平面圆盘- slab的整数占据切片的严格四层定理
A Sharp Four-Layer Theorem for Integer-Occupied Slices of a Planar Disk-Slab
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中文总结 AI 辅助
该研究针对平面圆盘-slab 的整数占据切片，证明了其格点最多占据四层的严格定理，通过构造显式实例达到该界，并利用直径估计、不等式与证书完成反例排除与模式归约。
中文摘要 AI 辅助
将满秩仿射陪格上的有理正定二次次水平集与闭 slab 相交，确定性二维 delta=3/4 归约选取本原对偶整数泛函与内椭圆。当对应 Babai 点位于椭圆外时，圆盘-slab 中的格点最多占据所选泛函的四个整数层。存在显式有理实例达到四层，故该界是严格的。严格直径估计将任何反例归约为五连续层的三个块，最近整数不等式排除中间块，对称性与单元定位将两侧块归约为八种模式：四种源于共享容量不等式，四种源于精确四变量 Bernstein 证书，该证书含 17640 个正系数，最小值为 625/2048。
英文摘要
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.
