Dynamics of the Bost–Connes State Space

We investigate whether the dynamics of the formation, motion, decay, and thermalization of matter in the physical world can be described within a unified and internally consistent mathematical framework. To this end, we start from the local decomposition of the Bost–Connes system and perform a finite p-adic reduction to construct the corresponding state space. Within this space, we characterize stable states relative to a background and establish a complete cycle consisting of free evolution, formation exchange, and pattern reconstruction. Excitation formation, motion, and decay, together with relaxation of the ensemble occupation distribution, are determined by the same joint evolution and its iterations, with the final state of each cycle serving as the initial state of the next. References, observation, and records are incorporated into the same joint construction, giving the evolution of a subject and its observable manifestations a common mathematical basis. Under the corresponding conditions, we derive event statistics, convergence to stationarity, and thermalization, and use joint conservation and entropy balances to relate reversible joint evolution to irreversible system relaxation.