Where Thermodynamics Meets Geometry: Critical-Radius Coincidences in Confining-NED Black Holes with Barrow Entropy

Abstract

We study a static, spherically symmetric black hole obtained from Einstein gravity coupled to a nonlinear electrodynamics model with a quark--antiquark confinement interaction. The metric extends Reissner--Nordstr\"om by a logarithmic correction controlled by $\zeta$, modifying both horizon structure and the near-singularity regime. The Hamilton--Jacobi tunneling method for Dirac fermions yields the Hawking temperature; the $\zeta$-dependent terms suppress the small-horizon divergence and signal a remnant. Quantum-gravitational fluctuations are incorporated through Barrow entropy with deformation index $\Delta$. Within the extended phase space we compute the internal energy, free energy, pressure, heat capacity, isothermal compressibility, and Joule--Thomson coefficient. The heat capacity locates $\Delta$-dependent stability regions; the compressibility stays negative across the domain analysed here, marking a mechanically rigid phase with no van der Waals criticality in this branch. The central result is a quadruple coincidence: the peak Hawking temperature, the heat-capacity divergence, the Joule--Thomson inversion, and the zero of the radial tidal force all sit at one radius $r_\star$ defined by $A''(r_\star)=0$, while the extremal horizon and the angular tidal-force zero coincide via $A'(r_h)=0$. These reduce the full critical-point analysis to two scalar equations on $A(r)$. Geometric tidal accelerations are mapped against the thermodynamic critical curves. Event Horizon Telescope observations of Sgr~A* translate into a constraint $\zeta\lesssim 0.7$ at $Q/M=0.5$, leaving a finite window open. The confinement term induces observable corrections to geodesic deviation.