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oai:figshare.com:article/34022592 2026-09-29T10:20:40Z category_29836 item_type_12 month_year_09_2026

The Exact Strict Spectrum of Integer Two-Level Sections of Boolean Cubes Akihiro Koide (24791941) Combinatorics and discrete mathematics (excl. physical combinatorics) Boolean cube integer two-level section Littlewood–Offord problem affine dimension extremal cardinality dimension-transfer theorem <p dir="ltr">This paper determines the exact strict extremal cardinalities for intersections of integer two-level sections of Boolean cubes in every dimension at least eight, with primitive noncoordinate normals in pairwise distinct projective directions. Below fifty percent of the cube's vertices, the three sharp maxima are three eighths of the total number of vertices plus two, plus one, or with no addition, according to whether the number of defining directions equals the dimension, exceeds it by one, or exceeds it by at least two.</p><p><br></p><p dir="ltr">The main result is a dimension-transfer theorem that propagates the three bounds from one dimension to the next. The proof combines large coordinate traces, control of the number of projected defining directions, and an exact local identity relating size to the number of carrier directions. Explicit constructions attain all three bounds. The finite cases in dimensions eight and nine are supplied by the author's earlier paper, Exact Boundary Layers for Integer Two-Level Sections of the Nine-Cube.</p><p><br></p><p dir="ltr">This deposit contains the manuscript PDF, LaTeX source, and computational supplement. The supplement provides two independent Python implementations, an exact local-table checker, reference data, and a reconstruction-and-comparison driver. The programs check binomial identities, explicit constructions, and local carrier tables using integer or rational arithmetic. The dimension-transfer argument is proved in the manuscript.</p> 2026-09-29T10:20:40Z Text Preprint 10.6084/m9.figshare.34022592.v1 https://figshare.com/articles/preprint/The_Exact_Strict_Spectrum_of_Integer_Two-Level_Sections_of_Boolean_Cubes/34022592 CC BY 4.0