Coherent Release Fronts in Krylov Chains and Thermal AdS$_2$ Response

Abstract: We study coherent release from a finite serial Krylov sector into an exact conformal tail with asymptotically linear Lanczos coefficients. We derive an exact finite-cut occupation kernel and its locally uniform moving-cut limit, whose derivative is the squared modulus of a filtered boundary-emission amplitude. The tail sets the filtering kernel and logarithmic transit scale, while the release profile also depends on the finite sector. Its early edge records the number of serial steps before emission, whereas its late edge reflects slow resonances, interference, and logarithmic enhancements from merged poles. A one- and two-site family can share the same leading continued-fraction pole while exhibiting inequivalent profiles. We further consider a thermal AdS$_2$ realization in which, in the weak-coupling limit, the dressed oscillator measure approaches a two-site shallow sector followed by the conformal tail, with the leading complex pole reproduced through second order.