Complex-gauge control of anomalous Floquet corner responses in a non-Hermitian physical-synthetic photonic lattice

We propose a non-Hermitian Floquet photonic lattice formed by a physical resonator coordinate and a synthetic frequency coordinate. A two-step modulation protocol realizes a chiral walk in this physical-synthetic plane, with a real synthetic flux controlling loop interference and imaginary gauge fields controlling non-reciprocal envelopes. We show that anomalous corner pairs at quasienergies zero and $\pi \mathrm{/}T$ exhibit three distinct layers of physics. A non-Bloch higher-order construction predicts whether the $0\mathrm{/}\pi$ corner pair exists under open boundaries. The imaginary gauge fields select where the right eigenmodes accumulate. The real flux controls the local interference matrix element that determines whether the doubled-period optical response is visible. As a result, the same topological coexistence sector can be bright, skin-dark, or flux-dark in a local optical measurement. We further show that the complex gauge can tune an exceptional point of the two-period corner propagator. At this point the anomalous response keeps its doubled-period sign alternation, but its envelope becomes algebraic because of a Jordan block. These results provide a photonic route to separate topological existence, skin-selected localization, optical visibility, and defective two-period dynamics in a non-Hermitian synthetic dimension.