PROJECT T : The Complete Algebraic Architecture of the 0-Hecke Monoid H₀(Ã₂)

This document presents the complete algebraic architecture underlying Project T. At its foundation lies a single mathematical object: the 0-Hecke monoid of the affine Weyl group of type Ã₂, denoted H₀(Ã₂). From this object alone, with zero free parameters and zero empirical inputs, emerges the entire structural content of fundamental physics. The central thesis is radical in its simplicity: three generators {a, b, c} satisfying two axioms — idempotence (x² = x) and braiding (aba = bab, bcb = cbc, cac = aca) — are sufficient to derive the complete architecture of matter, including three generations of fermions, color confinement, mass gap, Pauli exclusion, and the SU(3) gauge structure of the strong force. The key structural insight is the commutation deficit formula C(K) = K(K−3)/2, which counts the number of generator pairs that commute rather than braid. For K = 3, this gives C(3) = 0 — the unique value where every pair of generators interacts. This single algebraic fact forces confinement, maximal interference, and dimensional selection.