Breakdown of Axis Separability and Structured Inversion in Finite-Domain Normalized Position Measurements: A Quadrant Photodetector Case Study
Finite-domain normalized spatial measurements combine an incident field, geometric support, local response, and channel weighting. These operations can break an otherwise separable response, so accurate single-axis calibration alone does not establish separability of a multidimensional measurement. We represent normalized readouts as channel-weight expectations under a parameter-dependent effective measure. For fixed reception and channel operators, local sensitivity is the covariance between the channel weight and the source score. This representation distinguishes axis separability, parameter mixing, and local recoverability, and applies at the appropriate detector level to continuous-electrode position-sensitive detectors, Shack--Hartmann and pyramid wavefront sensors, back-focal-plane split detection, and finite pixel arrays. A quadrant photodetector (QPD) provides an analytically tractable case. Starting from the complete two-dimensional power integrals, we identify the spatial-moment origin of its lowest-order cross term. Axis calibration and system symmetry then constrain the inverse map, giving Axis-Anchored Cross-Residual Inversion (ACRI). Numerical tests of this QPD realization support the sensitivity relations and show that the constrained cross correction substantially reduces the off-axis bias retained by single-axis inversion. The results illustrate how the general measurement representation guides inverse construction, with performance bounded by the working domain, calibration state, and preserved symmetries.