BC State-Space Computer

We propose a next-generation paradigm of state-space computing based on the arithmetic covariance structure of Bost–Connes (BC), establishing a correspondence between the composition of data, the task constraints of programs, and the physical execution of computation within a common structure. Prime-factor composition is carried by mode occupations, multiplication changes the corresponding occupations, and the logarithmic energy spectrum relates these data changes to definite energy differences. Programs select admissible states and exchange processes through programmable boundaries, allowing task conditions to participate in the formation of results. The arithmetic module performs internal processing, while the interaction module brings external conditions into a run and passes the resulting records and states to subsequent computations. On finite carriers, we establish a compilation from divisibility conditions to occupation windows, derive the target distribution, convergence time, and joint error bound for an independent-exchange model, and provide correctness guarantees for condition updates and multistage runs. These results show how a common arithmetic structure can connect data representation, task compilation, and result formation, providing a unified mathematical description of component reuse and computational continuation, and a concrete structural and dynamical basis for the BC state-space computer.