{
  "abstract": "We study the behavior of the $k$-th-order nonhomomorphicity of Boolean functions under restriction to affine subspaces. For odd $k\\ge3$ and $0\\le t<n$, we introduce the $t$-restriction $k$-th-order nonhomomorphicity $P_{k,t}(f)$, defined as the minimum of $P_k(f_{\\mid A})$ over affine spaces of codimension at most $t$. For a direct sum of $m$ monomials of degrees at least two, we determine $P_{k,t}$ exactly. When $t<m$, a minimum is obtained by setting one variable equal to zero in each of the $t$ monomials of lowest degree. When $m\\le t<n$, the minimum is zero. The proof combines multiplicativity of even Walsh moments with a contraction argument for affine restrictions. We establish a general formula for the Walsh transform of a Boolean function restricted to an affine subspace of arbitrary codimension, yielding an exact spectral characterization of the $t$-restriction normalized nonlinearity $\\rnnl_t(f)$. As odd $k$ tends to infinity, $(1-2P_{k,t}(f))^{1/(k+1)}$ converges to $1-2\\rnnl_t(f)$. We further prove that every non-affine function has a hyperplane restriction with strictly smaller odd-order nonhomomorphicity. Finally, we determine $\\rnnl_1(T_{d,n})$ exactly for all threshold functions with $n\\ge3$. This proves the two threshold-function conjectures left open in~\\cite{SergeAnaPante2025}, including the previously unresolved odd-majority case, where a minimizing hyperplane is $x_1+x_2=0$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.15868",
  "author_total": 3,
  "authors": [
    {
      "name": "Serge Feukoua"
    },
    {
      "name": "Ana Sălăgean"
    },
    {
      "name": "Pantelimon Stănică"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/598f1nk6cv3bq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.15868",
  "fulltext_url": null,
  "id": "598f1nk6cv3bq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.15868",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.IT"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "openalex",
      "1702|Coding theory and cryptography"
    ]
  ],
  "openalex": "W7221758659",
  "pdf_url": "https://arxiv.org/pdf/2610.15868",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/598f1nk6cv3bq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.15868",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.15868"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Higher-order nonhomomorphicity under affine restrictions",
  "title_text": "Higher-order nonhomomorphicity under affine restrictions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/598f1nk6cv3bq",
  "venue": "arXiv",
  "versions": []
}