Information Transfer in the Trojan Universality Class II: Geometric Information from an Elastic Pendulum to a Fermi-Resonant Molecular Model

This paper investigates whether entire geometric structures of phase space, rather than single numerical observables, can be transferred between unrelated members of the Trojan universality class near a two-to-one resonance. The source is an elastic pendulum whose resonant twist is always positive. The target is a Morse–Fermi stretch–bend molecular model whose twist is negative. Both systems share equal Krein signatures, so energy exchange remains bounded. In the invariant coordinates that flatten the reduced phase space into a classical teardrop surface, the local geometry is completely determined by the same two numbers that governed energy exchange in the preceding paper: a scaled detuning and a resonant twist. A complete classification of the reduced phase portraits shows that, at small amplitude, the topological type depends only on the detuning and changes only when its absolute value crosses four. An exact mirror reflection of the teardrop conjugates the dynamics of any member at one pair of values to the dynamics of a member at the opposite pair, allowing a positive-twist laboratory to stand in for a negative-twist target. A level-set transfer bound is proved whose leading error is proportional to the square root of amplitude in raw coordinates and linear in amplitude once a first-order normal-form correction is applied. Periodic orbits, separatrices, invariant cells and their Liouville probabilities, and Poincaré sections all fall under the same bound. In a blind test the pendulum’s sections were reflected, rescaled and frozen before the molecular model was integrated. In the corrected coordinates the median distance between the frozen sections and the molecular sections lay between roughly two and seven thousandths, against distances of order one when the reflection was omitted and roughly two hundredths in raw coordinates. The five smooth periodic orbits, the cell probabilities measured from two hundred random initial conditions, and the stability boundaries of the pure stretch orbit all transferred with errors that followed the predicted amplitude scaling. The content that must cross from one member to the other remains two continuous numbers plus a single discrete bit for the sign of the twist. Once these are matched, residual geometric error is set by amplitude alone. All statements are classical, local, conservative and restricted to this resonance.