Unified Rigidity Theorem for PSC/UGP Universes

We show that three independently developed rigidity mechanisms jointly select the same canonical physical branch under an explicit and audited premise bundle. Layer I/II PSC theory-space selection~ narrows the space of 4D renormalizable gauge theories to the Standard Model gauge structure and three generations; this is supported by a finite exhaustive enumeration over 34,560 candidate universes across 12 gauge groups including Pati-Salam and E_6 (TE2.2 extended scan: SM ranks~\#1, only 0.03\% pass the hard PSC filters, all survivors are SM-like, all BSM candidates decisively eliminated~). UGP arithmetic and seed rigidity~ is fully machine-checked in -lean~: the RSUC theorem isolates the Lepton Seed (1,73,823) as the unique MDL-minimal survivor of the two-stage sieve (0~sorrys, 0~custom axioms). Dynamical basin structure places the Lepton Seed and its mirror in the canonical basin~A across all tested configurations, while off-residual seeds fall in basins~B/C; this is frozen and SHA-256 certified. A new Lean bridge library, -rigidity-lean~, formalizes the common admissibility framework and proves \_rigidity\_theorem under the explicit premise bundle recorded in its assumption ledger. Every claim is strictly typed: theorem-extracted, bridge, computational/certified, or residual/open. The result is a conditional synthesis theorem under an explicit premise bundle; every assumption is typed and audited. The formal PSC→SM step is now unconditional: the Residual Classification (RCC) is established as a theorem ( .RCCInfiniteFamilies, zero ~); the all-n UGP extension is now proved ( \_sparsity\_universal, zero sorry~); the neutrino mass-squared splitting ratio Δ m^2_21/Δ m^2_31 is predicted to 0.52\% from Braid Atlas b-values via the formula m_ν_g b_g^N_c + θ_ Koide = b_g^29/9, with normal ordering automatic and no external anchor (companion paper~). Any construction of a PSC-admissible theory outside the SM signature, a non-Lepton-Seed UGP survivor satisfying the full sieve, or a canonical-seed trajectory failing the predicted basin would falsify the result.