# Integral counterexamples to Gersten’s conjecture

> Two ramified rings are claimed to break Gersten's conjecture: nonzero K-theory classes vanish over their fractions. Plain-English explainer with sources.

Canonical: https://closedform.nishachay.tech/e/openai-math-2026-209/
Last updated: 2026-10-14

**Two ramified rings are claimed to break Gersten's conjecture: nonzero K-theory classes vanish over their fractions.**

**Trust:** Claimed, not checked · **Papers:** 2 · **First paper:** September 25, 2026 · **Field:** Algebra · **Entry:** 209

*Explained by AI.* Drafted by one AI model and checked sentence by sentence against the paper by a different one. Spot a mistake? Report an error.

K-theory attaches groups to a ring that record information about its matrices. Gersten's conjecture says, for regular local rings, that no nonzero class dies when you pass to fractions. These papers claim two explicit counterexamples. Both are ramified rings in mixed characteristic (0, 5), with failures in degrees 5 and 3. The rational-coefficient version of the conjecture is left open.

## At a glance

- **First step:** 1973 (Daniel Quillen, the regular-local conjecture)
- **Significance:** Major. It would answer the unrestricted integral form of a 50-year-old conjecture negatively, while leaving key variants open.
- **Trust:** Claimed, not checked
- **Field:** Algebra (2 papers)

## The explanation

Start with a ring of numbers and pass to its fractions. Take the integers and allow every denominator not divisible by 5. A number like 2 or 3/7 can be inverted in this ring. It stays a nonzero fraction when you pass to all of the rational numbers. Nothing is lost on the way.

K-theory attaches a sequence of groups K₀, K₁, K₂ and so on to every ring. In low degrees they are familiar. For rings like this one, K₁ is essentially the group of invertible elements. Higher groups record subtler information about matrices over the ring. Gersten's conjecture says that, for well-behaved rings, nothing is lost in any degree when passing to fractions. A class that dies in the fraction field must already be zero[1]. Well-behaved here means regular local: the ring describes the neighborhood of a smooth point.

Now look at the paper's ring. Take the ring of integers of an extension of the 5-adic numbers of degree eight. Add two variables x and y, impose 5 + x⁴ + y⁴ = 0, and look near the point where 5, x and y vanish[2]. In this ring, 5 equals −x⁴ − y⁴. So 5 lies deep inside the ideal generated by x and y, in its fourth power[3]. Rings where the prime 5 is this small are called ramified. This one is regular, but it is not smooth over its base[4].

The papers claim Gersten's conjecture fails for such rings. In this ring, some nonzero class in K₅ becomes zero in the fraction field[5]. A second paper builds a different ramified ring, from a degree-five curve, with the same failure in K₃[6]. Both rings have mixed characteristic (0, 5). That means the ring contains the integers, but 5 becomes zero at the special point.

Much was known before. Quillen formulated the conjecture and proved it for rings built from finite-type algebras over a field[7]. Panin proved the full case where ring and point share a characteristic[8]. In mixed characteristic, several positive results need smoothness over a base, or the unramified condition[9]. They also often allow only finite coefficients, such as working mod 5. The paper's ring is excluded from the unramified results, since 5 lies in the fourth power of the maximal ideal[10].

The single obstacle is integrality. Feld had found a failure with coefficients mod 3 at a ramified ring[11]. But a class mod 3 need not come from an honest integral class that dies[12]. The K₅ paper lifts a class from arithmetic to integral K₅. That needs a vanishing fact about integral K₄[13]. It then detects the class by reducing mod 5, using an elliptic curve hidden in the quartic equation.

The result has limits. The detection does not find the order of the integral class[14]. It does not say whether the class survives when one allows rational coefficients[14]. So the rational version of the conjecture stays open. Neither paper uses the other, and neither finds the smallest degree where failure happens[15].

How sure is this? There is no Lean formalization for these results in the openai/math release, so no computer has checked the proofs. The arguments rely on deep arithmetic K-theory. No specialist review of either paper has been published.

**What it doesn't show.** This does not settle Gersten's conjecture with rational coefficients, or for unramified rings.

## Where it came from

This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.

## Hype check

Major if it holds. There is no Lean check, the arguments use deep arithmetic K-theory, and no specialist review is published.

## Words used

- **Regular local ring:** An algebraic model of the neighborhood of a smooth point, such as power series in a few variables.
- **K-theory:** A sequence of groups K₀, K₁, K₂, … attached to a ring, built from its matrices and modules.
- **Fraction field:** All fractions a/b of ring elements with b nonzero.
- **Mixed characteristic (0, 5):** The ring contains the integers, but 5 becomes zero at its special point.
- **Ramified:** Here, the prime 5 lies in the square of the ideal of quantities that vanish at the point.
- **Gersten's conjecture:** The prediction that, for regular local rings, K-theory classes never die on passing to fractions, plus a longer exact sequence.

## The original papers (2)

- [An integral counterexample to Gersten's conjecture](https://github.com/openai/math/blob/main/preprints/An-Integral-Counterexample-to-Gerstens-Conjecture-September-25-2026/An-Integral-Counterexample-to-Gerstens-Conjecture-September-25-2026.pdf) - September 25, 2026
- [An integral degree-three Gersten counterexample](https://github.com/openai/math/blob/main/preprints/An-Integral-Degree-Three-Gersten-Counterexample-September-26-2026/paper.pdf) - September 26, 2026

## Quoted lines from the paper (15)

[1] From [An integral counterexample to Gersten's conjecture](https://github.com/openai/math/blob/main/preprints/An-Integral-Counterexample-to-Gerstens-Conjecture-September-25-2026/An-Integral-Counterexample-to-Gerstens-Conjecture-September-25-2026.pdf):

> The first assertion is generic injectivity: a class that vanishes after passage to the fraction field must already vanish on $A$.

[2] From [An integral counterexample to Gersten's conjecture](https://github.com/openai/math/blob/main/preprints/An-Integral-Counterexample-to-Gerstens-Conjecture-September-25-2026/An-Integral-Counterexample-to-Gerstens-Conjecture-September-25-2026.pdf):

> Let $V$ be the ring of integers in the unramified extension of $\Q_5$ of degree eight, and put \[ A=\bigl(V[x,y]/(5+x^4+y^4)\bigr)_{(5,x,y)}. \]

[3] From [An integral counterexample to Gersten's conjecture](https://github.com/openai/math/blob/main/preprints/An-Integral-Counterexample-to-Gerstens-Conjecture-September-25-2026/An-Integral-Counterexample-to-Gerstens-Conjecture-September-25-2026.pdf):

> The maximal ideal of $A$ is $(x,y)$, and $5\in(x,y)^4$.

[4] From [An integral counterexample to Gersten's conjecture](https://github.com/openai/math/blob/main/preprints/An-Integral-Counterexample-to-Gerstens-Conjecture-September-25-2026/An-Integral-Counterexample-to-Gerstens-Conjecture-September-25-2026.pdf):

> Thus the regular total ring is ramified and is not smooth over $V$.

[5] From [An integral counterexample to Gersten's conjecture](https://github.com/openai/math/blob/main/preprints/An-Integral-Counterexample-to-Gerstens-Conjecture-September-25-2026/An-Integral-Counterexample-to-Gerstens-Conjecture-September-25-2026.pdf):

> Consequently the unrestricted integral Gersten conjecture has a negative answer.

[6] From [An integral degree-three Gersten counterexample](https://github.com/openai/math/blob/main/preprints/An-Integral-Degree-Three-Gersten-Counterexample-September-26-2026/paper.pdf):

> We construct a ring for which this requirement fails in degree three.

[7] From [An integral counterexample to Gersten's conjecture](https://github.com/openai/math/blob/main/preprints/An-Integral-Counterexample-to-Gerstens-Conjecture-September-25-2026/An-Integral-Counterexample-to-Gerstens-Conjecture-September-25-2026.pdf):

> Quillen formulated it for arbitrary regular local rings and proved it for semilocalizations of finite-type algebras over a field at regular primes.

[8] From [An integral counterexample to Gersten's conjecture](https://github.com/openai/math/blob/main/preprints/An-Integral-Counterexample-to-Gerstens-Conjecture-September-25-2026/An-Integral-Counterexample-to-Gerstens-Conjecture-September-25-2026.pdf):

> Panin established the full equicharacteristic case.

[9] From [An integral degree-three Gersten counterexample](https://github.com/openai/math/blob/main/preprints/An-Integral-Degree-Three-Gersten-Counterexample-September-26-2026/paper.pdf):

> In mixed characteristic, smoothness over a DVR and the choice of coefficients lead to different positive results.

[10] From [An integral counterexample to Gersten's conjecture](https://github.com/openai/math/blob/main/preprints/An-Integral-Counterexample-to-Gerstens-Conjecture-September-25-2026/An-Integral-Counterexample-to-Gerstens-Conjecture-September-25-2026.pdf):

> In mixed characteristic $(0,p)$, unramified means $p\notin\mathfrak m^2$; our ring is excluded since $5\in\mathfrak m^4$.

[11] From [An integral counterexample to Gersten's conjecture](https://github.com/openai/math/blob/main/preprints/An-Integral-Counterexample-to-Gerstens-Conjecture-September-25-2026/An-Integral-Counterexample-to-Gerstens-Conjecture-September-25-2026.pdf):

> It gives a degree-two augmentation kernel modulo three at a ramified regular local ring whose integral $K_2$ map is injective.

[12] From [An integral counterexample to Gersten's conjecture](https://github.com/openai/math/blob/main/preprints/An-Integral-Counterexample-to-Gerstens-Conjecture-September-25-2026/An-Integral-Counterexample-to-Gerstens-Conjecture-September-25-2026.pdf):

> An integral generic kernel therefore requires an additional argument.

[13] From [An integral counterexample to Gersten's conjecture](https://github.com/openai/math/blob/main/preprints/An-Integral-Counterexample-to-Gerstens-Conjecture-September-25-2026/An-Integral-Counterexample-to-Gerstens-Conjecture-September-25-2026.pdf):

> Here a vanishing result for integral $K_4$ permits a cyclotomic coefficient class to lift to integral $K_5$.

[14] From [An integral counterexample to Gersten's conjecture](https://github.com/openai/math/blob/main/preprints/An-Integral-Counterexample-to-Gerstens-Conjecture-September-25-2026/An-Integral-Counterexample-to-Gerstens-Conjecture-September-25-2026.pdf):

> This detection does not determine the order of the integral class or whether its rationalization is nonzero.

[15] From [An integral counterexample to Gersten's conjecture](https://github.com/openai/math/blob/main/preprints/An-Integral-Counterexample-to-Gerstens-Conjecture-September-25-2026/An-Integral-Counterexample-to-Gerstens-Conjecture-September-25-2026.pdf):

> Neither proof uses results of the other paper, and neither determines a smallest degree of failure.

## Other sources

- [openai/math release notes (2026): model provenance and review status](https://github.com/openai/math/blob/main/README.md): “manuscripts and supporting proof artifacts produced by an internal OpenAI model”

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