed solutions can be verified programmatically, a design constraint that rules out problems where correctness is itself a matter of expert judgment.

The benchmark currently lists 15 problems across four tiers of significance. OpenAI is the only entity to have purchased access to the verifiers; it also funded the creation of the original FrontierMath: Tiers 1–4, though Open Problems is developed independently and owned solely by Epoch. The 2-adic Galois group problem is the second to be solved; the first was a combinatorics problem involving hypergraphs.

The problem's difficulty was calibrated carefully. One mathematician consulted before it was accepted into the benchmark said a solution "would certainly be publishable, probably in a pretty good journal." That same mathematician noted the problem was "basically attention-bottlenecked": not a deep enigma, but one that simply hadn't attracted the right sustained effort. Roe himself suggested the main difficulty was that "the answer is likely to be messy." That messiness is precisely where AI tools have an edge.

A signal worth watching

The FrontierMath: Tiers 1–4 benchmark, a separate set of harder structured problems, has seen dramatic score jumps recently. Fable 5 hits 88% accuracy on the hardest tier, up from Opus 4.5 which sat below 10% in early 2026. GPT-5.5 reaches about 75% on the same tier. Those are structured problems with known answers. The Open Problems benchmark is categorically different: at the time of release, none of these problems had been solved by humans or AI systems, so any solution represents a genuine advance in mathematical knowledge.

The 2-adic Galois group result also clarifies what human-AI collaboration looks like at this level. Neither Fable 5 nor GPT-5.5 Pro solved the problem in isolation; both required a mathematician to frame the problem, interpret outputs, run verification infrastructure, and drive the formalization. The full interactive paper and proof materials are publicly available, including a comparison of the two independent presentations each model produced. Epoch AI has also noted an expanded problem set is coming within the week.