From random fringes to deterministic response: statistical foundations of time-reversed Young interferometry

Young interference is usually read as the gradual statistical accumulation of random detection events. Here we show that a time-reversed Young (TRY) geometry has a different statistical character: instead of a detector-side first-order fringe from a fixed source, TRY reconstructs a source-conditioned response indexed by a programmed source coordinate at a fixed detector. With a fixed detector and a scanned source basis, the observable is an operational hybrid correlator between detector signal and source label. The resulting interference is deterministic at the response-function level, while photon and detector noise remain present in the precision with which that response is estimated. We distinguish explicitly between reciprocity of the optical Green function and equivalence of the measured statistical experiment: a reciprocal propagation kernel does not turn a detector-side likelihood into a fixed-detector source-label conditional distribution, nor does it require their Fisher informations to be equal. Together with the previously established hybrid second-order and diffraction-cancelled character of the ideal symmetric TRY response, this statistical distinction clarifies why TRY is more than a coordinate relabeling and why source-coordinate control can support calibration, lock-in readout, near-null sensing, and source-plane superresolution.