Optical signatures, particle dynamics, and epicyclic frequencies of Mod(A)Max black holes

We investigate optical observables, neutral-particle dynamics, and epicyclic frequencies in the spacetime of spherically symmetric black holes sourced by the ModMax and phantom ModMax, or Mod(A)Max, nonlinear electrodynamics sectors. The metric contains an effective charge contribution governed by the electric charge [Formula: see text], the ModMax parameter [Formula: see text], and the discrete branch parameter [Formula: see text]. This structure is simple enough to allow analytical progress, but rich enough to generate two qualitatively different phenomenological regimes. We provide a unified analysis of null geodesics, massive circular motion, and small oscillations about stable orbits. For photons, we derive the effective potential, the photon-sphere radius, the shadow radius, the weak-field deflection angle, and representative trajectories. For neutral massive particles, we obtain the Hamiltonian effective potential, the effective radial force, the specific energy and angular momentum of circular equatorial orbits, and the innermost stable circular orbit. We then compute the Keplerian, radial, and vertical epicyclic frequencies, together with the periastron precession frequency, and discuss their relevance for QPO phenomenology. The main result is a coherent branch-dependent pattern: the ordinary ModMax sector tends to shrink the photon sphere, shadow, and ISCO scales as the effective charge grows, whereas the phantom Mod(A)Max sector enlarges these scales and shifts stable circular motion outward. The parameter [Formula: see text] exponentially suppresses the effective charge contribution and drives both branches toward the Schwarzschild limit. This combined optical–dynamical treatment identifies the parameter combination [Formula: see text] as the key quantity controlling the deviations and provides a baseline for future constraints from black-hole shadows, accretion-disk observables, and QPO measurements.