PROJECT T Volume I : ALGEBRAIC FOUNDATIONS The Affine 0-Hecke Monoid H₀(Ã₂)

This volume establishes the algebraic foundations of Project T. We construct the affine 0-Hecke monoid H₀(Ã₂) from first principles, defining each structure precisely and proving the essential properties. The central object is a monoid with three generators {a, b, c} satisfying: • Idempotence: x² = x for each generator • Cyclic braid relations: aba = bab, bcb = cbc, cac = aca The cyclic structure of these relations—encoded in the triangular Dynkin diagram of type Ã₂—produces an infinite monoid with remarkable properties: every element is idempotent, the structure admits a natural Z₃ symmetry, and specific counting invariants (33 seed configurations, 42 root configurations) emerge from the combinatorics. This document is self-contained. Every theorem is proved. Every definition is motivated. A companion Lean 4 formalization provides machine-verified proofs. Keywords: 0-Hecke algebra, affine Coxeter group, Dynkin diagram, idempotent monoid, braid relations, trefoil knot, golden ratio MSC 2020: 20M05, 20F55, 05E16, 16T05