Manuscript OpenAI@2026-10-07 · OpenAI.tex · remark · 6a931f8f
Remark: 155-01. A translational tile with no fully periodic tiling in dimension three
Manuscript map · lines 5365–5374
OpenAI@2026-10-07#rem:oai-p-155-01
Authors: OpenAI. Family 155.
Title: A translational tile with no fully periodic tiling in dimension three.
Abstract: We construct a finite translational tile in ℤ3 that admits tilings but no fully periodic tiling. Its unit-cube thickening has the same property in ℝ3, even when arbitrary real translations are allowed. This gives a negative resolution of the periodic tiling conjecture in dimension three.
This abstract is a catalogue record. It is not a verification and not a premise. It does not close O.1. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted.
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