VibeMathedMath problems solved with AI
Sign in

Existence of bounded-degree F2 coboundary expanders in every dimension

A finite pure dd-dimensional simplicial complex XX is an ε\varepsilon-coboundary expander over F2\mathbb{F}_2 if, with the standard weighted norms, ∥δf∥≥ε dist(f,Bi(X))\|\delta f\|\ge\varepsilon\,\mathrm{dist}(f,B^i(X)) for every ii-cochain ff and all i<di<d; this forces vanishing cohomology below dd. Linial-Meshulam random complexes, Latin-square complexes (Lubotzky-Meshulam) and Steiner-system complexes (Lubotzky-Luria-Rosenthal) are coboundary expanders, but their vertex degrees grow. Bounded-degree cosystolic expanders exist in every dimension (Kaufman-Kazhdan-Lubotzky; Evra-Kaufman), and Chapman-Lubotzky (2025) built bounded-degree two-dimensional coboundary expanders. Gromov (2010) asked for large complexes with bounded degree and uniform filling, and Chapman-Lubotzky posed the problem in general: for every d≥3d\ge3, are there arbitrarily large bounded-degree dd-dimensional F2\mathbb{F}_2 coboundary expanders with uniform constant?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Construction
Field
High-dimensional expanders: coboundary expansion
Posed by
M. Gromov, Singularities, expanders and topology of maps, Part 2 (GAFA 2010), Sections 2.3-2.5 and 2.14; M. Chapman and A. Lubotzky, Adv. Math. 463 (2025), Problem 1.3
Year posed
2010
Years open
16y
Solved
2026-09-24
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Lean-checked, statement unaudited
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
33 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: for every d≥3d\ge3 there are DD, ε>0\varepsilon>0 and finite connected pure dd-dimensional complexes XmX_m with ∣Xm(0)∣→∞|X_m(0)|\to\infty, every vertex in at most DD top faces, and ∥δif∥≥ε dist(f,Bi)\|\delta_if\|\ge\varepsilon\,\mathrm{dist}(f,B^i) for all i<di<d. The complexes are dd-skeleta of congruence quotients modulo tmt^m of a coset complex for the Iwahori-type subgroup of SL2d+1(k[t])\mathrm{SL}_{2d+1}(k[t]) over a large odd prime field; cohomology vanishing comes from an affine twin building, and quantitative expansion from Oppenheim-Valentiner-Branth. Constants depend on dd and are not explicit; only F2\mathbb{F}_2 coefficients are treated.

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.

Verification

No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1 of the TeX source, read against the problem as cited from Gromov (2010) and Chapman-Lubotzky (2025); the proof was not refereed. Lean: the Comparator challenge CoboundaryExpanders (OAI.CoboundaryExpanders.main, solution module OAI/Combinatorics/Coboundary/Main.lean) is not in the release's formalization catalogue, but its JSON and solution file exist at the pinned commit. Its statement was read here: for every d >= 3 there are D, eps > 0 and a sequence of finite pure connected d-dimensional complexes with vertex counts tending to infinity, at most D top faces at each vertex, and eps times the weighted distance to coboundaries at most the weighted norm of the coboundary in every degree i < d. This is the headline claim; the graph and two-dimensional cases are prior work. Not rebuilt here. The construction uses the cosystolic expansion theorem of Oppenheim and Valentiner-Branth as an input.

Sources

Changelog1 change

Discussion