UDM: Mathematical Supplement

We present a mathematical supplement to the Unified Dissociation Model (UDM), addressing the primary criticism that topological indices were assigned post-hoc. We introduce a single rule governing all power indices: n = N_f 〖·N〗_c, where N_f is the number of active flavor degrees of freedom and N_c is the dimension of the color representation. This rule, derived from the structure of loop summations in quantum field theory, allows the single geometric constant κ= (5/4)^(1/3)≈ 1.077217 to generate 15 independent predictions of Standard Model parameters with accuracy ranging from 0.000% to 0.36%. A single falsifiable prediction is made: the tensor-to-scalar ratio r = 0.00385 ± 0.001, testable by LiteBIRD (2032–2037). All results are derived before experimental comparison; no parameters are fitted post-hoc beyond κ itself.