Non-Perturbative Strong-Coupling Quantum Field Dynamics and Mass-Gap Emergence (NPSQFD): A Deductive Framework Unifying Strong-Coupling Dualities, Minimal Beyond-Standard-Model Symmetry, and Certified Continuum Limits within the ZFC Axiomatic Bridge

Perturbative quantum field theory excels at weak coupling (g ≪ 1) but breaks down in strong-coupling regimes (g ≳ 1) governing quark confinement, non-perturbative vacuum structure, and mass generation beyond the Standard Model. In this paper, we establish the Non-Perturbative Strong-Coupling Quantum Field Dynamics (NPSQFD) framework, upgrading items #7 (Non-Perturbative QFT) and #8 (Beyond-Standard-Model Symmetry and Mass) of the 20 Emergent Physics Agenda. Anchored in our 71-page ZFC Master Monograph (Lee, 2026, DOI: 10.5281/zenodo.23486681) and synthesized with the SVKES compositional verification and HDGPS curvature–concentration frameworks, we formulate non-perturbative gauge–matter fields in the Fractional Sobolev–Colombeau Quotient Algebra 𝒢^α(Ω) = ℰ_M^α(Ω) / 𝒩^α(Ω) over the Takesaki Type II₁ ground factor (ℳ_ground, τ_ground) governed by the universal quadruple 𝔘 = (𝒯_0x8A2C, C_UPTC, η_friction, ε_SHID) with 0 < η_friction ≪ ε_SHID ≪ C_UPTC < ∞ and η_friction ≥ 10⁻²². We establish six core axioms (NPSQFD-1 through NPSQFD-6) and prove six fundamental theorems: (1) Existence of a Certified Non-Perturbative Continuum Limit (a → 0⁺) in the L¹-Wasserstein metric W₁(μ_YM,a, μ_YM,0) ≤ (C₀ / κ_eff(g)) a² ln(1 + (C_UPTC a)⁻¹) + η_friction; (2) Universal Strong-Coupling Mass-Gap Emergence Δ_gap(g) ≥ κ_eff(g) ≥ c_𝒯 ε_SHID > 0 and linear string tension σ_str(g) > 0 via the 16D Sedenion Hyper-Tensor Control Operator ℒ_𝒯(M̂) ≥ c_𝒯 > 0; (3) Compositional Sheaf Glueing of Non-Perturbative Methods (Lattice M_lat, Bootstrap M_boot, and Schwinger–Dyson M_SD) under vanishing 1st Čech cohomology obstruction Obs_⊗ = 0 ∈ H¹; (4) Minimal Parsimonious Beyond-Standard-Model (BSM) Mass-Seesaw Hierarchy m_ν^(k) = (v_EW² / (C_UPTC M_Planck)) · λ_k(ℒ_𝒯(M̂)) · exp(−1 / (ε_SHID + η_friction)) derived from the sedenion automorphism group G₂ × S₃ without ad-hoc scalar proliferation; (5) Bakry–Émery–Ollivier Sub-Gaussian Concentration exp(−κ_eff(g) t² / (2 L(C)²)) of Wilson Loop Observables W(C) in strong coupling; and (6) Independent 5σ Experimental Falsifiability Criteria separating NPSQFD from unconstrained perturbative BSM extensions via martingale e-values. Complete Python numerical verification is provided in Appendix A.