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Math-2618-A1 · revision r1 · on Math-2618

OpenAI claim summary for Math-2618

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status: open. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

Three-dimensional nonperiodic translational tile: The manuscript’s result has two distinct conclusions: a finite nonempty Z^3 tile with no finite-index-invariant complement, and its bounded positive-volume cube thickening with no full-rank-lattice-invariant real translation set despite an almost-everywhere tiling. The first clause supplies an exact counterexample to this question. The real conclusion explicitly permits arbitrary real translation vectors.

The claimed counterexample lies within the question’s hypotheses and contradicts its universal conclusion.

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. Periodic translational tiling in dimension three
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Mathematical claim 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

Periodic translational tiling in dimension three

OpenAI's claim

Three-dimensional nonperiodic translational tile: A finite lattice tile in $\mathbb Z^3$ is claimed to tile while admitting no tiling invariant under a finite-index translation group. Its thickening by unit cubes has the same obstruction in Euclidean space, including tilings by real translation vectors.

Mathematical statement of the claimed result

Mathematical claim 1.
The construction asserts a finite nonempty $T\subset\mathbb Z^3$ admitting a disjoint translational tiling $A\oplus T=\mathbb Z^3$, while no such translation set $A$ is invariant under any finite-index subgroup of $\mathbb Z^3$. Its bounded measurable thickening $\Omega=T+[0,1]^3$ tiles $\mathbb R^3$ up to null sets, but no translation set for such a tiling remains unchanged under every translation in any rank-three lattice.

Outline of the argument

Three-dimensional nonperiodic translational tile: A Sudoku-type word obstruction is encoded as several dependence and activation tests in a cyclic group. Stacking combines the tests into one tile, and an explicit model supplies a tiling. Marked component shapes rigidify any hypothetical fully periodic Euclidean tiling onto a grid, from which it would descend to a forbidden periodic solution of the encoded tests.

Formal verification

Family 155 describes Lean coverage for nonexistence of fully periodic lattice and Euclidean tilings for the three-dimensional construction.

Three-dimensional nonperiodic translational tile: The manuscript links PeriodicTilingThree.lean (declaration OAI.PeriodicTilingThree.periodic_tiling_counterexample_and_minimality).

The documented selection includes this manuscript’s main assertion; that assertion retains the restrictions stated above.

This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.

Status caveat

OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.

Sources