Energy-fed commitment: a minimal model of random selection followed by reinforcement

When a symmetric instability is crossed, fluctuations choose one of several equivalent branches. We study a minimal stochastic model of what can then make that choice progressively harder to undo. After selection, the stiffness of the selected state grows in proportion to the energy released each time the structure recovers from a perturbation; in this conservative model that energy equals, up to small and explicitly computed terms, the work absorbed from the perturbing agent. The model is a phenomenological closure with one coupling constant. That energy reinforces a branch already chosen, rather than choosing it, is an assumption of the model, which we state and do not derive. Its consequences are quantitative. The growth of commitment depends on how the structure is driven: geometric under displacement control, linear under energy control and sublinear under force control. Reinforcement suppresses reversal exponentially; beyond a finite commitment the alternative minimum disappears and reversals become rare crossings of an irreversibility threshold. With unbounded reinforcement a choice is retained forever with positive probability under stated conditions; with saturating reinforcement retention is finite, with a lifetime that grows exponentially with the saturation level. Exact first-passage calculations agree with stochastic simulations; these verify the closure and its analysis but cannot test the physical postulate. We identify the experiments that could: protocols that separate absorbed energy from the number of perturbations, and driven steady states with nonzero housekeeping heat, in which heat and absorbed work differ.