{
  "abstract": "We establish that the spectral mass gap is an emergent topological invariant of nonlinear self-consistent field theories, classify the conditions under which it arises, and analyze its transmission through the one-body projection. **Theorem 1** (Spectral gap — unconditional). The Dirac operator in any topologically non-trivial self-consistent soliton background ($B = 1$, hedgehog, monotone profile) has a strictly positive spectral gap: $0 < \\varepsilon_\\mathrm{val} < M_q$. The proof combines the non-normalizability of the zero-energy solution (Weyl limit-point classification via Wronskian analysis) with the monotonicity of the spectral flow. The topological argument extends to any compact simple Lie group $G$ via $\\pi_3(G) = \\mathbb{Z}$ (Bott periodicity). Two independent analytical verifications are given: (i) the exact transcendental equation $\\varepsilon/M_q = g(x) > 0$ for the step-function profile (Proposition 1); (ii) the exact self-consistency $\\varepsilon_\\mathrm{val} = \\omega$ in the 1D Soler model (Emergence Paper 0 [11]). The gap is an emergent property of the nonlinear substrate: it is absent in linear theories (no self-trapping) and in theories with trivial $\\pi_3$ (no topological protection). A **classification criterion** is established (Theorem 4, §8): a field theory has a spectral mass gap in the soliton sector if and only if (i) nonlinear self-trapping produces a stationary soliton with a bound valence mode, and (ii) $\\pi_k(M) \\neq 0$ provides spectral flow protection. Six test cases spanning two topological mechanisms (soliton-driven and instanton-driven), four space-time dimensions, and three topological classes are verified. The two mechanisms are unified by the Atiyah-Manton duality: the soliton is the holonomy of an instanton, and the heat kernel $K(\\tau)$ reads the gap identically in both cases. The CP$^1$ bridge identifies the rigorously proven 2D mass gap (Polyakov) as the lower-dimensional instance of the same topological structure. **Theorem 2** (Mass gap — conditional on existence). If a non-trivial quantum Yang-Mills theory exists on $\\mathbb{R}^4$ satisfying the Wightman axioms, then $\\Delta > 0$. The proof rests on three pillars: (I) the trace anomaly ($\\beta_0 > 0$, [24]) gives a physical scale $\\Lambda > 0$; (II) no-go theorems exclude massless glueballs in every spin channel $J^{PC}$; (III) dimensional analysis gives $\\Delta = c\\Lambda$ with $c > 0$. All proof steps are mathematically certain (Table 6). **Theorem 3** (Existence and mass gap). For any compact simple Lie group $G$, a non-trivial quantum Yang-Mills theory exists on $\\mathbb{R}^4$ satisfying the Osterwalder-Schrader axioms, with mass gap $\\Delta > 0$. The gap is a spectral property that survives the one-body projection: Class A observables (masses, charge radii) are reproduced within 2% (Emergence Paper VII [17]). The deficit hierarchy for Class B--D observables is fully explained by the triple screening structure (Emergence Paper VIII [18]) — the structural limitation of the one-body approximation in representing inter-sector correlations — without compromising the gap. The same heat kernel $K(\\tau) = \\mathrm{Tr}(e^{-\\tau D^2})$ that encodes the emergent gauge Lagrangian in its UV regime (Emergence Paper V [12]) yields the mass gap in its IR regime, with 0.08% agreement over three decades of proper time. The spectral zeta function $\\zeta_D(s)$ is meromorphically regular, with a functional determinant ratio $\\det(D^2_\\mathrm{sol}/D^2_\\mathrm{vac}) \\approx 1.61$.",
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  "article_type": null,
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  "authors": [
    {
      "name": "Fabien Déchery"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3rft9ksyr7y70.bib",
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  "landing_url": "https://hal.science/hal-05793310v2",
  "language": "en",
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  "pdf_url": "https://hal.science/hal-05793310/document",
  "pmcid": null,
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  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3rft9ksyr7y70.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:02Z",
      "native_id": "hal-05793310",
      "source": "hal",
      "url": "https://hal.science/hal-05793310v3"
    }
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  "subjects": [
    {
      "field": "mathematics",
      "origin": "hal:1.math.math-fa",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "hal:1.phys.mphy",
      "subfield": "mathematical-physics"
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    {
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      "subfield": "high-energy"
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    {
      "field": "physics",
      "origin": "hal:1.phys.nucl",
      "subfield": "nuclear"
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  "title": "Spectral Mass Gap in Self-Consistent Soliton Backgrounds: Emergence, Classification and Screening",
  "title_text": "Spectral Mass Gap in Self-Consistent Soliton Backgrounds: Emergence, Classification and Screening",
  "type": "preprint",
  "updated_at": "2026-11-07T12:20:25Z",
  "url": "https://research.draconitic.com/paper/3rft9ksyr7y70",
  "venue": "HAL",
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