PROJECT T FOUNDATION II : The Bulk The Infinite Structure of Collapsed Dimensions

Foundation I established the fundamental object of Project T: the 0-Hecke monoid H₀(Ã₂), uniquely selected by C(K) = K(K−3)/2 = 0. This second Foundation volume asks: what lies beyond K = 3? If K = 3 is the unique stable rank, what is the structure of the infinitely many collapsed ranks K ≥ 4 that surround it? We call this structure the bulk — the complete hierarchy of collapsed dimensions encapsulating the stable K = 3 universe. This volume proves that the bulk is not featureless. It possesses exact arithmetic structure, converges to a precise rational constant, and exhibits a crystalline pattern governed by a classical Pell equation. Every result in this volume is either algebraically derived from the commutation deficit formula C(K) = K(K−3)/2, or formally verified in Lean 4 with Mathlib on La Forge. No free parameters are introduced.