Kinematic Bounds on Charged Energy Extraction in the Purely Electric Branch of Euler--Heisenberg Type <i>f</i> ( <i>R,T</i> ) Black Holes

We derive and test kinematic upper bounds on charged energy extraction in the purely electric branch of a static [Formula: see text] black hole family sourced by an Euler–Heisenberg type nonlinear electromagnetic sector. The analysis is restricted to external, minimally coupled probe particles on the fixed background. In this electric solution the field remains Coulombic up to an additive gauge constant; the nonlinear electromagnetic and matter-coupling coefficients therefore affect extraction indirectly through the metric, the event-horizon position, and the global causal structure. We obtain a general local bound retaining nonzero angular momentum and radial velocity of the negative energy fragment, and identify the commonly used zero angular momentum turning-point choice as an upper envelope rather than a generic decay configuration. A co-moving split supplies a locally four momentum-conserving benchmark. We classify horizons with the double root conditions [Formula: see text], distinguish asymptotically flat, de Sitter, and anti-de Sitter branches, and impose the corresponding global outward-accessibility criterion. In the de Sitter static patch the electrostatic energy is referenced to the cosmological horizon, so the relevant scale is [Formula: see text] rather than [Formula: see text]. Reproducible scans compare Reissner–Nordström, Einstein–Euler–Heisenberg type, [Formula: see text]–Maxwell, and full [Formula: see text]–Euler–Heisenberg type backgrounds. The results show that the branch-referenced horizon potential controls the local upper envelope, whereas the asymptotic structure determines whether an outward trajectory can reach infinity, a cosmological horizon, or only a finite outer turning point.