Exact rapid maintenance behaves singularly at the boundary of quantum state space. Let a finite-dimensional target rho have support projector P, and let an uncontrolled quantum Markov semigroup have outward support-leakage rate a = Tr[(1-P)L(rho)]. We prove that, whenever a > 0, the nonequilibrium free-energy loss has the universal short-time form F(rho) - F(exp(tL)rho) = k_B T a t log(1/t) + O(t). Under an explicit fresh-resource-cell ledger, the optimal period-t exact-refresh power therefore diverges as k_B T a log(1/t). For GKLS generators, a is a positive sum of squared support-crossing jump amplitudes. We prove the complete dichotomy: a > 0 gives logarithmically infinite rapid power, while a = 0 gives finite rapid power even when Hamiltonian rotation produces population outside the fixed initial support at second order. We also define a periodic threshold-reset corridor and prove its sharp k_B T a log(1/r) + O(1) small-corridor law.The general singularity has an unexpected geometric consequence. In a fixed-parity dual-rail SSH code, a boundary transfer has desired logical weight w_x^2, but its exact summed zero-mode-to-bulk weight is w_x(1-w_x). At the remote boundary, these scale respectively as Theta(zeta^(4 ell)) and Theta(zeta^(2 ell)). Thus a soft spectral tail changes the exponential rate of bounded-latency refresh from 4|log zeta| to 2|log zeta| unless the tail itself is suppressed at least as zeta^(2 ell). We prove the general rate formula min{4m, 2m+q}, where m = -log zeta and q is the exponential suppression rate of the spectral tail. Exact rapid maintenance and the wide-membrane limit do not commute: every nonzero tail gives infinite rapid power at fixed width, while fixed-period power still vanishes exponentially with width. The result turns perfect filtering from a technical convenience into the sharp boundary between finite and singular exact maintenance.