Pair Creation in Space–Time Modulated Media IV: Synthetic Lattices, Gauge Flux and Band Topology

A ring of resonators pumped with a phase pattern is a lattice in space and frequency, with the pump phase as a gauge field. The screw that rotates the ring and shifts time is a magnetic translation, whose label fixes the open pair and conversion channels at every depth. Its sector chains are Harper chains, and a vortex of golden-ratio charge gives an Aubry–André transition of the frequency comb. A staggered pump is a signed graph, and a seam delay of half a pump period turns an odd ring into a frequency-space Möbius ring without a defect. Pair links reverse the frequency axis: a network keeps a bin-diagonal charge exactly when its signing is balanced, and sublattices decide the stability of a bin chain closed by a pair link. In a reciprocal ring a pump of flux one half is its own mirror image and doubles pair channels. As a circuit, the ring sorts harmonics, intermodulation products and noise-born frequency-bin pairs into sequence components; single ports carry everything, and single-element errors generically open the forbidden products linearly. In a crystal with a primed twofold axis the Floquet operator has a real form, whatever its reciprocity class. Frame charges sit only at crossings of equal Krein sign; a pair channel is a tongue bounded by exceptional points and carries none. A forced doublet with that axis is a pinned node in every second gap. A half turn of the pump pattern sweeps a forced conversion node across a resonance curve, flipping the relative charge of the enclosed node. A drive can move 2 static nodes into adjacent Floquet gaps, where they cannot annihilate. In a glide-symmetric waveguide a layer misalignment opens the zone-edge gap linearly, at the first-order matrix element. Design rules, a measurement protocol and refutable predictions follow.