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

Motivated by the low-energy effective action of heterotic string theory, where the Kalb-Ramond (KR) two-form and nonlinear gauge corrections arise simultaneously, we investigate a static, spherically symmetric black hole (BH) in Einstein gravity coupled to a KR field and ModMax nonlinear electrodynamics (NED). The solution depends, beyond mass and charge, on the Lorentz-symmetry-breaking (LSB) parameter $\ell$, the ModMax deformation parameter $\gamma$, and a discrete branch selector $\zeta=\pm1$. We show that the ordinary branch admits extremal and non-extremal configurations, while the phantom branch generically supports a single-horizon geometry. BH thermodynamics is analyzed within the Tsallis non-extensive framework, revealing branch-dependent stability and Joule-Thomson (JT) behavior. Weak gravitational lensing is computed via the Ono-Ishihara-Asada (OIA) extension of the Gauss-Bonnet (GB) theorem, yielding a negative topological correction that reduces light bending relative to the Schwarzschild baseline, opposite in sign to Barriola-Vilenkin (BV) monopole backgrounds. Photon sphere (PS) properties in plasma environments and tidal forces through geodesic deviation are also studied, revealing a universal tidal balance ratio $R_{\rm rad}/R_{\rm ang}=3/2$ in the ordinary branch. These multi-channel signatures provide concrete observational handles for constraining the KR-ModMax framework through Event Horizon Telescope (EHT) data and next-generation interferometric arrays.