Photon-Ring Retarded- Time Tomography: The Mahakal Phenomenon

Abstract

Near-critical null geodesics generate image-order channels with distinct mappings, delays, and weights, so one observer-time data set samples multiple source epochs. We organize the inverse problem by historical reach, dimension, and held-out recovery. Using continuum-noise-consistent, matrix-free Kerr/Schwarzschild operators for orders n = 0, 1, 2, we audit 12 spin-inclination geometries. At normalized SNR0 = 100, resolved orders extend anchor-connected history beyond direct imaging in all cells by 40 M, 24 M, and one 4 M grid step at inclinations 20 degrees, 50 degrees, and 75 degrees. Operational rank rises from 153 to 202 and old-age innovation from 8.29 M to 40.72 M; complete order summation retains only 9.66 M. Compact temporal support gives an exact result: temporal basis functions outside the direct retarded-time footprint generate identically zero direct columns, while higher orders can lift those old-epoch blind blocks. Direct old-structure rank is zero, while resolved orders support 38, 93, and 185 operational directions across three classes. Yet source enrichment leaves reach nearly fixed while rank fraction and singular-value stability collapse; this is the Mahakal phenomenon. Held-out tests show bounded benefit, not a movie. Resolved orders extend stable baseline-inclusive emissivity-level span from 48 M to 80 M, but that endpoint is level dominated. On 60 histories, physical morphology error falls by 13.3 percent with TSVD and 16.4 percent with ridge. Reliable two-feature recovery fails and the stable morphology interval is zero. Complete order summation yields no material morphology gain. The positive results are therefore best-case, known-geometry image-domain benchmarks under ideal order separation; SNR0 is an operator normalization, not an observation.