{
  "abstract": "The topological gap $Δ$ -- the excess $H_1$ total persistence of a critical point cloud over a density-matched null -- scales as $Δ\\sim L^{d+η}$. We derive this analytically: the spectral integral $I(α) = \\sum_{k\\neq 0} S_{\\mathrm{conn}}(k)\\,|k|^α$ scales as $L^{2-α-η}$ when IR-dominated, giving $I(-2η) \\sim L^{d+η}$. The decomposition $I(-2η) = I_0 \\cdot I_{\\mathrm{shape}}$ separates volume ($I_0 \\propto N(1-m^2) \\sim L^d$) from anomalous dimension ($I_{\\mathrm{shape}} \\sim L^η$); the volume factor accounts for the magnetization-driven per-configuration variance of $Δ$. We prove the mechanism requires $d < 2 + η$ (IR dominance), confining it to $d = 2$ for physical systems; in $d = 3$ the spectral integral is UV-dominated, explaining why density normalization is needed. An $α$-sweep for Potts $q = 4$ at $L = 32$--$256$ finds $α_{\\mathrm{opt}}$ in $[-0.75, -0.5]$, consistent with $-2η_{\\mathrm{Ising}}$ and inconsistent with $-2η_{q=4} = -1$; we flag this as tentative pending $L \\geq 1024$ confirmation. The $\\langle m^2 \\cdot I(-2η)\\rangle$ hyperscaling product is dominated by the correlation $r(m^2, I) \\approx -0.98$ via the shared $I_0$ amplitude, so we report it as a covariance-correction analysis. Under a heuristic argument extending Divol--Polonik to inhomogeneous Poisson intensities, the bare PH kernel is flat; the effective kernel acquires $k$-dependence only at criticality. The per-configuration agreement between $Δ$ and $I(-0.5)$ is primarily a magnetization correlation: $R^2 = 0.91$ at $L = 256$ collapses to $R^2 \\approx 0$ once $|M|$ is partialed out. Per-configuration evidence corroborates the $I_0$ Parseval identity but not the $|k|^{-2η}$ shape factor; the latter is established by ensemble $L$-scaling.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.09159",
  "author_total": 1,
  "authors": [
    {
      "name": "Matthew Loftus"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dr7wj3wj0jdvf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.09159",
  "fulltext_url": null,
  "id": "dr7wj3wj0jdvf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.09159",
  "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.stat-mech"
    ],
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "math.AT"
    ],
    [
      "openalex",
      "3104|Theoretical and Computational Physics"
    ]
  ],
  "openalex": "W7212186252",
  "pdf_url": "https://arxiv.org/pdf/2609.09159",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dr7wj3wj0jdvf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:12Z",
      "native_id": "2609.09159",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.09159"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AT",
      "subfield": "geometry-topology"
    },
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.stat-mech",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "Spectral origin of the topological gap exponent d + η: mechanism, kernel, decomposition, and scope",
  "title_text": "Spectral origin of the topological gap exponent d + η: mechanism, kernel, decomposition, and scope",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:15Z",
  "url": "https://research.draconitic.com/paper/dr7wj3wj0jdvf",
  "venue": "arXiv",
  "versions": []
}