Additive–Multiplicative Duality and the Unique Equilibrium of the Critical Line

This paper develops a unified framework for the natural numbers based on the coexistence of two fundamentally distinct scaling structures: the additive structure and the multiplicative (prime-based) structure. Through the logarithmic map, these structures are embedded into a common additive space, where primes emerge as linearly independent frequency modes. Within this framework, the Riemann zeta function is interpreted as a spectral object that simultaneously encodes both structures. We introduce a dual balance functional that measures the imbalance between the multiplicative weighting and its functional-equation dual. We show that this functional attains its unique minimum at σ = 1/2, providing a characterization of the critical line as the unique equilibrium point where neither scaling structure dominates. An information-theoretic interpretation is also developed: primes act as incompressible codewords, while composite numbers represent compressed encodings. The zeta function then appears as the generating function of the full compression spectrum. This perspective connects naturally to spectral interference, entropy growth under coarse-graining, and prime-frequency dynamics.