A Smoothness Principle for Branch Selection Across Black-Hole Horizons

Many black-hole wave equations reduce to radial ordinary differential equations with regular singular points at horizons, where local solutions split into distinct branches. We propose a simple branch-selection principle to isolate the physically distinguished branch: removing the leading horizon behavior and imposing $C^{\infty }$ smoothness on the residual field. Using an exactly solvable scalar field in $JT/AdS_{2}$, we demonstrate that this single principle successfully organizes three seemingly distinct structures: smooth continuation across the black-hole horizon, the ambiguity of the boundary retarded Green function (pole-skipping), and the lowest-weight structure of an $SL(2,\mathbb{R})$ representation. Rather than treating these phenomena as isolated boundary or horizon properties, our results unify them into a cohesive bulk-to-boundary dictionary, proving that the boundary pole-skipping ambiguity is the precise holographic manifestation of horizon-residual smoothness. Furthermore, the group-theoretic identification reveals that this physical branch selection is not a mere mathematical artifact, but is fundamentally governed by the underlying spacetime symmetries. Finally, we apply this framework as a diagnostic tool to a static holographic superconductor model, revealing that a genuine pole-skipping point requires both simultaneous residual smoothness and strict linear independence of the boundary branches to prevent deceptive branch-collapse artifacts.