立方截断的Hadamard单纯形的完整性、光滑性与正规性界
Integrality, smoothness and normality bounds for cube-truncated Hadamard simplices
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中文总结 AI 辅助
该研究解决了扩张 Hadamard 单纯形与立方体交集的完整性、光滑性与正规性问题，给出了维度十一的非完整示例、光滑性完全刻画及正规性界，并用 Lean 验证了所有编号结果。
中文摘要 AI 辅助
Santos 提出了一个问题：在给定的仿射格中，扩张的 Hadamard 单纯形与立方体的交集何时是完整的、光滑的或正规的。我们在维度十一构造了一个非完整示例，并证明了不存在更小维度的此类示例。我们完全刻画了光滑性，并证明了该族中每个光滑成员都是正规的。维度十五的一个显式示例表明，完整性本身并不蕴含正规性。对于阶数至少为十六的 Sylvester 单纯形，我们建立了一个尖锐的均匀完整性界，并在其下方立即构造了反例。我们还获得了针对一般 Hadamard 单纯形的充分正规性界，以及针对 Sylvester 族的更强界。证明使用了带有分离角切割的盒子整数分解以及三个有符号平板约束下的舍入。所有编号结果在 Lean 中都有形式化对应验证。
英文摘要
Santos asked when intersections of dilated Hadamard simplices with cubes are integral, smooth, or normal, in a prescribed affine lattice. We construct a nonintegral example in dimension eleven and prove that no smaller-dimensional example exists. We characterize smoothness completely and show that every smooth member of this family is normal. An explicit example in dimension fifteen shows that integrality alone does not imply normality. For Sylvester simplices of order at least sixteen, we establish a sharp uniform integrality bound and construct counterexamples immediately below it. We also obtain sufficient normality bounds for general Hadamard simplices and stronger bounds for the Sylvester family. The proofs use integer decomposition for boxes with separated corner cuts and rounding under three signed slab constraints. All numbered results have formal counterparts verified in Lean.
