{
  "abstract": "Motivated by Eliahou's 64-modular Hadamard construction at the smallest open Hadamard order n=668, we introduce an explicit half-flip family of 32-modular Hadamard matrices at orders n=4L for L = 3 (mod 8). A master identity reduces the four-sequence Golay-quadruple condition under the half-flip ansatz (s, s*, sq, (sq)*) to a single-sequence type-restricted autocorrelation c_k^tau(s). The construction yields a closed-form expression for c_k^tau, a true Hadamard matrix at L=11, and 32-modular matrices at every L = 3 (mod 8), including the open orders n=716 and n=1132. Existence of 32-modular at L = 3 (mod 4) is due to Eliahou-Kervaire (2001); Eliahou's 2026 follow-up in J. Algebraic Combin. gives 64-modular matrices at L = 3 (mod 16) and L = 7 (mod 32) via the same ansatz and correlation identity we call the master identity, subsuming our construction on that residue subclass. Our contribution is therefore primarily structural. Applying Barrera Acevedo-O Cathain-Dietrich (2019) and Alvarez et al. (2020), the family is non-cocyclic over any group at every prime L in {11, 19, 59} in the YES set, yet pseudococyclic over the Goethals-Seidel Moufang loop GS_{4L} at every L. A half-flip H-set decomposition theorem parameterizes the symmetric difference of any two family elements by a single sequence flip set, giving the family the structure of a length-L Hamming cube. A symbolic mod-tower verifier (mod-8 is F_2-linear) classifies true Hadamards in the family through k=14 (L <= 115): the YES set is empirically bounded by k=7, refuting four H4 predictions and excluding L in {179, 283} within the ansatz. A Grobner basis at L=11 exhibits a previously unrecorded even-T0-block linear identity. All code and JSON certificates: github.com/michelkulhandjian/hadamard-halfflip-structural",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.25543",
  "author_total": 1,
  "authors": [
    {
      "name": "Michel Kulhandjian"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6sqaw4ty69d8n.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.25543",
  "fulltext_url": null,
  "id": "6sqaw4ty69d8n",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.25543",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1702|Coding theory and cryptography"
    ]
  ],
  "openalex": "W7214111945",
  "pdf_url": "https://arxiv.org/pdf/2609.25543",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6sqaw4ty69d8n.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.25543",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.25543"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "An explicit half-flip family of 32-modular Hadamard matrices at L = 3 mod 8: structural placement and mod-tower analysis",
  "title_text": "An explicit half-flip family of 32-modular Hadamard matrices at L = 3 mod 8: structural placement and mod-tower analysis",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/6sqaw4ty69d8n",
  "venue": "arXiv",
  "versions": []
}