The Stability Axiom for the Prime Population: Centrality of Integers, Invariant Logarithmic Windows, and Additive Consequences

This article introduces and develops a population-level axiom governing the distribution of prime numbers, referred to as the Axiom of Logarithmic Stability. The axiom formalizes the empirical and analytic observation that the global prime population imposes invariant logarithmic windows around every integer, within which local fluctuations are absorbed and structural symmetry is enforced. We show that this axiom implies the centrality of integers within prime gaps, establishes tightness properties for symmetric prime offsets, and yields additive symmetry as a direct corollary. In particular, Goldbach’s conjecture emerges naturally as a consequence of population stability rather than as an isolated additive hypothesis. The framework is supported by extensive empirical validation and is positioned within the broader historical and theoretical context of analytic number theory.