Branch Structure and Non-Extensive 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 ζ = ±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, photon propagation in plasma, and tidal forces are then studied, revealing clear optical and strong-field signatures that distinguish the two branches. In particular, the ordinary branch exhibits finite-radius tidal inversion, absent in the phantom sector. Our results demonstrate that the combined Kalb-Ramond and ModMax effects lead to a rich and observationally distinguishable black-hole phenomenology.