From Atomic Relations to Persistent Molecular Closure: An SFM 2.0 and RCG Analysis of Chemical Bonding, Molecular Geometry, Competing Pathways, Dissociation, and Cross-Scale Handover

Paper 5 of 8 in the Recursive Continuity Geometry Across Physical Scale series. This paper applies the Structural Fingerprint Method Version 2.0 and Recursive Continuity Geometry to the transition from persistent atomic units to persistent molecular closure. Established molecular physics and quantum chemistry remain authoritative for molecular Hamiltonians, electronic structure, potential-energy surfaces, spectra, reaction pathways and dissociation thresholds. The PATON framework provides a shared structural account of recurrence, persistence locking, admissibility, competing pathways, disturbance, recovery, ordered traversal and cross-scale handover. The paper distinguishes compositional, charge-state, connectivity, stereochemical, conformational and quantum-state identities. A worked hydrogen-molecule demonstration uses the H2 equilibrium distance, Morse potential and dissociation margin to model molecular deformation, loss of admissibility and recovery. The resulting molecular closure may then be handed forward as a constituent of materials, fluids, polymers, membranes or biological networks.