We introduce an empirical framework treating the positive integers as a compressible arithmetic fluid, where composite structure induces effective density, pressure, viscosity, entropy, and shock-like behavior. Using operationally defined observables derived from prime factorization statistics, we construct a discrete fluid state over N and analyze its large-scale behavior. We show that these quantities are finite, structured, and strongly coupled, with singular release events associated with primes. This work establishes a diagnostic foundation for arithmetic dynamics without invoking probabilistic randomness assumptions.