Information Transfer in the Trojan Universality Class IV: Weak Forcing, Thin Chaos, and Transfer across a Mismatched Krein Gate

This paper completes the series by asking what information still transfers when the reduced motion is no longer integrable and when the Krein signatures of the two modes disagree. For members that share equal signatures, a weak periodic modulation of the detuning is added. The change in reduced energy over a single separatrix excursion is given to first order by a Melnikov amplitude that depends only on the two content numbers, the scaled forcing strength, and the ratio of forcing frequency to the linear growth rate. That amplitude is largest when the forcing frequency is slightly larger than the growth rate and admits a closed form when the resonant twist vanishes. A slow modulation of finite amplitude sweeps a band of levels through the separatrix and switches a calculable fraction of a uniform ensemble from one invariant cell to the other. Blind tests from a balanced nonlinear circuit to a micromechanical model confirmed these results. Frozen predictions matched the onset of the thinness ratio to within one percent and the switched fraction to within one standard error. The Melnikov law itself held in both members to within a few percent of its predicted amplitude at modest forcing. When the signatures are opposite, the reduced model is simply the earlier difference-type model continued to negative action. The onset window and the linear growth rate remain unchanged, yet the separatrix excursion no longer terminates at a finite action set by the detuning alone; instead it turns near an action of order one that does not shrink with amplitude. That turning point therefore lies outside the domain controlled by the degree-four truncation, so the subsequent fate cannot be predicted from the content. Within the opposite-signature class a planar Penning trap, frozen as the source, predicted the instability window of the triangular Lagrange point to within a few tenths and its transit times to a root-mean-square error of fifteen percent. Across the gate the same-signature elastic pendulum correctly located the window but predicted only bounded exchange at every point where the three-body problem escaped. The gate, treated as a precondition in earlier papers, is thereby shown to be the datum that separates transferable statements from those that cannot cross.