Rankine Vortex Fold Cascade as a Deterministic Generator of Molecular Geometries: Analytical Derivations, Variational Principle, and Quantitative Predictions

We present a self-contained, analytically grounded derivation of stable molecular bond geometries from the fold-cascade equation of the Tensioned Continuum Resonance (TCR v2) framework, without recourse to quantum-mechanical probability amplitudes. Three new elements beyond the initial preprint (v1) are added: (i) a rigorous derivation of the Sailly angle I3 = arctan(1/2) = 26.565 deg as the unique angular stasis of a Rankine vortex satisfying the PPN closure condition beta_PPN = 0.5; (ii) a variational (Rayleigh-Lagrange) derivation of the fold-cascade equation from a dissipative action functional; (iii) quantitative predictions of bond angles for ten common molecules, with residuals below 0.22 deg for seven of them and exact agreement for CO2. The Collective Sailly Lock (Run B, N=137 folds, 45 STASE events, residual 0.595 deg) is further shown to be the global minimum of the Willmore energy functional of the Rankine torus, by a factor of 820 relative to Run A. An anticipatory Q&A; appendix addresses the most likely objections from referees. Keywords: Rankine vortex; deterministic molecular geometry; fold cascade; fine-structure constant; Sailly angle; variational principle; VSEPR; Willmore energy; stasis attractor; PPN closure