Mark Tilden’s BEAM robotics demonstrated that adaptive behaviours—phototaxis, gait stabilization, obstacle recovery, and environmental exploration—can arise from simple analog circuits lacking processors, memory stores, or symbolic logic. Although these behaviours have traditionally been described topologically, as stable limit cycles on toroidal manifolds, BEAM systems have lacked a unifying energetic account. Here we apply the Deferred-Dissipation Transform (DDT) to show that BEAM agents operate, in a strictly geometric and dynamical sense, as embodied τ-systems: dissipative machines in which short-horizon, path-dependent behaviour emerges from asymmetric charge–discharge pathways in the persistence coordinate τ = E / (T Ṡ). Distinguishing between substrate-bound τ-machines (e.g., RC/LC circuits) and mobile embodied τ-machines, we show that locomotion continually reshapes a robot’s dissipative geometry. Incorporating phase-dependent dissipation—reflecting the higher energetic cost of sensory accumulation (charging) versus the inertial efficiency of motion (discharging)—reveals that identical energy states acquire different persistence values depending on their recent dynamical pathway. Simulations of a minimal two-reservoir phototaxis agent exhibit three hallmark signatures of this mechanism: (1) robust τ-loops generated by asymmetric dissipation, (2) consistently positive persistence surplus indicating retention of recent dissipative history, and (3) reliable long-range approach trajectories—including spiraling, orbital passes, and repeated near-source engagements—across diverse initial conditions. Sensitivity analysis shows that these features are structural invariants, persisting across ±20–25% variation in all key physical parameters. These results demonstrate that the classical structural stability of BEAM nervous-net oscillators, long described via Peixoto stability, is physically expressed through gradient ascent in τ-space rather than through circuit heuristics alone. DDT thereby unifies BEAM robotics with non-equilibrium thermodynamics and establishes BEAM agents as minimal embodied systems whose behaviour emerges from the geometry of dissipative flow—revealing an energetic organisation that parallels gradient-following dynamics in biological taxis. Keywords: BEAM robotics; dissipative systems; Deferred-Dissipation Transform (DDT); persistence coordinate; τ-geometry; hysteresis; non-equilibrium thermodynamics; embodied intelligence; analog robotics; memristive dynamics; Peixoto stability; phototaxis; dynamical systems; gradient flows; energy-based modelling.