UDM explanations

We present a rigorous mathematical extension to the Unified Dissociation Model (UDM). First, we derive the fundamental compactification constant κ = (5/4)1/3 ≈1.077217 dynamically as the global minimum of the K¨ahler modulus potential for a quintic Calabi-Yau threefold. Second, we establish a gauge-invariant definition of the topological loop index n = Nf ·Nc, demonstrating that the scale of dissociation (µ = Mt) naturally protects the model from heavy-flavor decoupling. Third, wederive the signs of composite hadron indices from the Witten index in supersym-metric quantum mechanics. Finally, we show that the fermion mass hierarchy is not an arbitrary input but is geometrically quantized, with generational mass tran-sitions corresponding to exact integer steps (∆n ∈ Z) of a cascading dissociation operator.