{
  "abstract": "On a one-dimensional exactly solvable Ising ring, this paper constructs the com\n\nposite difference of two coarse-graining paths in order to characterize explicit co\n\nordinates of the commutation obstruction. The two paths are, respectively, spatial\n\nblocking followed by temperature stepping and temperature stepping followed by\n\nspatial blocking. The effective energy difference is denoted by ∆(b), and the gauge\n\nobstruction is defined as δ∗(b) = ∆(b)−lnfib(b), where fib(b) is the cardinality of the\n\nfiber of block configurations b under the coarse-graining map. The gauge decompo\n\nsition gives ∆(b) = lnfib(b) +δ∗(b), δ∗(b) = ln[Mb(βJ)2/Mb(2βJ)], where Mb(µ) is\n\nthe uniform fiber mean. When βJ ̸ = 0, δ∗(b) ≤ 0 follows from the Cauchy–Schwarz\n\ninequality on the uniform distribution over the fiber, with equality if and only if\n\nthe bond variable is constant on the fiber; when βJ = 0, δ∗ degenerates to the zero\n\nfunction and provides no diagnostic of fiber constancy. In the small-coupling ex\n\npansion, the first-order term cancels identically, the degeneracy term is eliminated\n\nfiber by fiber in the gauge baseline, and the first surviving term is −λ2κ2(b), where\n\nκ2(b) is the variance of the bond variable on the fiber; if C(R) = √︁ ∑︁ b κ2(b)2 > 0,\n\nthen ∥δ∗∥2 = C(R)λ2(1 + O(λ)). For the fully enumerated parameterized simula\n\ntion of the N = 8 periodic Ising ring, the leading coefficients of the two rules are\n\nC(D) = 35.7771 and C(S) = 17.6677, and the L2 ratio is about 2.025, showing that\n\nthe leading coefficient of the quadratic scaling is a functional of the coarse-graining\n\nrule. In the componentwise vertical cell coordinate framework, the commutation ob\n\nstruction is carried by ∆ as the vertical coordinate; the conjugacy criterion uses the\n\ndiagonal-operator similarity modulo additive constants, the mean-centered compo\n\nnent multisets of the eighteen cases are pairwise distinct, and neither relabeling-type\n\nnor general linear-similarity-type 1-morphism corrections exist. Two-dimensional\n\nscaling, all-order Laurent coefficients, and higher-level carriers of multi-rule cyclic\n\nconsistency are left as open problems.",
  "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/09met0axy5we1.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23529425",
  "fulltext_url": null,
  "id": "09met0axy5we1",
  "issns": [],
  "landing_url": "https://zenodo.org/records/23529425",
  "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/23529425/files/Self-Referential%20Fibration%20Figure%201;%20Naturality%20Failure%20of%20Coarse-Graining---Explicit%202-Morphism%20Coordinates%20of%20the%20Commutation%20Obstruction%20-%20%E5%89%AF%E6%9C%AC.pdf/content",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/09met0axy5we1.ris",
  "sources": [
    {
      "first_seen": "2026-11-04T12:21:16Z",
      "native_id": "23529425",
      "source": "zenodo",
      "url": "https://zenodo.org/records/23529425"
    }
  ],
  "subjects": [],
  "title": "Self-Referential Fibration Figure 1: Naturality Failure of Coarse-Graining—Explicit 2-Morphism Coordinates of the Commutation Obstruction",
  "title_text": "Self-Referential Fibration Figure 1: Naturality Failure of Coarse-Graining—Explicit 2-Morphism Coordinates of the Commutation Obstruction",
  "type": "preprint",
  "updated_at": "2026-11-04T12:21:16Z",
  "url": "https://research.draconitic.com/paper/09met0axy5we1",
  "venue": "Zenodo",
  "versions": []
}