A Sharp Four-Layer Theorem for Integer-Occupied Slices of a Planar Disk-Slab

Let a rational positive-definite quadratic sublevel set on a full-rank affine lattice coset be intersected with a closed slab. A deterministic two-dimensional delta=3/4 reduction selects a primitive-dual integer functional and an inner ellipse. In the branch where the corresponding Babai point lies outside that ellipse, the lattice points in the disk-slab occupy at most four integer levels of the selected functional. An explicit rational instance attains four, so the bound is sharp. A strict diameter estimate reduces any counterexample to three blocks of five consecutive levels, and a nearest-integer inequality excludes the central block. Symmetry and cell localization reduce the two side blocks to eight modes: four follow from a shared-cap inequality, and four from an exact four-variable Bernstein certificate. The certificate contains 17,640 positive coefficients, with minimum 625/2048.

Source: arXiv:2608.12204v1 - http://arxiv.org/abs/2608.12204v1 PDF: https://arxiv.org/pdf/2608.12204v1 Original Link: http://arxiv.org/abs/2608.12204v1