Source authenticated

The periodic tiling conjecture in dimension three

Theorem 1.1: there is a finite nonempty $T\subset\mathbb Z^3$ that tiles $\mathbb Z^3$ by translations, but no translation set of a tiling is invariant under a finite-index subgroup; and $T+[0,1]^3$ tiles $\mathbb R^3$ 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.

Exact FrontierDelta

Prior state unknown→disproved

Scope and record

Occurred: Sep 23, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: The periodic tiling conjecture in dimension three · Periodic tiling conjecture, dimension 3

Confidence: Not scored

Registry verification: lean checked · announcement · candidate

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Attribution

VibeMathed
registry · event recorded by

Unreleased internal OpenAI model
model · ai model contributor · OpenAI

Evidence vector

SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review

Named evidence

VibeMathed record: The periodic tiling conjecture in dimension three

specialist registry · passed

No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the conjecture; it gives a finite tile of $\mathbb Z^3$ that tiles but admits no fully periodic tiling, and its unit-cube thickening has the same property in $\mathbb R^3$ 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 $T\subset\mathbb Z^3$ with a tiling and no finite-index-invariant tiling, an almost-everywhere tiling of $\mathbb R^3$ by its thickening with no tiling invariant under a full-rank lattice, and that 3 is the least $d$ admitting such a lattice tile. This states the headline claim, plus minimality. Not rebuilt here. Permitted axioms: propext, Quot.sound, Classical.choice.

Source ↗

The periodic tiling conjecture in dimension three

witness · pending

Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

Source ↗

Lean proof (OAI.PeriodicTilingThree.periodic_tiling_counterexample_and_minimality)

lean artifact · pending

lean-proof link supplied by the VibeMathed registry.

Source ↗

Comparator statement: PeriodicTilingThree.lean

lean artifact · pending

lean-proof link supplied by the VibeMathed registry.

Source ↗

Greenfeld and Tao, A counterexample to the periodic tiling conjecture (high dimension)

specialist registry · pending

problem-record link supplied by the VibeMathed registry.

Source ↗

Artifacts and verifiers

Lean proof (OAI.PeriodicTilingThree.periodic_tiling_counterexample_and_minimality)

lean artifact · pending

Artifact ↗

Comparator statement: PeriodicTilingThree.lean

lean artifact · pending

Artifact ↗

Compute record

No linked compute attempts recorded.

Lineage and corrections

Comparator statement: PeriodicTilingThree.lean evidence for this event

This event attributed to Unreleased internal OpenAI model

Lean proof (OAI.PeriodicTilingThree.periodic_tiling_counterexample_and_minimality) evidence for this event

The periodic tiling conjecture in dimension three evidence for this event

The periodic tiling conjecture in dimension three parent of this event

Greenfeld and Tao, A counterexample to the periodic tiling conjecture (high dimension) evidence for this event

VibeMathed record: The periodic tiling conjecture in dimension three evidence for this event

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