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 SpivackPhysicsPortal,Spivack2025_FirstPrinciplesSM 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 SpivackPSCConcordance). UGP arithmetic and seed rigidity Spivack2025_UGPUniqueness,spivack2025main_ugp_paper is fully machine-checked in ugp-lean ugp-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, unified-rigidity-lean unified-rigidity-lean, formalizes the common admissibility framework and proves unified\_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 PSCSM step is now unconditional: the Residual Classification (RCC) is established as a theorem (PSC.RCCInfiniteFamilies, zero sorry ugp-lean); the all-n UGP extension is now proved (asymptotic\_sparsity\_universal, zero sorry Spivack2026_Uniqueness25); the neutrino mass-squared splitting ratio m^2_21/ m^2_31 is predicted to 0.52\% from Braid Atlas b-values via the formula m__g b_g^N_c + _Koide = b_g^29/9, with normal ordering automatic and no external anchor (companion paper Spivack2026_SM_UGP). 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.