Joint large-time, large-depth limit for the response-derived physical chain

Establish the joint large-time, large-depth scaling limit of the physical Krylov chain reconstructed from the thermal AdS$_2$-coupled oscillator measure, including the coherent-front regime at fixed $N_Te^{-2\alpha t}$ and release windows of order $\alpha/\lambda^2$.

Background

The paper derives the coherent moving-front profile rigorously for a prescribed Jacobi model consisting of a finite shallow sector attached to an exact conformal tail. It then constructs a physically motivated thermal AdS$_2$ realization in which a boundary oscillator is coupled to a conformal bath. In the weak-coupling limit, the oscillator’s spectral measure reproduces a two-site shallow sector followed by the conformal tail at fixed Krylov depth, and the resulting reference chain matches the oscillator’s leading complex pole through order $\lambda^2$.

The unresolved issue is whether this fixed-depth, weak-coupling matching controls the genuinely dynamical regime relevant to the coherent front: simultaneously large time and large depth with $N_Te^{-2\alpha t}$ held fixed, as well as parametrically long release windows of order $\alpha/\lambda^2$. Resolving this would determine whether the front obtained for the prescribed-tail model also describes the response-derived physical chain.