Information Transfer in the Trojan Universality Class III: Temporal Information, from Reduced Periods to Swept Passage through a 2:1 Resonance

This paper extends the transfer of information within the Trojan universality class from static numbers and shapes to times and time-dependent protocols near a two-to-one resonance between modes of equal Krein signature. Once a parameter is swept slowly, the truncated normal form continues to reduce to the same one-degree-of-freedom Hamiltonian studied earlier, now evaluated along a path in the plane of scaled detuning and resonant twist. A physical ramp rate in one member maps to a scaled speed set by the coupling strength and action scale, so that two gate-matched systems that follow identical paths at the same scaled speed share the same reduced dynamics. All periods, small-oscillation frequencies and spectral sideband spacings of the reduced motion are fixed by the two content numbers and the energy level, measured in units of the exchange time that serves as the natural clock of every member. When a sweep forces an orbit across the separatrix that divides the reduced phase space into two invariant cells, the orbit leaves one cell and enters the other carrying an action equal to the complement of its former action. This area-complement rule follows from conservation of phase volume and requires no special smoothness at the conical point of the reduced space. Consequently, a sufficiently slow sweep that begins on the pure faster mode carries the energy into the slower mode along the anti-phase periodic orbit. A blind test transferred conversion data from a balanced nonlinear circuit to a two-mode micromechanical model whose exchange times differ by a factor of roughly two and a half. The circuit’s conversion, averaged over seed phase and frozen at twelve distinct sweep rates, predicted the target’s conversion to a root-mean-square error of about five thousandths, compared with more than a third for transfer at the same raw ramp rate. The area-complement rule itself predicted the final action of individual test orbits to within one hundredth and their final conversion to within two thousandths. All results remain conservative, two-dimensional and local; long-time confinement is not tested.