Weak Cosmic Censorship as a Vorticity-Controlled System: Penrose Inequality, Self-Correcting Dissipation, and the Navier-Stokes Structural Transfer

Penrose's weak cosmic censorship conjecture (WCC) — that gravitational collapse from generic initial data cannot produce naked singularities visible to distant observers — remains one of the central open problems in general relativity. We analyze WCC through the lens of the Vorticity-Controlled System (VCS) framework [1], in which the Kerr spin parameter \(a^* = J/M^2\) serves as a vorticity functional with critical threshold \(\Omega_c = 1\) (extremality). The Penrose process provides the dissipation mechanism \(\delta_P > 0\).

We formalize three structural constraints on WCC:

1. Penrose inequality. The Riemannian Penrose inequality (Huisken-Ilmanen [2], Bray [3]) bounds the irreducible mass \(M_{\mathrm{irr}} \geq M/\sqrt{2}\), limiting extractable energy to \(\leq 29.3\%\) of the black hole mass.

2. Area theorem protection. Hawking's area theorem [4] makes \(M_{\mathrm{irr}}\) monotonically nondecreasing. To reach extremality from a sub-extremal state, the black hole must gain mass — but mass gain raises the threshold, creating a moving target.

3. Self-correcting dissipation. As \(a^ \to 1\), the ergosphere (where the Penrose process operates) grows: the gap \(r_{\mathrm{ergo}} - r_+ = M(1 - \sqrt{1-a^{2}}) \to M\). Higher spin means more efficient extraction — a negative feedback mechanism that resists threshold crossing.

The third point is structurally analogous to the Navier-Stokes perturbation regularity [5], where the curvature-radius product \(\kappa r_0 \to 0\) near blowup, causing the perturbation ratio to improve as the system approaches the pathological regime. We formalize this "NS-WCC structural transfer": both systems exhibit self-correcting dissipation in the VCS framework, suggesting a common proof architecture for Normal-branch VCS problems.

All 11 theorems are machine-verified in a companion proof repository and exportable to Lean 4.