Dark matter deformations of photon-ring echoes in horizon-scale interferometry

arXiv:2608.23591v1 Announce Type: new Abstract: Strong lensing around a rotating black hole produces delayed and rotated higher-order images even in Kerr. We ask how a dark matter (DM) environment changes these photon-ring echoes after the leading ring scale is matched to Kerr. We follow the signal from rotating DM geometry to finite-order photon transfer, slow-light emission, interferometric visibilities, and source-population tests. For a strong central-spike benchmark with $a/M=0.8$ and $i=70^\circ$, a $1.49\%$ critical-parameter fingerprint remains after ring matching. The order-1 and order-2 delay medians are $13.33M$ and $33.06M$, and the 230-GHz static and dynamic visibility contrasts are $0.070F_0$ and $0.112F_0$. The differential signal also remains distinct in a shared-source oracle test. The harder step is attribution when the source is unknown. Raw EHT-like closures give AUC $0.658$, while fresh validation with total-flux and compact spatial source drivers gives $0.507$ and $0.500$. We therefore use the loss of separation on new sources to derive observing requirements rather than a detection claim. At the pre-defined Kerr threshold, 75\% power requires $d=2.319$, compared with $d=0.553$ for the raw independent-source test. Under a fixed-mean diagnostic, the residual source scatter would need to be about $0.238$ of its present value, although the bootstrap range $0.022$--$0.444$ is broad. Daily gaps can also miss the smaller spike--Kerr delay difference. Finding a photon echo and attributing a small deformation to DM are therefore different tasks. Robust attribution needs source control and suitable delay coverage, not sensitivity alone.