Second-law check through the inner Cauchy horizon of regular black holes with nonlocal fakeon-regulated mass inflation

Abstract

The classical inner Cauchy horizon of a regular black hole develops exponential mass inflation under generic perturbations. The recently proposed fakeon Pauli-Villars regulator from the spectral causal theory (SCT) - a one-loop effective gravitational theory with entire-function nonlocal form factors \(F_1,F_2\) acting on the Weyl- and Ricci-squared operators - truncates this divergence at the level of the effective Ori-Poisson-Israel equation. Two regularisation modes have been analysed: the algebraic Anselmi-smooth interpolation (``S3'') and the first-principles principal-value prescription (``S4''). Working in the S3 regime, where the regulated mass aspect \(\delta m_{\mathrm{MS}}(v)\) grows monotonically and saturates, we study whether the canonical entropy current of the spectrally fixed Hayward+SCT background grows monotonically along the inner-horizon evolution. Given the working ansatz that the canonical entropy is the sum of Bekenstein-Hawking area terms applied to both horizons (including the dynamically perturbed inner horizon) plus the spectral-action logarithmic correction at each, that the higher-derivative nonlocal entropy correction \(S_{\mathrm{NL}}\) is suppressed by \((\Lambda/M_{\mathrm{Pl}})^2\), and that the Hayward functional form is the leading-curvature SCT vacuum ansatz, the chain rule yields opposite signs for the two contributions (\(\mathrm{d} A_+/\mathrm{d} v > 0\) and \(\mathrm{d} A_-/\mathrm{d} v < 0\)), so the second law is not automatic but reduces to a non-trivial inequality between two opposite-sign terms. An asymptotic argument shows the inner-horizon contribution is suppressed as \(\mathcal O(1/M^2)\) in the Schwarzschild-like limit (a sharper suppression than naive expectation, traceable to the spectral lock \(r_-^2 - 1/(\Lambda^2 z_1)\sim 1/(2M)\) near the asymptote), and a 60-cell production sweep spanning \(M\Lambda\in[0.84,\,10^6]\) (including five near-extremal cells just above the spectrally fixed subextremal bound \(M\Lambda = \sqrt{27/(16 z_1)} \simeq 0.83595\)) and five Price-tail exponents verifies \(\mathrm{d} S_{\mathrm{total}}/\mathrm{d} v \ge 0\) at machine precision in every cell. The S4 principal-value regime, in which \(\delta m_{\mathrm{MS}}(v)\) acquires a sign-flip past the crossover advanced time, requires a separate analysis of the indefinite-metric fakeon flux and is not covered here.