Observable Signatures of Lorentz-Violating Black Holes: Lensing, Shadows, and Thermodynamic Properties in ModMax-Kalb-Ramond Gravity

We study electrically charged black holes in which ModMax nonlinear electrodynamics is coupled to a Kalb-Ramond two-form whose vacuum expectation value breaks Lorentz symmetry. The exact metric function carries the ModMax parameter [Formula: see text], which controls the electromagnetic nonlinearity, and the Lorentz-violating parameter [Formula: see text], and it interpolates between the Reissner-Nordström geometry and configurations whose asymptotic structure is permanently deformed. We first map the horizon structure, which admits extremal, non-extremal, and naked-singularity branches, and compute the curvature invariants. The Hawking temperature is raised by the nonlinear electromagnetic factor [Formula: see text] and reshaped by [Formula: see text], while a generalized-uncertainty-principle correction lowers it through the factor [Formula: see text] and shuts off the tunneling of probe quanta as their mass approaches the bound set by the deformation parameter, which is one route to a stable remnant. Restoring the cosmological constant as a thermodynamic pressure, we obtain the Joule-Thomson coefficient and the inversion curve in closed form; the ratio of the minimum inversion temperature to the critical temperature keeps the Reissner-Nordström-AdS value [Formula: see text] for every [Formula: see text] and [Formula: see text], while the critical pressure depends on [Formula: see text] and [Formula: see text] alone. Weak gravitational lensing is treated with the Gauss-Bonnet method in vacuum and in a dispersive plasma, where both [Formula: see text] and the plasma density raise the bending for small impact parameters. Finally, the photon sphere and the shadow radius are obtained analytically and confronted with the Keck and VLTI bounds on the shadow of Sgr A[Formula: see text], which restrict [Formula: see text] to a few percent when the charge is close to extremality and admit charges up to [Formula: see text] at [Formula: see text].