Periods and affine structure of two-trace masks
Kuppusamy Ravindran · Zenodo (CERN European Organization for Nuclear Research) · 2026
A translation period preserves a set of cube points, but need not preserve the coordinate labels attached to those points. I study how a subspace of such periods restricts the size of a two-trace mask, a set whose points carry complementary coordinate traces with the required disjointness between opposite traces. The paper gives a general quotient bound. For a period of corank one, meaning rank d − 1 in dimension 2d, it proves the sharp bounds 12, 24 and 48 in dimensions six, eight and ten. For every d ≥ 2, it also gives a bound and an equality classification for sets lying in one sphere of radius d and invariant under a rank-(d − 1) period subspace. This sphere result does not require a mask witness. Further results concern cancellation of affine factors, intersections under a fixed coordinate pairing, and nonspherical examples. The unrestricted three-coset bound remains open for d ≥ 6. This 33-page author preprint includes both technical appendices: the local counts for the ten-dimensional proof and the complete crossing table for the eight-point witness. No external dataset or computer-generated certificate is required for the proofs. The manuscript includes the funding, competing-interest and AI-assistance statements. This deposit is a preprint, not a journal version of record. The earlier framework is credited in the article, including the preprints at https://doi.org/10.5281/zenodo.22174995 and https://doi.org/10.5281/zenodo.21471327.