Certified Statistical Inverse Readout for Singular Phase Kicks: Weak-Topology Certification, Time-Validity Gates, and Numerical Trace

This manuscript develops a certified finite-resolution statistical inverse-readout framework for singular density-selected phase kicks. The framework constructs a weak weighted topology directly from noisy pixelized data, certifies stability through a computable spectral gap, and returns either a strictly certified reconstruction or a certified obstruction under declared assumptions. The draft includes an initial proof-of-concept numerical trace for a two-dimensional logarithmic Coulomb benchmark and a reproducibility package. No external experimental datasets were used. Numerical material is synthetic and generated from the equations and parameters described in the manuscript and reproducibility package.