{
  "abstract": "In real-space renormalization, the two operations of aggregating spin configura\n\ntions by a block map and adjusting source parameters generally do not commute.\n\nThis paper splits this noncommutativity into two separable bookkeeping objects:\n\nthe fiber baseline depends only on the fiber cardinality and is a purely combi\n\nnatorial quantity of the block map; the gauge obstruction is defined by the fiber\n\nmoment-generating function and carries coupling-dependent energy-distribution in\n\nformation. For an arbitrary surjective block map on a one-dimensional Ising ring,\n\nthe n-th order λ-coefficient of the gauge obstruction equals negative (2n − 2)/n!\n\ntimes the n-th cumulant of the fiber energy distribution; the first-order cancella\n\ntion follows from 2 − 2n vanishing at n = 1. The conversion between cumulants\n\nand moments is accomplished by Bell polynomials, and the moments are written as\n\nweighted sums of fiber Walsh coefficients over multisets of bond occupations. When\n\nthe block map is a GF(2)-linear map, the fiber Walsh coefficients degenerate into\n\nthe product of the indicator function of the row space and a sign factor; at general\n\nrank this sign factor is independent of the choice of representative of the solution,\n\nand full row rank is only the special case in which the solution is unique. Conse\n\nquently, the computation of moments reduces to signed pure counting grouped by\n\nthe mod-2 sum of bond incidence vectors. For the dual-bit extraction rule, odd\n\norder cumulants vanish, and even-order cumulants are the Rademacher even-order\n\ncumulants multiplied by power sums of adjacent sums of block values; the explicit\n\nexpression for κ2 agrees pointwise with this closed form on all fibers for N = 8. The\n\nnon-positivity of the gauge obstruction is given by the Cauchy–Schwarz inequality,\n\nwith equality at nonzero source parameters if and only if the fiber energy is con\n\nstant. This non-positivity does not constrain the signs of the coefficients order by\n\norder; higher-order coefficients may change sign. At the algorithmic level, coeffi\n\ncient enumeration for fixed order is polynomial in N, but the overall complexity for\n\nan arbitrary block map involves 2N enumeration or a 2r row-space table, and does\n\nnot constitute a polynomial-time scheme in the general case.",
  "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/7szh11j34skha.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23557760",
  "fulltext_url": null,
  "id": "7szh11j34skha",
  "issns": [],
  "landing_url": "https://zenodo.org/records/23557760",
  "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/23557760/files/Self-Referential%20Fibration%20Figure%204;%20Full-Order%20Combinatorial%20Enumeration%20Formula%20for%20Gauge%20Obstructions%20on%20Coarse-Graining%20Fibers.pdf/content",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7szh11j34skha.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T12:20:41Z",
      "native_id": "23557760",
      "source": "zenodo",
      "url": "https://zenodo.org/records/23557760"
    }
  ],
  "subjects": [],
  "title": "Self-Referential Fibration Figure 4: Full-Order Combinatorial Enumeration Formula for Gauge Obstructions on Coarse-Graining Fibers",
  "title_text": "Self-Referential Fibration Figure 4: Full-Order Combinatorial Enumeration Formula for Gauge Obstructions on Coarse-Graining Fibers",
  "type": "preprint",
  "updated_at": "2026-11-05T12:20:41Z",
  "url": "https://research.draconitic.com/paper/7szh11j34skha",
  "venue": "Zenodo",
  "versions": []
}