Nothing in Common but the Mathematics

This paper brings together six studies that test the Trojan universality class, a framework in which physical systems that reduce to the same normal-form type near a resonance or frequency collision share quantitative predictions after each is rescaled by a few invariants computed from its own Hamiltonian. The six pairs were deliberately chosen to share no physics: a micromechanical beam and a Fermi-resonant molecule, a triangular Lagrange point and a Penning trap, a spinning top and a Penning trap, two coupled pendulums and a Bose–Einstein condensate, a child’s swing and a superconducting cat qubit, and Halley’s comet and a highly excited hydrogen atom in a microwave field. They differ by many orders of magnitude in size and period. The work restates a transfer protocol and proves a transfer inequality that bounds the discrepancy between a dimensionless observable measured in two members of the same type by the sum of their individual truncation errors plus a term that measures how well their invariants match. The inequality isolates three independent sources of error, each of which appears in the numerical record. Across twenty headline tests, transfers that match every invariant of a resonant type produce errors ranging from parts per thousand to about ten percent, with known origins and known dependence on amplitude. Leaving one invariant unmatched raises the error by one to two orders of magnitude, by an amount the inequality predicts in advance. A further proposition shows that the pumped swing is the prescribed-pump limit of the two-to-one resonant type, so that five of the six pairs fall under three resonant types. The sixth pair lies outside the class and tests the transfer of an exit mechanism; only scaling laws and orders of magnitude survive, because one invariant is a function that cannot be matched by a single parameter. In every case the labor of transfer consists of computing at most three numbers from each Hamiltonian and converting units. Quantum corrections, dissipation, and global structure do not transfer unless they are inserted into the reduced model. The practical consequence is that an easily studied system can serve as a quantitative laboratory for a far more difficult one, with the expected accuracy known before any computation is performed.