Integral counterexamples to Gersten's conjecture

This page concerns Gersten's conjecture in algebraic K-theory. For Gersten's conjecture on one-relator groups, see Gersten's conjecture and virtual specialness of one-relator groups.

Integral counterexamples to Gersten's conjecture concerns Gersten's conjecture in algebraic K-theory, which asserts that for a regular local ring the Quillen K-groups map injectively into those of the fraction field and extend to an exact complex of residue maps. Two manuscripts in OpenAI's public mathematics collection, dated 25 and 26 September 2026, claim two-dimensional ramified regular local rings of mixed characteristic (0, 5) carrying integral classes in K₅ and in K₃ that vanish over the fraction field but not on the ring. This would give a negative answer to the unrestricted integral form of the conjecture. The collection's Lean library does not formalize these results.

Background

As recounted in the manuscripts' introductions:
- The conjecture. Gersten's questions on transfer and local higher K-theory, and his work on exact sequences, placed the conjecture in the study of localization and residue maps. Quillen formulated it for arbitrary regular local rings and proved it for semilocalizations of finite-type algebras over a field at regular primes, and Panin established the full equicharacteristic case.
- Mixed characteristic. Gillet and Levine reduced the smooth local case to the case of discrete valuation rings. Geisser and Levine proved Gersten exactness with finite coefficients for semilocal rings essentially smooth over a discrete valuation ring, and Skalit proved it with finite coefficients for unramified regular local rings. A preprint of Druzhinin states a result for essentially smooth local Henselian schemes over a discrete valuation ring.
- A finite-coefficient example. Feld gave, in 2026, a degree-two kernel modulo three at a ramified regular local ring whose integral K₂ map is injective; the degree-five manuscript describes it as a close methodological predecessor.

The original results

Degree five, 25 September 2026. The main theorem of An integral counterexample to Gersten's conjecture states that for the ring of integers V in the unramified extension of degree eight of the 5-adic numbers, the local ring A, at the ideal generated by 5, x and y, of the quotient of the polynomial ring over V in x and y by 5 + x⁴ + y⁴, is a Noetherian regular local domain of dimension two, and the kernel of K₅(A) → K₅(Frac A) is nonzero.
Degree three, 26 September 2026. The main theorem of An integral degree-three Gersten counterexample states that for the local ring A at the point [0:0:1] of the cyclotomic quintic model (X − Z)⁵ − XY⁴ + Y⁵ + ζZ⁵ = 0 over the integers of the field generated by a primitive fifth root of unity ζ over the 5-adic numbers, there is an integral class in K₃(A) that vanishes over the fraction field and whose reduction modulo five stays nonzero after inverting a uniformizer, so that K₃(A) → K₃(Frac A) is not injective.
Scope, as stated: both rings are ramified, with 5 or the uniformizer deep in the maximal ideal, and are not smooth over the base, so the positive smooth and unramified results do not apply. The manuscripts state that they do not determine the order of the classes or whether their images with rational coefficients are nonzero, so the rational variant of the conjecture is left open, and that neither determines the smallest degree of failure. Each proof states that it does not use the other.
Method, as described: a class pushed forward from a closed divisor of the spectrum of A vanishes over the fraction field, and the work is to show that it is nonzero. In degree five an integral cyclotomic class in K₅ of the cyclotomic integers with 5 inverted is lifted to the valuation ring of the divisor and detected modulo five through a specialization criterion on the Fermat quartic and its elliptic quotient w² = s⁴ + 1. In degree three the class comes from a Steinberg relation and is detected by a norm valuation test on coherent K-theory.
Appendix: the degree-five manuscript treats a vanishing theorem in version 8 of a preprint of S. Mochizuki, Theorem 5, which would annihilate the pushforward used. It gives a counterexample to the functor assertion of the preprint's Lemma 13(III), and states that this concerns the supplied proof and does not by itself refute the theorem's statement; the main proof does not depend on the appendix.
Sources. Commit adc7f12 of the openai/math repository: An integral counterexample to Gersten's conjecture and An integral degree-three Gersten counterexample.
Formalization. The collection's Lean library has no scope note or Comparator statement for these manuscripts.

Public follow-up work

Date	Work	Relation	Public claim	Public status (checked 1 November 2026)
2026-10-08	HayamaKazuma/gersten-counterexamples, Rational Gersten kernels in ramified regular local rings	Parallel work on the rational variant	A 121-page article claims infinite-order elements in the kernels of the rational generic restriction maps for ramified regular local rings of mixed characteristic, which is the rational variant that both manuscripts leave open. Its basic examples have dimensions two and three, each with a K₃ class that vanishes over the fraction field and stays of infinite order after completion; it extends the construction to every residue prime and to higher degrees, and it also examines Lemma 13(III) of Mochizuki's version-8 preprint. Its bibliography cites Feld's finite-coefficient example and Mochizuki's preprint but not the manuscripts of this collection.	GitHub repository, created 8 October 2026. The repository states that its OpenAI byline denotes user-requested preparation with ChatGPT and Codex and that it is not an official OpenAI publication. It reports reviews by independent language-model checks and twelve small Lean lemmas that do not formalize the K-theory arguments. Read on 1 November 2026.
2026-10-16	J. Jardine, Lattice24, referee reviews rank_323 (degree five) and rank_380 (degree three) in JJardine919/openai-math-audit (archived at Zenodo 10.5281/zenodo.23422995)	Review (AI-assisted audit)	Both reviews report no gap and one minor note each. For degree five, the note is a consistency flag: accepting the main theorem means accepting that the statement of Mochizuki's Theorem 5 is false, which the paper itself does not refute. For degree three, the note is that one step relies on the multiplicativity of the motivic spectral sequence for a mixed integral and mod-five pairing, cited from Friedlander–Suslin and Suslin rather than proved. The reviewer states that it read every section of both papers and lists hand checks, among them the vanishing of the 5-torsion of K₄ of the cyclotomic integers with 5 inverted for degree five and the regularity of the ring for degree three.	Part of an audit of the collection's manuscripts that the audit describes as complete. The reviews are published under CC BY 4.0.
2026-10-17	AI Proved What?, episode 209 Dead over F, alive on A?, remade version of 24 October 2026	Reading (exposition)	Video explainer of both preprints, of 25 and 10 pages. It recalls the positive results recounted by the papers and Feld's mod-three example, states that the papers claim nothing about rings containing a field, smooth or unramified rings, the rational variant, the order of the classes or the smallest degree that fails, and outlines both detection arguments. It reports its own checks: point counts on w² = s⁴ + 1 over the fields with 5, 25, 625 and 5⁸ elements, the weighted sum 4Q modulo five, and the special fibre of the degree-three model. It also notes that a 2015–2020 preprint of Mochizuki, one version of which claimed Gersten's conjecture for all regular local rings, has a vanishing theorem that would rule out such classes, and that this theorem cannot hold as stated if the new papers are right.	Episode status, last checked 30 October 2026: posted as preprints, not revised; under expert review, not yet; peer reviewed, not yet; not formalized in Lean, the release's Lean library having no higher K-theory; no expert reaction found. The site states that it is made with Anthropic's Claude. Read on 1 November 2026.
2026-10-18	AI Proof Watch, record OpenAI reports an integral counterexample to Gersten's conjecture	Catalogue record, assessment	Rates the importance as major, as a disproof of the unrestricted integral Gersten conjecture with a ramified regular local ring, and the AI involvement as AI-led. It lists as limitations that the counterexample uses a specific ramified ring of mixed characteristic in degree five and does not contradict the smooth or unramified positive cases, and that no completed proof check was documented. It rates the technical verification as reported and the human understanding as few, the defaults when no higher level is documented.	Curated public tracker; record published 18 October 2026, assessment reviewed 31 October 2026. Read on 1 November 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.