{
  "abstract": "This paper establishes higher-order snake equations within the framework of\n\nadjoint chains in tensor categories, and strictly restricts their physical carrier to\n\nthe encoding–recovery composite of concatenated quantum codes. The core con\n\nclusions have a three-layer structure. First, the snake equations induced by duality\n\nhold automatically by the coherence theorem for compact closed categories and\n\ncarry no error-correction information. Second, the encoding–recovery composite\n\nholds in the exact-recovery case if and only if the Knill–Laflamme condition is sat\n\nisfied; in the broken case, it is characterized by two-sided inequalities between the\n\nlogical error rate and the deviation of the snake diagram in diamond norm. Third,\n\ntaking the Steane [[7,1,3]] code as the component code, within the model of a\n\nsingle bit-flip sector, ideal syndrome measurements, and per-syndrome minimum\n\nweight decoding, we exactly enumerate the decoding failure set of all 128 errors,\n\nobtain a degree-seven polynomial for the logical error rate and the complete er\n\nror spectra for concatenation depths one through six; the threshold fixed point is\n\n0.0645962393 (single sector), and the leading coefficient 21 is determined by the\n\ncombinatorial fact that none of all two-qubit errors escapes. Under complement\n\nsymmetry, the fixed-point ledger is completed to five points (including the end\n\npoint 1), and the basin-of-attraction structure is closed accordingly. The recursion\n\nstructure of the cascaded error spectra is an exact equality under independent\n\nand identically distributed Pauli noise, blockwise independent decoding, and per\n\nsyndrome minimum-weight strategies; the corresponding relations under full Pauli\n\nnoise, correlated noise, measurement noise, and non-Pauli effective channels are\n\nexplicitly registered as open boundaries, and partial results are not used to replace\n\ngeneral conclusions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 2,
  "authors": [
    {
      "name": "changzheng zhou",
      "orcid": "0009-0002-4186-7877"
    },
    {
      "name": "ziqing zhou",
      "orcid": "0009-0005-6883-6300"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6zw101jd5kyr3.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23493971",
  "fulltext_url": null,
  "id": "6zw101jd5kyr3",
  "issns": [],
  "landing_url": "https://zenodo.org/records/23493971",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [],
  "openalex": null,
  "pdf_url": "https://zenodo.org/api/records/23493971/files/Self-Referential%20Fibration%20Snake%202;%20Higher-Order%20Snake%20Equations%20and%20Cascaded%20Error%20Spectra%20of%20Topological%20Codes.pdf/content",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6zw101jd5kyr3.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:19Z",
      "native_id": "23493971",
      "source": "zenodo",
      "url": "https://zenodo.org/records/23493971"
    }
  ],
  "subjects": [],
  "title": "Self-Referential Fibration Snake 2: Higher-Order Snake Equations and Cascaded Error Spectra of Topological Codes",
  "title_text": "Self-Referential Fibration Snake 2: Higher-Order Snake Equations and Cascaded Error Spectra of Topological Codes",
  "type": "preprint",
  "updated_at": "2026-11-03T12:21:19Z",
  "url": "https://research.draconitic.com/paper/6zw101jd5kyr3",
  "venue": "Zenodo",
  "versions": []
}