ALGEBRAIC RESOLUTION — Volume III :The Complete Chain From Two Axioms to Five Predictions via Confinement

This volume unifies two previously separate formalizations into a single, self-contained proof chain. Part A establishes quantum lattice confinement: the commutation deficit C(K) = K(K−3)/2 uniquely selects K = 3, the S₃ Cayley graph Laplacian has verified spectrum {0, 1, 1, 3, 3, 4} with spectral gap Δ = 1, and string tension σ = ln(3) follows from plaquette order 3. Part B traces the complete derivation chain: from two axioms (idempotence T² = T and braid relations aba = bab), through eight algebraic steps, to five falsifiable predictions about neutrino physics—with zero free parameters and zero sorry. Part C synthesizes both into a single Master Theorem verified in Lean 4 with 70+ machine-checked theorems. Keywords: 0-Hecke monoid, confinement, commutation deficit, neutrino mass ordering, Koide formula, PMNS mixing, spectral gap, formal verification, Lean 4