Dimension-saturating local tomography of off-diagonal Fourier-interferometer errors from displaced dark events

Dark output events are useful null tests for linear-optical devices, but their unprobed probabilities have no signed linear response to a coherent error. We append positive and negative calibrated output displacements and use their count difference as a phase-sensitive local observable. We prove that number-state inputs leave an exact \((m-1)\)-dimensional gauge at an \(m\)-mode Fourier interferometer, whereas a phase-calibrated fixed-number multimode cat removes that continuous obstruction. For every \(m\geq2\) and every \(n>2\) divisible by \(m\), an explicit family of dark outcomes and two fixed probes gives a full-rank local Jacobian for all \(m(m-1)\) off-diagonal Hermitian error coordinates using four programmed settings. The scalar response count saturates a dimension bound within the declared weak-probe model. We also give exact four-mode certificates, a weighted local estimator, finite-angle and finite-shot tests, and an augmented analysis showing that unknown cat amplitudes as well as phases are confounded with device errors. The construction is an identifiability result, not a scalable replacement for conventional optical characterization: at \(n=m\) its reference probabilities decay as \(O(\epsilon^2m^{2-m})\) for unit-spectral-norm probes.