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The finite lattice representation problem: is every finite lattice the congruence lattice of a finite algebra?

Gratzer and Schmidt (1963) proved that every algebraic lattice is the congruence lattice of some algebra, but the representing algebra may be infinite even when the lattice is finite. The finite lattice representation problem asks whether every finite lattice LL is isomorphic to Con(A)\mathrm{Con}(A) for some finite algebra AA (any finite signature). Palfy and Pudlak (1980) showed this holds for all finite lattices if and only if every finite lattice is an interval [H,G][H,G] in the subgroup lattice of a finite group. Positive results cover small lattices and lattices of width at most two, and the decision version (given a lattice table, decide representability) was recorded separately. Is every finite lattice the congruence lattice of a finite algebra?

Result
Disproved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Universal algebra; congruence lattices and subgroup intervals
Posed by
Finite form of the Gratzer-Schmidt representation theorem; the manuscripts do not name who first posed it
Year posed
—
Years open
—
Solved
2026-09-24
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
42 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Principal manuscript, Theorem 1.1: (i) a finite lattice is representable iff it admits a finite colored-graph witness; (ii) no Turing machine decides representability from the order table, so not every finite lattice is representable; (iii) there is a nonrepresentable lattice of minimum size, larger than 7, specified by a finite description involving a Busy Beaver constant (its size is not computed); (iv) recognizing full subgroup intervals of finite groups is undecidable. The companion constructs a finite lattice that is no interval [D,G][D,G] in a finite group, hence (Palfy-Pudlak) a nonrepresentable one. No explicit small counterexample is exhibited.

What the AI did

The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The family has two manuscripts dated September 24, 2026: the principal one gives a colored-graph characterization, undecidability of representability and of full subgroup-interval recognition, and a minimal counterexample described via a Busy Beaver constant; the companion gives a direct construction of a nonrepresentable lattice through subgroup intervals.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the principal manuscript and Theorems 1.1-1.2 of the companion were read against the problem; both claim a finite lattice that is not the full congruence lattice of any finite algebra, the companion via a lattice that is no finite subgroup interval. Proofs (over 170 and 80 pages, using the classification of finite simple groups and Larsen-Pink) were not refereed. A Lean challenge exists, lean/ComparatorChallenges/FiniteCongruenceGraph.json (solution module OAI.Algebra.Universal.GraphCriterion present), not in the formalization catalogue; its statement, read here, is only the colored-graph criterion (a finite lattice is representable iff it has a finite colored-graph witness). That criterion does not imply the negative answer, so the entry is left unreviewed. Not rebuilt here.

Sources

Changelog1 change

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