Spectral Mass Gap in Self-Consistent Soliton Backgrounds: Emergence, Classification and Screening

@misc{dechery2026spectral,
  title = {{Spectral Mass Gap in Self-Consistent Soliton Backgrounds: Emergence, Classification and Screening}},
  author = {Fabien Déchery},
  howpublished = {HAL},
  year = {2026},
  month = oct,
  url = {https://hal.science/hal-05793310v2},
  note = {CC BY-NC-ND 4.0},
}

TY  - UNPB
TI  - Spectral Mass Gap in Self-Consistent Soliton Backgrounds: Emergence, Classification and Screening
AU  - Fabien Déchery
PY  - 2026
DA  - 2026/10/08
PB  - HAL
UR  - https://hal.science/hal-05793310v2
AB  - 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 < ε_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 π_3(G) = ℤ (Bott periodicity). Two independent analytical verifications are given: (i) the exact transcendental equation ε/M_q = g(x) > 0 for the step-function profile (Proposition 1); (ii) the exact self-consistency ε_val = ω 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 π_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) π_k(M) ≠ 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(τ) 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 ℝ^4 satisfying the Wightman axioms, then Δ > 0. The proof rests on three pillars: (I) the trace anomaly (β_0 > 0, [24]) gives a physical scale Λ > 0; (II) no-go theorems exclude massless glueballs in every spin channel J^PC; (III) dimensional analysis gives Δ = cΛ 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 ℝ^4 satisfying the Osterwalder-Schrader axioms, with mass gap Δ > 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(τ) = Tr(e^−τ 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 ζ_D(s) is meromorphically regular, with a functional determinant ratio det(D^2_sol/D^2_vac) ≈ 1.61.
ER  -