An Analytic Proof of $\kappa_+ < \kappa_-$ for Non-Degenerate Rotating Kerr-Newman-de Sitter Black Holes via Vieta Root Relations

For every non-degenerate rotating Kerr-Newman-de Sitter (KNdS) black hole — any configuration whose radial polynomial \(\Delta_r\) admits four simple real roots \(r_n < 0 < r_- < r_+ < r_c\) — we prove analytically that the outer-horizon surface gravity is strictly smaller than the Cauchy-horizon surface gravity, \[\kappa_+ < \kappa_-.\] The proof uses a single Vieta root-sum identity on the \(\Delta_r\) quartic and contains no numerical computation. Combined with the standard leading-order identification \(\alpha_{\mathrm{NH}} = \kappa_+/2\) of the near-horizon QNM damping rate, this gives the strong-cosmic-censorship (SCC) ratio \(\beta_{\mathrm{NH}} = \kappa_+/(2\kappa_-) < 1/2\) uniformly over the KNdS family — strictly below the Christodoulou \(C^0\)-extension threshold. To our knowledge, this is the first analytic proof of the surface-gravity ordering across the full rotating KNdS family, as opposed to numerical scans or near-extremal asymptotic expansions.

The core algebraic steps are machine-verified independently in two formal systems: the Platonic proof kernel (Python, four theorems) and Lean 4 under Mathlib v4.28 (six theorems, zero axioms beyond the standard library). A \(7 \times 6 \times 5 = 210\)-point numerical scan over \((a^, q^, \Lambda) \in [0.01, 0.99] \times [0, 0.99] \times [0.001, 0.15]\) yields \(131\) non-degenerate configurations; \(\max(\kappa_+/\kappa_-) = 0.856\), giving \(\max \beta_{\mathrm{NH}} = 0.428\) and a uniform safety margin of at least \(0.072\) below the Christodoulou threshold.