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Howie's conjecture: every nonsingular system of equations over a group is solvable in an overgroup

Let GG be a group and w1,…,wm∈G∗F(x1,…,xn)w_1,\dots,w_m\in G*F(x_1,\dots,x_n) a system of equations wi=1w_i=1 with coefficients in GG. Its exponent-sum matrix A∈Mm×n(Z)A\in M_{m\times n}(\mathbb Z) records the exponent of each xjx_j in each wiw_i; the system is nonsingular if AA has rank mm over Q\mathbb Q. Gerstenhaber and Rothaus (1962) solved nonsingular systems over finite groups and compact Lie groups, Howie (1981) over locally indicable groups, and Nitsche-Thom over hyperlinear groups. The one-equation, one-variable case is the Kervaire-Laudenbach conjecture (a nonzero exponent sum should guarantee solvability). Does every finite nonsingular system over an arbitrary group GG have a solution in some group containing GG, that is, is G→(G∗Fn)/⟨⟨w1,…,wm⟩⟩G\to(G*F_n)/\langle\langle w_1,\dots,w_m\rangle\rangle injective?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Combinatorial group theory; equations over groups
Posed by
James Howie (1981), On pairs of 2-complexes and systems of equations over groups; name and formulation as in Klyachko, Mikheenko and Roman'kov (2024)
Year posed
1981
Years open
45y
Solved
2026-10-05
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
45 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Claims Theorem 1: for every group GG and every finite system w1,…,wm∈G∗Fnw_1,\dots,w_m\in G*F_n whose exponent-sum matrix has rank mm over Q\mathbb Q, the canonical map G→(G∗Fn)/⟨⟨w1,…,wm⟩⟩G\to(G*F_n)/\langle\langle w_1,\dots,w_m\rangle\rangle is injective, so the system has a simultaneous solution in an overgroup. This contains the Kervaire-Laudenbach conjecture (one equation, one unknown, nonzero exponent sum) and the Kervaire conjecture. It does NOT control the overgroup (no finiteness, solvability or other structure of the solution group), and says nothing about singular systems.

What the AI did

The release README says the manuscripts were produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure using on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier results produced by the models. Its named exceptions to that procedure (the zeta zero-free region work, whose Re(s) > 11/12 write-up was human edited, and the Hodge conjecture for CM abelian varieties) do not concern this family. The manuscript is credited to OpenAI with no human author named. It states that its spectral-phase and planar-surface strategy comes from the family's earlier Kervaire manuscript (24 September 2026), extended by a multi-block fixed-space theorem for products of unitary groups.

Verification

No independent mathematician has checked this yet. Theorem 1 of the principal manuscript was read against the conjecture as the manuscript formulates it: for any group GG and 1≤m≤n1\le m\le n, full rational row rank of the exponent-sum matrix gives injectivity of GG into the quotient, with Remark 5.1 extending to infinite systems with independent rows. This manuscript has no Lean formalisation. The family's Lean challenge (ComparatorChallenges/Kervaire.lean) covers only one relator with exponent sum ±1\pm1, which is much narrower than this theorem. The release README warns that some unformalised results could have issues.

Sources

Changelog1 change

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