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oai:figshare.com:article/34012974 2026-09-28T21:40:34Z category_29836 item_type_12 month_year_09_2026

Exact Boundary Layers for Integer Two-Level Sections of the Nine-Cube Akihiro Koide (24791941) Combinatorics and discrete mathematics (excl. physical combinatorics) Boolean cube integer two-level section Littlewood–Offord problem affine rank exact enumeration computer-assisted classification <p dir="ltr">An integer two-level section of the Boolean cube is the set on which a primitive, noncoordinate integer linear functional takes one of two consecutive integer values. We study intersections whose defining normal directions are pairwise nonparallel. In dimension nine, the largest intersection size strictly below 50% of the cube’s vertices is 194 for nine defining directions, 193 for ten, and 192 for every larger number. We describe all extremizers: the 194-point layer has twenty-two signed-coordinate orbits, the 193-point layer has seventeen, and the 192-point layer with at least eleven available defining directions has eight. The first two layers have full affine span and a unique pair or point whose deletion lowers affine rank; the resulting 192-point core determines an explicit canonical representation. The minimum numbers of defining sections are respectively five through seven and six through eight. The proof combines integer coefficient bounds, quadratic vanishing spaces, nonnegative relations among receiver quadratics, and complete finite intersection calculations. All orbit representatives are given explicitly, together with the complete finite carrier families of the full-span models. The coefficient and presentation bounds reduce the completeness assertions to finite computations.</p><p dir="ltr"><br></p><p dir="ltr">This deposit contains the manuscript PDF, LaTeX source, and computation supplement. The supplement provides exact representative verifiers, 37 standalone C++20 enumeration programs, finite catalogues and retained computation outputs, a proof guide, and a regeneration driver. Representative verification checks the listed models and their properties; completeness combines the finite-domain reductions in the manuscript with the complete enumeration calculations.</p> 2026-09-28T21:40:34Z Text Preprint 10.6084/m9.figshare.34012974.v1 https://figshare.com/articles/preprint/Exact_Boundary_Layers_for_Integer_Two-Level_Sections_of_the_Nine-Cube/34012974 CC BY 4.0