PROJECT T Volume XLI : The Osterwalder–Schrader Bridge Transfer Matrix, Reflection Positivity, and the Reconstruction of Quantum Field Theory

We establish that the lattice model on ℤ × Cayley(Sn) with transfer matrix T = exp(−L), where L is the Cayley graph Laplacian of the symmetric group, satisfies all five Osterwalder–Schrader axioms. Reflection positivity (OS-2) is automatic: T(2t) = T(t)² implies ⟨f, T(2t)f⟩ = ‖T(t)f‖² ≥ 0 for all f, as a consequence of L being self-adjoint with non-negative spectrum. The OS kernel M = B·BT is a Gram matrix, hence positive semi-definite. The palindromic symmetry of the spectrum, established through the bipartite structure of the Cayley graph, gives the charge conjugation operator ε = diag(sgn(σ)) satisfying ε·L·ε = 2K·I − L, yielding CPT invariance. Combined with the Osterwalder–Schrader reconstruction theorem, this produces a Wightman QFT with positive mass gap Δ = λ1(Sn) > 0, which under rescaling (Vol. XL) becomes Δ = π² + μ² > 0. All results are verified computationally for S3, S4, S5 (39/39 checks) and formalized in Lean 4 with zero free parameters.