Parameter-Decoupled Superresolution with Airy-Disk-Informed Learning: Application to NGC 4486B

Estimating the separation of two unresolved or nearly unresolved sources from recorded intensity images is difficult when source brightness, mutual coherence, and phase are unknown and the point-spread function (PSF) departs from a Gaussian. We formulate Hermite--Gaussian (HG) projections of one- and two-dimensional Airy amplitudes and prove that ratios of mode probabilities with equal longitudinal parity exactly cancel the brightness, coherence, and phase nuisance parameters under a symmetric two-source model. For a one-dimensional Airy slice we achieve an explicit multi-ratio separation inversion. For the physical two-dimensional circular-pupil PSF we obtain exact finite-pupil overlap integrals and locally invertible separation responses. Motivated by these analytical invariants, we examine Airy-informed convolutional neural networks (CNNs) on synthetic Airy images with known separations. At 150,000 training images, the approximate mean absolute errors are 0.018 in plotted separation units, 10 times better comparing to 0.18 for non-Airy Gaussian PSF approximation. Applied to an archival Hubble Space Telescope image of the double nucleus of NGC 4486B, the Airy-informed CNN gives estimates of approximately 9.6--12.0 pc for training sets of 100,000--150,000 images, broadly consistent with the historical approximately 12 pc value. We distinguish exact optical-mode nuisance cancellation from empirical robustness of an image-only learned estimator and identify the limits imposed by PSF mismatch and extended galaxy structure.