Integrality, smoothness and normality bounds for cube-truncated Hadamard simplices

@misc{lebedev2026integrality,
  title = {{Integrality, smoothness and normality bounds for cube-truncated Hadamard simplices}},
  author = {Nikita Lebedev},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.32950},
  eprint = {2609.32950},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.32950},
}

TY  - UNPB
TI  - Integrality, smoothness and normality bounds for cube-truncated Hadamard simplices
AU  - Nikita Lebedev
PY  - 2026
DA  - 2026/09/26
PB  - arXiv
DO  - 10.48550/arxiv.2609.32950
UR  - https://arxiv.org/abs/2609.32950
AB  - 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.
ER  -