Dark matter halo inference with higher order photon rings and closure data in multi frequency VLBI

Higher order photon rings form a universal, near geometric substructure in optically thin black hole emission: an exponentially demagnified sequence of subrings that asymptotes to the critical curve and imprints slowly decaying, oscillatory structure in long baseline VLBI visibilities. We develop a closure first forecasting and inference framework that targets these oscillations directly in the data domain, using multi frequency closure phases and closure amplitudes to infer weak gravitational environments of supermassive black holes. The environmental contribution is modeled as a perturbative, spherical dark matter halo (NFW/Dehnen/Hernquist) that coherently deforms the near critical ring geometry and produces correlated, order dependent shifts across the subring hierarchy. The inferred halo imprint is defined physically as the fractional ring scale response, h = κ ( M ) M e x t ( 3 M ) / M , in the weak environment limit. Using synthetic EHT/ngEHT like observing configurations for both M87* and Sgr A*, we quantify (i) when long baseline closure data enter the photon ring regime, (ii) the resulting constraints on an effective halo imprint parameter, and (iii) halo model distinguishability via Bayesian evidence ratios. The forecasts identify the baseline and frequency conditions required to separate halo imprints from emission and propagation systematics, and provide design level guidance for multi band observing strategies that maximize sensitivity to near critical lensing structure.