The Unification of Formation, Motion, Decay, and Thermalization of Matter from the Bost–Connes System

The formation, motion, decay, and thermalization of matter have each been established on the state space of the Bost–Connes system, but the complete cycle map has not yet been constructed. On the finite joint space taking part in the exchange, this paper composes free evolution, formation exchange, and pattern reconstruction into a unitary cycle, with the exchange matrix elements and the environment preparation given as conditions of the construction. A standing wave exchanges continuously with the environment during its propagation, and the system state at the end of the cycle and the outgoing records are determined by the same joint evolution; the environment is refreshed cycle by cycle, and the system continues to evolve under the same map. Retrieval, transit, and transport together determine the maintenance of the standing wave, the change of identity, and the formation of records, while the reverse exchange replenishes formation under the corresponding incident occupation. Record weights are determined, under explicit statistical conditions, by the squared norms of the components of the joint state, and energy and the corresponding conserved charges are balanced between system and environment. Under a specified coherent detection preparation, a conserving cycle realizes position records with controlled error and gives error bounds for the interference visibility; the corresponding finite model gives conditional thermalization during identity maintenance and the equilibrium of formation and exit. Under fixed supply and convergence conditions, the steady total occupation of a linear ensemble equals the product of the production rate and the mean waiting time to the first exit. The evolution of a single standing wave, the occupation of many standing waves, and the statistics of actual records are thereby connected within the same cycle.