The Uniqueness of K=3 Topological Collapse Analysis for K=4 and K=5

We demonstrate that K=3 is the unique algebraic rank compatible with stable physical structure. The affine 0-Hecke monoid H₀(Ã₂) with three generators {a, b, c} admits a coproduct preserving idempotence if and only if the coupling constant λ = -1. Higher ranks K=4 and K=5 collapse due to topological obstructions: the loss of locality (non-planar graphs), broken symmetry (no cyclic closure), and algebraic decoherence (λ becomes complex or multi-valued). This establishes K=3 as the minimal and unique rank for coherent physical reality.