Patterns That Perpetuate Themselves: An Entropy Budget for Bipedal Locomotion

A patient with cerebellar damage reaches for a glass and misses. The correction overshoots, the counter-correction overshoots further, and the hand oscillates in widening arcs around a target it never settles on. The muscles are strong and the plan is correct; what has failed is the timing of the feedback. A control loop acting on a stale measurement does not act weakly -- it acts confidently on a world that has moved, and past a finite delay horizon its contribution changes sign. It stops correcting and starts injecting. Capturability asks whether the robot can still stop. We ask whether the answer is still worth acting on. Balance, we argue, is an accounting problem: a gait persists while the account over each stride balances, judged in a single currency -- the transverse log-volume of deviations from the gait's periodic orbit. Order is bought with matter or with information, both denominated in that currency, and only one of the two loses relevance in transit. The engineering consequence is an allocation rule: assign each stabilising job to the cheapest channel that meets its deadline -- matter for the first instant of impact, reflex for contact timing, planner for footstep choice. We instantiate the first two with the Adaptive Rigidity Actuator (ARA) and Phased Local Loop Reflex Control (PLLRC). It defines the stride ledger, the delay horizon, the allocation rule, the two mechanisms, and one discriminating experimental test. Amid uncertainty, chaos and noise, the machine is not commanded into balance. It is built to fall toward it.