What was already known
Greenfeld and Tao had disproved periodic tiling in sufficiently high dimension. The manuscript cites positive results in lower dimensions and seeks the smallest dimension of failure.
AI involvement
OpenAI credits itself as manuscript author and states that its internal frontier model produced the results in this release. Individual human input is not detailed enough to establish task-only autonomy.
OpenAI is named as author on the manuscript, and its release attributes the results to an internal frontier model. The individual human input is not detailed enough to establish autonomy.
Importance
Lowers a known periodic-tiling counterexample to the least possible lattice dimension, three.
Limitations
Fully periodic means invariance under a finite-index lattice subgroup; the result does not prohibit all nontrivial translation symmetries.
Technical verification
OpenAI reports a Lean proof matching the selected claim.
Formal checking
The pinned OpenAI Lean scope note identifies this selected claim.
- Checker
- OpenAI-reported Lean 4 / Comparator
- Scope
- A finite tile in ℤ³ admits a translational tiling but no fully periodic tiling; its unit-cube thickening has the same property in ℝ³ even with arbitrary real translations.
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
Few is the default when evidence for a higher level is insufficient. No named person is documented as explaining the central argument for this assessed contribution.
Correction and publication history
Primary manuscript, attribution, prior-work, and scoped checking review for non-notifying local inclusion.
Read this published revision