Is there a strongly aperiodic monotile in three dimensions?

Claimed yes: Chair44 (R44), a rational polyhedral 3-ball built from seven cubes in a chair, whose 24 exposed unit panels carry small square-pyramid features. The claim is that it tiles $\mathbb R^3$ by congruent copies with reflections allowed, that every such tiling has no translational period and a symmetry group of order at most 24, and that every tiling is homochiral and carries a unique infinite hierarchy of nested supertiles. The features force each tiling onto a registered lattice; the finite facts - dissection, the 44-contact atlas, 33 first shells, a 6,862 to 44 companion census with 299,975 collision boxes, mesh and angle audits - are enumerated exhaustively and checked in Lean, with a second independent Python implementation replaying them.

Exact FrontierDelta

Prior state unknown→proved

Scope and record

Occurred: Sep 16, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: preprint. 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: Is there a strongly aperiodic monotile in three dimensions? · 3D strongly aperiodic monotile

Confidence: Not scored

Registry verification: lean checked · preprint · resolved

Open the source record ↗

Attribution

VibeMathed
registry · event recorded by

GPT-6 Astra
model · ai model contributor · OpenAI

Ioannis Tsiokos
human · human collaborator

Evidence vector

SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review

Named evidence

VibeMathed record: Is there a strongly aperiodic monotile in three dimensions?

specialist registry · passed

Audited here on 27 September 2026 from a clone at HEAD. 85 Lean files, 29,815 lines on Lean 4.31.0. Zero real sorry - the single grep hit sits inside a comment - zero axiom declarations, zero unsafe or implemented_by. But thirty-five native_decide calls across twelve files, which the author discloses rather than hides: AXIOMS.md lists the named compiler hook per theorem, README carries a tiered trust ledger (T1 standard axioms, T1n additionally native hooks), the negative controls are diagnostic-checked, and verify/replay.py reconstructs the same finite facts in standard-library Python. The unconditional spine is r44_einstein: existence, no translational period, and symmetry group of order at most 24. Recorded lean-checked rather than the submitted lean-verified, for two reasons. First, native_decide discharges goals through Lean's compiled evaluator, so the trusted base includes the compiler and runtime and not the kernel alone; the catalog's Dittert and composites-among-xi-7-n entries are labelled the same way for the same reason. Second, lean-verified additionally requires the formal statement to be anchored outside the prover's own repository - a comparator configuration, a Palomar entry or an audit by someone with no stake - and none is present. The repository's only CI workflow builds a reader notebook and is currently failing, so no automated build checks the Lean at all. A comparator or Palomar statement check plus a green Lean build would lift this to lean-verified.