Arithmetic Ecology: A Constraint-Dissipation Framework for Prime Structure

We propose an ecological interpretation of arithmetic in which integers are not static objects but participants in a dynamic coherence system governed by irreversible operations, latent structure, and periodic constraint relief. Multiplication is treated as a many-to-one, entropy-like process, while factorization is gradient-opposed. From this asymmetry emerges a landscape of M_latent—dormant arithmetic potential encoded in residue-class structure—that accumulates through compositional pressure and dissipates through admissible configurations. Classical problems are reinterpreted not as isolated puzzles but as structural diagnostics: observables measuring different aspects of constraint-dissipation dynamics. Goldbach measures compositional pathway connectivity; bounded gaps measure coarse dissipation capacity; twin primes specify minimal optimal relief geometry; the Riemann Hypothesis bounds maximal turbulence in the constraint field; P̸=NP reflects the substrate’s intrinsic operational gradient; Collatz tests parity-dominance in dissipation channels. These conjectures resist traditional proof because they concern global system stability rather than element-local properties. We argue that several are already structurally resolved—their violation would contradict observed operational gradients defining arithmetic’s substrate. The stable functioning of arithmetic across all measurable scales constitutes evidence these requirements hold.