Reconstruction of function fields from Milnor K-theory and Galois data
From OpenAI Aftermath
Reconstruction of function fields from Milnor K-theory and Galois data concerns Bogomolov's program in birational anabelian geometry: a function field of transcendence degree at least two over an algebraically closed field should be determined by very small Galois-theoretic or cohomological invariants. Three manuscripts in OpenAI's public mathematics collection, dated 23 September and 5 October 2026, claim the Bogomolov–Pop reconstruction theorem for the pro-ℓ abelian-by-central quotient of the Galois group, reconstruction from the first two Milnor K-groups modulo a prime ℓ different from the characteristic, and reconstruction from Milnor K-theory modulo the characteristic. The collection's Lean formalization covers the uniqueness half of the Galois statement.
Background
- As recounted in the manuscripts' introductions:
- Bogomolov's program. Bogomolov proposed in 1991 to reconstruct higher-dimensional function fields over algebraically closed constants from the pro-ℓ quotient of the absolute Galois group in which commutators are central. Bogomolov and Tschinkel proved reconstruction for surfaces over algebraic closures of finite fields and then in higher dimension, and Pop independently completed reconstruction over those constants. Over more general algebraically closed constants, Pop obtained reconstruction from pro-ℓ data with divisorial inertia supplied, in transcendence degree greater than two.
- Milnor K-theory. Topaz proved a Milnor-theoretic isomorphism theorem in transcendence degree at least five when all rational subgroups are supplied together with these groups. Bogomolov and Tschinkel, Cadoret and Pirutka, and Topaz used algebraic dependence or rational Milnor K-theory as further routes to reconstruction.
The original results
- Galois data, 23 September 2026. The main theorem of The Bogomolov–Pop reconstruction theorem states that for algebraically closed fields k and l of characteristic different from ℓ and function fields K/k and L/l of transcendence degree at least two, the isomorphisms of perfect closures that carry k to l, taken modulo powers of Frobenius, correspond bijectively to the bracket-compatible isomorphisms of the abelian-by-central pro-ℓ Galois data, taken modulo ℓ-adic units. The statement includes surfaces and ℓ = 2, and assumes neither equal characteristics nor equal transcendence degrees.
- Milnor K-theory modulo ℓ, 5 October 2026. The main theorem of Reconstruction of Function Fields from Mod-ℓ Milnor K-Theory states the same kind of bijection for the first Milnor K-group modulo ℓ together with the Steinberg relations of the second, with one global scalar in the multiplicative group of the field with ℓ elements as the only ambiguity besides Frobenius.
- Milnor K-theory modulo the characteristic, 5 October 2026. The main theorem of Reconstruction from Milnor K-theory modulo the characteristic states that for function fields of transcendence degree at least two over algebraically closed fields of characteristic p, every isomorphism of the first Milnor K-groups modulo p that preserves the Steinberg relations is a nonzero scalar multiple of the map induced by a unique field isomorphism.
- Scope, as stated: all three statements assume transcendence degree at least two and algebraically closed constants. The mod-ℓ manuscript states that it uses no pro-ℓ field-reconstruction theorem as a premise.
- Method, as described: alternating pairs of characters, visible in the Milnor product, are turned into valuations by established local theory, and Pop's density theorem for minimized inertia recognizes residual curves over algebraic closures of finite fields. A bounded-support estimate and an incidence argument recover the subfields of curves, rational one-variable subfields are found from ratios of local parameters, and the field operations are rebuilt from them. The manuscript modulo the characteristic recovers projective lines over the subfield of p-th powers by a calculation with derivations.
- Sources. Commit
adc7f12of the openai/math repository: The Bogomolov–Pop reconstruction theorem, Reconstruction of Function Fields from Mod-ℓ Milnor K-Theory and Reconstruction from Milnor K-theory modulo the characteristic. - Formalization. The scope note states that the Lean development proves injectivity of the Galois reconstruction correspondence: two isomorphisms of perfect closures that preserve the constant fields and induce the same Galois-data isomorphism modulo ℓ-adic unit scaling agree modulo Frobenius. Existence of a field isomorphism for every admissible Galois-data isomorphism is outside it, and the Milnor K-theory statements are not formalized. Comparator statement:
BogomolovPopInjectivity.lean; the proof is inlean/OAI/Algebra/BogomolovPop, five files of 3,339 lines.
Public follow-up work
| Date | Work | Relation | Public claim | Public status (checked 31 October 2026) |
|---|---|---|---|---|
| 2026-10-08 | MathVet, a GitHub review of openai/math, tables at repository commit c76f1e4 of 14 October 2026 |
Review, fidelity of the formal statement | Rates this family "partial": the challenge states only the injectivity half of the Bogomolov–Pop reconstruction, on the Galois side, with no existence statement, no Milnor K-theoretic reconstruction and no recovery of the perfect closure and constants. Among the definitions to check it lists the encoding of ℓ-adic unit scaling through approximants, which it calls custom but plausible, and records a sandboxed Comparator pass on 8 October 2026. | Review repository. Read on 31 October 2026. |
| 2026-10-10 | cert-machine, openai/math lane K, run 38051470892 | Verification, two kernels | Records the challenge BogomolovPopInjectivity as certified: Lean's default kernel and the nanoda kernel both accept the solution, in 205 s. C. Toledo's report lists the lane's results. |
Repository. The run concerns the formalized statement only. |
| 2026-10-10 to 2026-10-14 | con-leche development journal, task #348 | Verification, second kernel | In its run over all Comparator challenges of the collection at commit fd4aeeb, the journal's table of exceptions does not list this family's challenge, so it was accepted by con-leche and by Lean's official kernel. |
Development journal. |
| 2026-10-11 | AI Proved What?, episode 009 The whole field from a shadow, remade version of 24 October 2026 | Reading, exposition | Video explainer of the three preprints, of 36, 26 and 11 pages, starting from functions in several variables kept only up to ℓ-th powers. It recalls the work of Bogomolov, Bogomolov and Tschinkel, Pop and Topaz, and states that nothing is claimed for curves, that the constants must be algebraically closed, and that the result says what the data determine, not that there is an algorithm to rebuild the field. | Episode status, last checked 30 October 2026: posted as preprints, unchanged since the repository's first public version; under expert review, not yet; peer reviewed, not yet; checked in Lean, partly, the uniqueness half of the Galois version, which the site did not run; no public critique specific to this result found. The site states that it is made with Anthropic's Claude. Read on 31 October 2026. |
| 2026-10-16 | J. Jardine, Lattice24, referee reviews rank_188 of the Galois manuscript, rank_381 of the mod-ℓ manuscript and rank_346 of the manuscript modulo the characteristic, in JJardine919/openai-math-audit | Review, AI-assisted audit | All three reviews report no gap; each states that the reviewer read the whole manuscript. The reviews of the Galois and mod-ℓ manuscripts each list two minor points of dependence on cited results that could not be checked against their sources: the recognition of residual curves through Pop's 2015 criterion, and, respectively, Pop's results on decomposition graphs and the application of Topaz's local theory to residue characteristic ℓ. The review of the manuscript modulo the characteristic checked every lemma line by line and lists no issue. | Part of the completed Lattice24 audit; the reviews are published under CC BY 4.0. |
| 2026-10-25 | Taming the Flow: Critical reports on AI-generated mathematics, an online seminar of the Arithmetic and Homotopic Galois Theory research network, autumn 2026, organized by B. Collas, RIMS Kyoto; announcement page last updated 25 October 2026 | Reading, seminar talks announced | Announces three talks on these manuscripts: A. Topaz, University of Alberta, on reconstruction from mod-ℓ Milnor K-theory, date to be announced; A. Pirutka, Courant Institute, on reconstruction from Milnor K-theory modulo the characteristic, on 18 December 2026; and a talk on the Bogomolov–Pop reconstruction theorem, speaker and date to be announced. The page links copies of the three manuscripts as AI-generated target notes. | Announced; no talk on this family had taken place by the date checked. Read on 31 October 2026. |
Notes
- The background summarizes the manuscripts' own account of earlier work.
- The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.
- Related manuscripts of the collection in anabelian geometry are described on The p-adic section conjecture and Uchida's conjecture on open homomorphisms of Galois groups.
