Detector-conditioned source-space nulls and null-mask loss in a programmable two-slit interferometer

Abstract Afshar's double-slit experiment probes wave--particle complementarity by placing a wire grid at downstream interference minima while retaining a path-resolving imaging basis. We formulate a theoretical time-reversed Young--Afshar (TRY--Afshar) configuration in which the corresponding null test is implemented by programming the source basis while a detector mode is held fixed. A detector-conditioned null is a source label (more generally, an input mode) whose coherent two-path transfer amplitude to the selected output mode vanishes. Suppressing such labels produces ideally no detector loss with both slits open, but produces finite loss when either slit is used alone or when which-path marking fills the null. We derive the scalar Fresnel model, the null-mask loss, and its relation to the standard Englert visibility--distinguishability law. The implementation analysis is extended beyond point-source and point-detector idealizations: a coherent Gaussian launch mode is propagated at the field level, finite detector area is included through an output-mode averaging factor, and finite slit width, mask width, stochastic source-coordinate error, phase noise, partial coherence, finite source-mask extinction, position-dependent coupling, and count statistics are treated explicitly. For a representative 633-nm geometry, a $5~\mu$m-rms coherent source mode leaves a calibrated minimum leakage below $10^{-3}$ for point detection, whereas a $50~\mu$m detector aperture gives a full-model leakage of about $6\times10^{-3}$. A calibration and interleaved acquisition protocol is given to separate shot noise from drift. The resulting framework identifies the TRY null as a mode-to-mode transfer minimum and provides quantitative requirements for an experimental null-mask test.