PROJECT T FOUNDATION I :The Object H₀(Ã₂) — The 0-Hecke Monoid of the Affine Weyl Group

This document identifies the fundamental algebraic object of Project T: the 0-Hecke monoid of the affine Weyl group of type A₂, denoted H₀(Ã₂). Every result presented here is either algebraically forced by the structure of this monoid or formally verified in Lean 4 with Mathlib. No free parameters are introduced at any stage. The central claim is that a single algebraic structure—selected by a unique combination of mathematical criteria—generates all the constraints needed to derive fundamental physics. This first Foundation volume establishes what the object is and why it is uniquely selected.