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Prescribed-Difference Matchings with Four and Eight Holes: Fourier Filters for Compatible Boundaries — arXiv:2608.27547

Source: arXiv (en)
Original: https://arxiv.org/abs/2608.27547
License (as archived): CC0-1.0
Archived: 2026-10-17T11:22:55.883Z
SHA-256 of the archived text: 2ea3dd91f82c8ea788e153892d52b584074f13c560872718f1feae547b089fe6
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Prescribed-Difference Matchings with Four and Eight Holes: Fourier Filters for Compatible Boundaries — arXiv:2608.27547 — arXiv (en), https://arxiv.org/abs/2608.27547 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_385a2eff504a (Metraipolis archive dc_385a2eff504a, sha256 2ea3dd91f82c8ea788e153892d52b584074f13c560872718f1feae547b089fe6)

BibTeX

@misc{dc385a2eff504a,
  title = {Prescribed-Difference Matchings with Four and Eight Holes: Fourier Filters for Compatible Boundaries — arXiv:2608.27547},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2608.27547},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_385a2eff504a; sha256 2ea3dd91f82c8ea788e153892d52b584074f13c560872718f1feae547b089fe6},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2608.27547. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

Let $s\geq 2$ and let $v_1,\ldots,v_{2^{s-1}}\in F_2^s\setminus\{0\}$ have sum zero. The prescribed-difference matching problem asks whether $F_2^s$ can be partitioned into pairs whose differences, counted with multiplicity, are exactly these vectors. Fix a hyperplane $H\leq F_2^s$. An $h$-hole instance is one in which exactly $h$ prescribed differences lie in $H$, counted with multiplicity. We prove that every four-hole instance has a solution for $s\geq 3$ and every eight-hole instance has a solution for $s\geq 4$. Thus no restriction on the multiplicities or on the sum of the holes is needed beyond the global zero-sum condition. After the internal pairs are placed, their endpoints are deleted from the two affine halves determined by $H$, and the remaining vectors must be paired with the prescribed crossing differences. A locally valid placement of the internal pairs need not admit such a completion. We encode the possible completions by coefficients of signed determinants and use the Walsh transform to sum over all disjoint legal placements while keeping the crossing profile fixed. Nonvanishing of this sum guarantees that at least one placement extends to a full matching. The four-hole and zero-sum eight-hole theorems are proved theoretically. The general eight-hole proof uses three finite exact verifications, independent of the crossing profile: a local filter calculation, a classification of the remaining eight-hole configurations, and integer certificates for those configurations. The verification software, exact inputs, and recorded outputs are archived in a versioned Zenodo record.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-08-31 in math.CO).
Categorie arXiv: Combinatorics (math.CO).
Pagina del lavoro: https://arxiv.org/abs/2608.27547 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-17T11:06:51.755Z dalla pagina https://arxiv.org/catchup/math.CO/2026-08-31?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).