Cauchy-horizon flux coefficients in the reduced Polyakov model

We derive the leading Cauchy-horizon flux coefficient in the stationary reduced Polyakov sector of spherically symmetric charged black holes. For a nonextremal inner horizon with affine coordinate $V_{-}=-e^{-{\kappa }_{-}v}$, a finite late-time Eddington–Finkelstein flux $F_{-}^{(\mathrm{\infty })}={\lim }_{v\to +\mathrm{\infty }}⟨T_{vv}⟩$ is amplified as $⟨T_{V_{-}{V}_{-}}⟩\sim F_{-}^{(\mathrm{\infty })}\mathrm{/}({\kappa }_{-}^2{V}_{-}^2)$. In the stationary reduced Polyakov model, $F_{-}^{(\mathrm{\infty })}=t_v-N{\kappa }_{-}^2\mathrm{/}(48\pi )$. Thus the leading pure $V_{-}^{-2}$ Polyakov coefficient is absent precisely on the inner-horizon cancellation surface $t_v=N{\kappa }_{-}^2\mathrm{/}(48\pi )$. The future event horizon determines the distinct outgoing condition $t_u=N{\kappa }_{+}^2\mathrm{/}(48\pi )$, so the two horizons select different loci in the stationary $(t_u,t_v)$ state space. Standard outer prescriptions, such as the asymptotically flat Unruh prescription and the outer-horizon thermal/KMS prescription, generically lie away from the inner-horizon cancellation surface and generate nonzero inner-horizon coefficients. We then analyze the total flux hierarchy $T_{vv}^{\mathrm{t}\mathrm{o}\mathrm{t}}=F_0+A{v}^{-p}+o(v^{-p})$: cancellation of the pure quadratic coefficient is the constant-level condition $F_0=0$, while nonzero Price-tail terms give logarithmically weakened divergences. This state-space formulation gives an exact characterization of Cauchy-horizon flux amplification in the anomaly-induced radial sector and shows that, when the total coefficient is nonzero, the corresponding radial null curvature diverges.