# Three-dimensional tiling feasibility and periodicity audit

Family 155: A counterexample to periodic tiling in dimension three. First application triage, 9 October 2026 Australia/Brisbane.

## Problem and potential new use

Packing researchers can distinguish tileability from availability of a periodic pattern in translational three-dimensional models.

## Applicability and commercial boundary

The counterexample concerns infinite space and one special tile, not finite manufacturing layouts or a generic packing algorithm.

## Initial business decision

Research or conditional engineering only until an effective implementation and a recurring buyer problem are identified. No commercial demand or profitability is established. Where a direct product bridge is weak, the legitimate use is a research reference or evidence adapter, rather than a new standalone company.

## Next verification action

Extract the explicit tile and decide whether a bounded visualization explains an actual design assumption.

## Evidence scope

The catalog statement was individually reviewed. The manuscript proof and selected formal statement have not yet been compared in depth for this family. No independent proof verification was run. Pinned source revision `fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb`. See [source metadata](source.json).

[A translational tile with no fully periodic tiling in dimension three](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/A-translational-tile-with-no-fully-periodic-tiling-in-dimension-three-September-23-2026/paper.pdf).