Static regular black holes in Horndeski theories: analytic no-go and nonanalytic obstructions

Regular black holes in Horndeski theories must have stable horizons and regular centers. We study static, spherically symmetric, asymptotically flat configurations with a time-independent scalar. The horizon branch on which the scalar kinetic term $X$ remains nonzero is generically obstructed by divergent propagation speeds or ghost/gradient instabilities, aside from special degeneracies. On the regular branch, where $X$ vanishes at the horizon, analyticity at the relevant $X=0$ endpoints reduces the leading scalar equation to finite sets of Taylor coefficients. For nondegenerate shift-symmetric theories this gives a nonperturbative current no-hair theorem: the scalar is constant and the metric is Schwarzschild, hence centrally singular for nonzero ADM mass. For non-shift-symmetric positive-power couplings, the corresponding exclusion applies to the perturbative branch continuously connected to Schwarzschild. We also classify marginal nonanalytic departures: covariant regularity fixes the scalar-Gauss-Bonnet chain as the unique marginal nonanalytic completion. Hairy black holes in this completion evade the analytic current step but remain centrally singular.