First-Law Entropy and a Degenerate Extremal Remnant in a Minimal-Length Simpson--Visser-Type Regular Black Hole: Geometrothermodynamics, Phase Structure, and Observational Discriminants

Abstract

We construct the first-law-consistent entropy of a geometrically minimal-length-deformed Schwarzschild spacetime, obtained via the areal-radius substitution $R(r)=\sqrt{r^2+{\ell }^2}$ on $r\in [0,+\infty )$ with $f(R)=1-2M/R$, whose nonvanishing Einstein tensor sources an effective geometric fluid with no classical matter counterpart. Integrating the first law gives $S=\pi [r_h{R}_h+{\ell }^2\ln ((r_h+R_h)/\ell )]$. This coincides in functional form with the semiclassical term found independently by Joshi and Joshi, but we fix its boundary condition $S(r_h=0)=0$ on independent physical grounds and adopt it, rather than the Bekenstein--Hawking area law, as the complete entropy of the model, building the free energy, geometrothermodynamics, and mode-stability analysis on it. Evaporation, governed by the Helmholtz free energy $F(M)$, terminates at $M_{\min }=\ell /2$ in a previously unrecognised endpoint: a degenerate extremal regular black hole, where the regular centre coincides with a degenerate Killing horizon of quadratic order, $f\approx r^2/(2{\ell }^2)$, at areal radius $\ell$, with $T_H\to 0$, $S\to 0$, $C\to 0^{+}$, and finite curvature everywhere; we display its Penrose--Carter structure for the first time. The same entropy, with $\ell$, defines the equilibrium state space of a Legendre-invariant geometrothermodynamic (GTD) description whose curvature scalar diverges independently at the Davies-type transition $M^{\ast }=\ell /\sqrt{2}$ and at $M_{\min }$, a divergence with no counterpart in the minimal-length black hole literature, specific to a genuine horizon at $S=0$. We embed this remnant within the observational discriminant noted qualitatively by Tsukamoto: exact shadow degeneracy combined with a measurable photon-ring flux enhancement $r_n=e^{-2\pi /a}>e^{-2\pi }$, accessible to next-generation very-long-baseline interferometry.