Dynamics of the Bost–Connes State Space

On the state space obtained by finite p-adic reduction of the Bost–Connes system, we characterize stable states relative to a background and construct a complete cycle of free evolution, formation exchange, and pattern reconstruction, giving a unified description of formation, motion, decay, and thermalization. Exchange between excitations and the background realizes coherent retrieval, exit, reverse formation, and background restoration within the same joint evolution. The state entering the next cycle is reconstructed from these components with their retained mutual coherence, providing a common dynamical basis for excitation motion, formation, and decay, and for relaxation of the ensemble occupation distribution. On this basis, we construct internal observation and recording processes, derive event statistics and convergence to stationarity under the corresponding conditions, and use joint conservation and entropy balances to relate reversible joint evolution to irreversible system relaxation.