The Exact Strict Spectrum of Integer Two-Level Sections of Boolean Cubes

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.