Zipf's Law as a Consequence of the Axiomatica Universalis: Language at the Optimal Operating Point

Zipf's Law — the inverse power-law distribution of word frequencies — has resisted a universal explanation for 90 years. We derive it as a direct consequence of the Axiomatica Universalis: any conforming receiver of depth D operating at C* = 1/(D+1) produces output distributions following a power law with exponent α ≈ D+1. For natural language (D≈1), α ≈ 2, consistent with observed Zipf exponents. The result unifies Zipf's Law with 1/f spectra under a single axiomatic framework.