Project T Volume VIII : Self-Correcting Mechanisms The Eight Layers of Algebraic Stability

This volume presents a complete taxonomy of the eight self-correcting layers in the affine 0-Hecke monoid H₀(Ã₂). The central organizing principle is the commutation deficit formula C(K) = K(K-3)/2, which uniquely selects K=3 as the sole rank admitting zero commutations — the algebraic signature of perfect confinement. We establish the Deficit Closure Theorem: the three structural deficits (Catalan: 19, Parity: 3, Causal: 2) sum to exactly 24 = |H₀(A₃)|, demonstrating that spacetime volume absorbs all numerical mismatches. This framework explains why matter exists, time flows, and coherence persists.