Author-submitted preprint version of the manuscript submitted to Journal of Physics A: Mathematical and Theoretical on 17 May 2026 under manuscript reference JPhysA-124826. This is not a journal-formatted or publisher-produced version.
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.
Finite-resolution density imaging is a central diagnostic in ultracold atomic gases and quantum simulators. In density-selected phase-imprint protocols, an unknown interaction or response kernel is encoded into a phase kick, whose subsequent short-time dynamics produces a measurable density contrast. This work develops a scaling-law theory for the resulting many-body-to-pixel inverse-readout problem, from microscopic density data to elliptic phase reconstruction and finite-band kernel recovery. The paper addresses a protocol-level question: when can finite particle number, finite frame averaging, and finite optical resolution still support certified phase or kernel recovery? Rather than reproving a propagation-of-chaos theorem, the analysis treats a controlled many-body interface as input and derives the certifiability and design theory that follows. The resulting bounds identify no-go regimes, optimal readout times, many-body-limited saturation thresholds, and a conservative pilot-calibrated rule for Richardson order selection. A finite-band kernel recovery theorem adds a bandwidth-conditioning trade-off for stable kernel recovery. This is the submitted preprint version of a manuscript submitted to The European Physical Journal D.