Persistence Coordinates and State Representation: Electromagnetic Structures in the Persistence Space

We investigate the dependence of persistence coordinates on the choice of physical state representation in electromagnetic systems. The persistence framework (Paper I, doi:10.5281/zenodo.20838334) assigns to each coherent structure a triple (P_N, P_S, P_L) of norm, shape, and localization persistence. We show that shape persistence P_S and the oscillation index O = limsup P_S - liminf P_S depend not only on the governing PDE but also on the physical quantity chosen to represent the state. Three representative examples are examined: (1) The NLS soliton is representation-invariant: field and energy representations yield identical persistence coordinates. (2) The Kuznetsov-Ma breather is representation-sensitive: different representations produce different values of O while preserving the same persistence class. (3) The ideal 1D Fabry-Perot cavity is representation-discontinuous: field, energy, and Poynting representations place the same Maxwell solution in different persistence classes. For the ideal cavity, O^(E) = 2/3 exactly, independently of cavity length and mode number. A complete analytical derivation is provided. This is Paper II of the FAP (Field-Attractor-Particle) Program.Paper I: doi:10.5281/zenodo.20838334