When a material's properties change abruptly in time — refractive index jumping at a specific moment — electromagnetic fields must satisfy continuity conditions across the temporal boundary, just as they satisfy boundary conditions across spatial interfaces. The conventional assumption: the electric displacement D and magnetic flux B are continuous across the temporal jump. These conditions determine how much of the wave is transmitted forward, how much is reflected backward, and what frequencies emerge from the discontinuity.
These continuity conditions are not physical laws (arXiv:2603.21622). They are design degrees of freedom. By engineering the material's response at the moment of the parameter jump — specifically, by using mixed continuity conditions where some field components are continuous while others are not — entirely new wave phenomena become accessible.
Under conventional conditions, temporal modulation produces momentum bandgaps: frequency ranges where waves are amplified or attenuated resonantly. Breaking the conventional continuity rules enables non-resonant wave amplification without these bandgaps. The amplification becomes broadband rather than frequency-selective.
More dramatically, the mixed conditions enable reversible conversion between propagating waves and static fields. A propagating electromagnetic wave can be frozen into a stationary field pattern — stored — and then released back into a propagating wave by a second parameter jump. This is optical memory without a cavity: the wave is stopped not by trapping it between mirrors but by converting it to a non-propagating state through a tuned temporal boundary.
The structural insight: what appeared to be a constraint was a choice. The continuity conditions at temporal boundaries are analogous to constitutive relations — they describe the material's response, not a fundamental law. Changing them changes what temporal modulation can do, the same way changing a spatial boundary condition changes the modes of a cavity. The limitation of conventional time-varying media was not physical but conventional: the assumption that one set of continuity rules was the only set.