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Gersten's conjecture for integral K-theory of arbitrary regular local rings (Quillen's form), in ramified mixed characteristic

For a Noetherian regular local domain AA with fraction field FF, Gersten's conjecture asserts exactness of the augmented complex 0→Kn(A)→Kn(F)→⨁ht p=1Kn−1(κ(p))→⋯0\to K_n(A)\to K_n(F)\to\bigoplus_{\mathrm{ht}\,\mathfrak p=1}K_{n-1}(\kappa(\mathfrak p))\to\cdots of integral Quillen K-groups; its first assertion is that Kn(A)→Kn(F)K_n(A)\to K_n(F) is injective. Quillen formulated the conjecture for all regular local rings and proved it for regular local rings essentially of finite type over a field; Panin proved the full equicharacteristic case. In mixed characteristic it was known for rings smooth over a DVR modulo the DVR case (Gillet-Levine), with finite coefficients for rings essentially smooth over a DVR (Geisser-Levine) and for unramified regular local rings (Skalit), while Feld (2026) found a finite-coefficient failure at a ramified ring. Is Kn(A)→Kn(Frac A)K_n(A)\to K_n(\mathrm{Frac}\,A) injective, and the Gersten complex exact, for every regular local ring AA, including ramified rings of mixed characteristic?

Result
Disproved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Construction
Field
Algebraic K-theory of regular local rings
Posed by
D. Quillen, Higher algebraic K-theory I, Lecture Notes in Math. 341 (1973), Section 7, Conjecture 5.10; following S. M. Gersten, Problems about higher K-functors (same volume)
Year posed
1973
Years open
53y
Solved
2026-09-25
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Contested
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
30 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1 (degree five): with VV the integers of the unramified degree-eight extension of Q5\mathbb Q_5 and A=(V[x,y]/(5+x4+y4))(5,x,y)A=(V[x,y]/(5+x^4+y^4))_{(5,x,y)}, AA is a two-dimensional regular local domain and ker⁡(K5(A)→K5(Frac A))≠0\ker(K_5(A)\to K_5(\mathrm{Frac}\,A))\ne0; the class also gives generic noninjectivity with Z/5\mathbb Z/5 coefficients. Companion Theorem 1.1 (degree three): the local ring of a cyclotomic quintic model over Z5[ζ5]\mathbb Z_5[\zeta_5] at a special point has a nonzero class in ker⁡(K3(A)→K3(Frac A))\ker(K_3(A)\to K_3(\mathrm{Frac}\,A)) not divisible by 5. Not shown: the order of these classes, whether they survive rationally (the rational Gersten conjecture is untouched), anything about unramified or smooth-over-DVR rings, or the least degree of failure.

What the AI did

The OpenAI math release (github.com/openai/math, commit adc7f12) states that its results were produced by an unreleased internal OpenAI model under one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's stated exceptions (the Re(s) > 11/12 zero-free region write-up and the Hodge conjecture for CM abelian varieties). The manuscript is credited to OpenAI alone and names no human author. The family has two independent manuscripts (degree five and degree three) whose proofs do not use each other.

Verification

No independent mathematician has checked this yet. Checked here: the introductions and main theorems of both manuscripts against Quillen's Conjecture 5.10. The degree-five paper gives an explicit two-dimensional ramified regular local ring of mixed characteristic (0,5)(0,5) with nonzero kernel on K5K_5; the companion gives one with nonzero kernel on K3K_3. The proofs were not refereed. Both papers stress that the counterexamples concern the unrestricted integral form, lie outside the unramified and smooth-over-a-DVR settings where positive results hold, and do not settle the rational-coefficient variant. Marked contested because a 2020 preprint of S. Mochizuki asserts a vanishing theorem that would kill the class constructed here; see the claim-issue note. No Lean formalization is supplied for this family.

Claim issue

S. Mochizuki, Local Gersten's conjecture for regular system of parameters (arXiv:1503.07966v8, 2020), abstract: 'we show local Gersten's conjecture for regular system of parameters. As its consequence we obtain Gersten's conjecture for a commutative regular local ring and smooth over a commutative discrete valuation ring.' The release's degree-five paper says: 'Mochizuki's Theorem 5 in version 8 ... asserts that, for a commutative Noetherian local ring B and a nonzerodivisor g, the inclusion from finitely generated projective B/gB-modules to finitely generated B-modules of projective dimension at most one induces zero on K-theory. This statement reaches the exact support in the present paper', so it 'would annihilate' the class constructed. The release gives a counterexample to Lemma 13(III) used in that proof, and says this 'by itself does not refute its theorem statement'. Mochizuki's headline corollary (rings smooth over a DVR) does not cover the ring used here.

Sources

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