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

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**Problem statement:** The classical inner Cauchy horizon of regular black holes undergoes exponential mass inflation under generic perturbations, threatening spacetime predictability. The fakeon Pauli-Villars regulator from spectral causal theory (SCT) truncates this divergence via entire-function nonlocal form factors. **Method:** In the S3 regularization mode—where the regulated mass aspect grows monotonically and saturates—the second law is tested by examining monotonicity of the canonical entropy current along the inner-horizon evolution. The entropy ansatz sums Bekenstein-Hawking area terms for both horizons with spectral-action logarithmic corrections; the nonlocal correction is suppressed by \((\Lambda/M_{\mathrm{Pl}})^2\). The chain rule yields opposite signs for the outer and inner horizon area derivatives, reducing the second law to a nontrivial inequality. **Main results:** An asymptotic argument shows the inner-horizon contribution is suppressed as \(\mathcal{O}(1/M^2)\) in the Schwarzschild-like limit, stronger than naive expectation due to a spectral lock. A numerical sweep over 60 parameter cells spanning \(M\Lambda\in[0.84,\,10^6]\) (including five near-extremal points just above the subextremal bound) and five Price-tail exponents confirms \(\mathrm{d}S_{\mathrm{total}}/\mathrm{d}v \ge 0\) at machine precision in every cell. **Limitations:** The analysis is restricted to the S3 regime; the S4 principal-value prescription, where the mass aspect acquires a sign-flip, requires separate treatment of the indefinite-metric fakeon flux and is not covered. The entropy ansatz also relies on specific assumptions about the form of spectral-action corrections.

AI-generated (deepseek-v4-flash) · created 2026-05-21