Photon-Tagged Energy Flow in Inclusive Endpoint $B$ Decays

We introduce a $B$-decay tagged energy correlator (BTEC) to resolve the angular structure of energy flow in inclusive $B$ decays in the endpoint region, focusing on the direct-photon contribution to $\overline B\to X_s\gamma$. At leading power and at the natural collinear angular scale $\tau\sim s\sim Q\Lambda_{\rm QCD}$, we derive a factorization relation involving the standard hard coefficient and $B$-meson shape function together with a new measured quark jet function; no new leading-power nonperturbative function is introduced. We calculate the measured jet function at one-loop accuracy and verify that its angular integral reproduces the standard inclusive quark jet function. For the central Bosch--Lange--Neubert--Paz parameter set, an illustrative benchmark gives $3 - 16\%$ migration outside the fixed angular cuts $\tau_c=2 - 4~{\rm GeV}^2$. With an independently constrained shape function, the angular cumulatives provide a closure test of leading-power endpoint factorization and are sensitive to direct-$O_7$ power corrections and resolved-photon effects. The BTEC thereby adds information on the angular structure of the inclusive recoil jet beyond the ordinary photon spectrum and may improve signal--background discrimination when their energy-flow profiles differ.