{
  "abstract": "Can excitation structure classify topological quantum codes up to finite-depth local circuits? For translation-invariant topological Calderbank--Shor--Steane (CSS) codes in arbitrary dimensions, we introduce excitation superselection profiles $\\mathscr E_σ$ ($σ=X,Z$). Their cohomology layers $E_σ^\\ell:=H^\\ell(\\mathscr E_σ)$ encode excitation types, fusion rules, and translation action, while $\\mathscr E_σ$ retains cross-degree gluing. Pointlike, looplike, and higher-dimensional excitations correspond to degrees $\\ell=1,2,\\ldots$, respectively. Our matrix-level Schanuel-lemma method proves single-degree superselection rigidity: if $\\mathscr E_σ$ is supported in a single degree $\\ell$, then $E_σ^\\ell$ and $\\ell$ fix the stable translation-invariant Clifford class under a finite-resolution hypothesis automatic for prime qudits. If $E_σ^\\ell$ is also finite, prime-qudit codes decouple into toric-code stacks by Clifford circuits after coarse graining, with the exact minimal threshold set by the translation kernel. We develop a unified topological theory of Koszul-complex models (encompassing toric codes, bivariate-bicycle codes, and Haah's cubic code): for regular models, we compute $E_σ^\\ell$, prove $\\mathscr E_σ$ is single-degree, express finite-torus logical modules as $\\mathrm{Tor}$ groups, and obtain coarse-grained toric-stack normal forms at full length. Two constructions extend this framework to realize any admissible single-degree profile and build multi-degree extensions. The latter yields eight 3D qubit toric-code extensions with identical layers in both Pauli sectors and trivial translation action, yet distinct size-dependent ground-state degeneracies. The seven nonsplit models require blocking to decouple. Thus, cross-degree gluing carries translation-symmetry-enriched topological information invisible to individual layers.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.09913",
  "author_total": 1,
  "authors": [
    {
      "name": "Hao Song"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cxtf2jg9e3v98.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.09913",
  "fulltext_url": null,
  "id": "cxtf2jg9e3v98",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.09913",
  "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",
      "cond-mat.str-el"
    ],
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "1702|Quantum Computing Algorithms and Architecture"
    ]
  ],
  "openalex": "W7202189273",
  "pdf_url": "https://arxiv.org/pdf/2608.09913",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cxtf2jg9e3v98.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:28Z",
      "native_id": "2608.09913",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.09913"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.str-el",
      "subfield": "condensed-matter"
    },
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Superselection Rigidity of Topological Calderbank--Shor--Steane Codes: From Koszul-Complex Models to Nonsplit Extensions",
  "title_text": "Superselection Rigidity of Topological Calderbank--Shor--Steane Codes: From Koszul-Complex Models to Nonsplit Extensions",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:53Z",
  "url": "https://research.draconitic.com/paper/cxtf2jg9e3v98",
  "venue": "arXiv",
  "versions": []
}