Zero-point length as a topological protection of black hole regularity

Abstract

We investigate the thermodynamic topology of regular black holes with zero-point length using an extended first law that includes the zero-point length stored in the geometry. By treating the regularization scale $l_0$ as a thermodynamic variable, we analyze the Hessian geometry of the thermodynamic manifold and demonstrate that the vector field $\vec{\phi} = (T, \Psi)$, where $T$ is the temperature and $\Psi$ is the conjugate to $l_0$, never vanishes in the physical parameter space for $l_0 > 0$. This implies the absence of Morse critical points and a vanishing winding number ($W = 0$), indicating topological protection against the formation of naked singularities. Crucially, we show that in the singular limit $l_0 \to 0$, a non-zero winding number ($W = 1$) emerges, characterizing the Schwarzschild singularity as a topological defect. The conservation of this topological invariant under smooth evolution provides a rigorous topological formulation of the weak cosmic censorship conjecture: the presence of zero-point length not only regularizes the spacetime background but also enforces topological protection against the formation of singularities, preventing black hole-to-naked singularity transitions.