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The periodic tiling conjecture in dimension three

A finite nonempty F⊂ZdF\subset\mathbb Z^d is a translational tile if A⊕F=ZdA\oplus F=\mathbb Z^d for some AA; the tiling is periodic if AA is invariant under a finite-index subgroup. The periodic tiling conjecture (Lagarias and Wang, 1996, also in a Euclidean form for bounded measurable tiles of Rd\mathbb R^d) asserts that every translational tile admits a periodic tiling. It holds in Z\mathbb Z (Newman) and Z2\mathbb Z^2 (Bhattacharya), and Greenfeld and Tao disproved it in sufficiently high dimension. Does every finite translational tile of Z3\mathbb Z^3, and every bounded tile of R3\mathbb R^3, admit a periodic tiling?

Result
Disproved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Construction
Field
Translational tilings of lattices and Euclidean space
Posed by
Jeffrey C. Lagarias and Yang Wang
Year posed
1996
Years open
30y
Solved
2026-09-23
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Lean-checked, statement unaudited
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
38 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: there is a finite nonempty T⊂Z3T\subset\mathbb Z^3 that tiles Z3\mathbb Z^3 by translations, but no translation set of a tiling is invariant under a finite-index subgroup; and T+[0,1]3T+[0,1]^3 tiles R3\mathbb R^3 up to null sets, with no tiling whose translation set is invariant under a full-rank lattice, even allowing real translations. With Bhattacharya's planar theorem, three is the least lattice dimension where periodic tiling fails. The tile is not claimed to be connected, and the result concerns full periodicity only, not the absence of every individual period.

What the AI did

The release README says every result in it was produced by an unreleased internal OpenAI model following 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. Single-manuscript family dated September 23, 2026. The paper says it follows the Sudoku strategy of Greenfeld and Tao and adds a new encoding keeping the finite factor cyclic.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the conjecture; it gives a finite tile of Z3\mathbb Z^3 that tiles but admits no fully periodic tiling, and its unit-cube thickening has the same property in R3\mathbb R^3 with arbitrary real translations. lean/formalization.yaml lists a main result for this manuscript (comparator PeriodicTilingThree, declaration OAI.PeriodicTilingThree.periodic_tiling_counterexample_and_minimality). The comparator statement was read here: it asserts a nonempty finite T⊂Z3T\subset\mathbb Z^3 with a tiling and no finite-index-invariant tiling, an almost-everywhere tiling of R3\mathbb R^3 by its thickening with no tiling invariant under a full-rank lattice, and that 3 is the least dd admitting such a lattice tile. This states the headline claim, plus minimality. Not rebuilt here. Permitted axioms: propext, Quot.sound, Classical.choice.

Sources

Changelog1 change

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