A Rigorous Mass Gap for SU(2) Lattice Yang–Mills at Strong Coupling via Spin-Foam Recoupling and Vertex Contraction

We provide a rigorous, fully gauge-invariant construction of pure SU(2) lattice Yang–Mills theory in four Euclidean dimensions, formulated on the surface (spin-foam) representation of the Wilson partition function, and prove a strictly positive mass gap in the strong-coupling regime β < β*. The value of β* is explicit and depends only on the bare plaquette activity and the combinatorial entropy of the lattice. The proof combines three structural ingredients: (i) a vertex-amplitude contraction mechanism derived from the Biedenharn–Elliott orthogonality of SU(2) 6j symbols, yielding a uniform estimate E[|A_v|] ≤ δ(β) < 1 for the {15j} vertex amplitude; (ii) a polymer block-spin renormalization group on a Banach space of effective surface interactions, with all combinatorial constants controlled by explicit bounds (lattice-animal entropy via Eden–Madras peeling, dependency-graph independence ratio via Bissacot–Fernández–Procacci–Scoppola); (iii) the Osterwalder–Schrader reconstruction with reflection positivity inherited from the Wilson measure via the exact Peter–Weyl character expansion. We prove unconditionally, for β ∈ (0, β*): convergent Fernández–Procacci polymer expansion, monotone surface-tension growth with an explicit rate c0 > 0, geometric contraction of the irrelevant sector, exponential decay of truncated correlations of bounded local gauge-invariant observables, and all four Osterwalder–Schrader axioms for the infinite-volume measure μβ. The resulting physical Hilbert space H_phys carries a positive self-adjoint Hamiltonian H_phys whose spectral gap satisfies gap(H_phys) = gap(ℒ) ≥ m∞(β) > 0, where ℒ is the Markov generator of the associated equilibrium dynamics. The mass gap m∞(β) is explicit in terms of the bare coupling β, the lattice geometry, and the contraction constant δ(β). All constants in the construction are either bounded by elementary combinatorial expressions or computable to certified precision (numerical evaluation of the {15j} symbol via interval arithmetic). The result is, to our knowledge, the most explicit closed-form construction of the mass gap for SU(2) lattice Yang–Mills at strong coupling in the surface-ensemble literature. For SU(2) in d = 4, our threshold β* improves on the Shen–Zhu–Zhu bound β_SZZ = 1/48 by a factor of roughly 20, while operating in a manifestly gauge-invariant surface-ensemble framework complementary to the stochastic Langevin approach. We do not claim resolution of the Clay Millennium Prize formulation; the construction is restricted to the fixed-lattice strong-coupling regime. The extension to all β > 0 (the physical continuum limit) is formulated as a precise structural problem, and we discuss its connection to the Balaban, Magnen–Rivasseau–Sénéor, and Chatterjee–Cao–Sheffield programmes.