Dissipative Numeral System, Paper 0: Axiomatic Foundations — A Formal Framework for Resource-Conscious Number Systems

This paper establishes the axiomatic foundations of the Dissipative Numeral System (DNS). Five axioms define a number system where resource accumulation, structural formation, cost, memory, and history are primitive features. The axioms formalize the separation of resources from structures, the requirement of cost for structural generation, the preservation of residual resources, the creation of independent hierarchical objects, and the non-equivalence of structures with different formation histories. From these, we derive the Latent Carry State and Structured Zero, and prove foundational results including the existence of latent states, the uniqueness of the classical limit, and the closure of the state space. The Classical Limit of DNS is precisely ordinary positional notation, establishing DNS as a proper generalization. This work serves as the logical foundation for subsequent developments, particularly a calculus of structural decomposition. The framework is inspired by the meta-theoretical principles of Reset Dynamics, with a detailed correspondence provided in an appendix.