Branch structure and nonextensive thermodynamics of Kalb-Ramond-ModMax black holes: observational signatures

We investigate a static, spherically symmetric black hole arising in Einstein gravity coupled to a Kalb-Ramond field and ModMax nonlinear electrodynamics, both of which are independently well motivated extensions of standard electrovacuum gravity. The solution depends, beyond mass and charge, on a Lorentz-violating parameter, a ModMax deformation parameter, and a discrete branch selector $\zeta =\pm 1$. We show that the ordinary branch admits extremal and non-extremal configurations, while the phantom branch generically supports a single-horizon geometry. Black-hole thermodynamics is analyzed within the Tsallis non-extensive framework, revealing branch-dependent stability and Joule-Thomson behavior. Weak gravitational lensing is analyzed using the Ono-Ishihara-Asada extension of the Gauss-Bonnet theorem, which is required by the non-Euclidean asymptotic structure of the Kalb-Ramond optical geometry and yields a negative topological correction that reduces light bending relative to the Schwarzschild baseline. Photon propagation in plasma and tidal forces are also studied, revealing clear optical and strong-field signatures that distinguish the two branches.