Ramanujan's Inequality Asymptotic Principle A Unified Framework for Analytic, Geometric, and Informational Inequalities

This paper introduces a unified framework for expressing mathematical inequalities through structured decomposition. Inspired by Ramanujan’s asymptotic expansions, the principle asserts that many inequalities—across number theory, geometry, and information theory—can be written as: $$ F(x) = M(x) + E(x), \quad |E(x)| \leq \Phi(x)\Psi(x) $$ Here, the main term M(x)M(x) captures dominant behavior, the decay factor Φ(x)\Phi(x) controls tightness, and the remainder Ψ(x)\Psi(x) encodes oscillation or structural deviation. The framework applies to both analytic functions and symbolic inequalities, introducing a classification system to describe dominance, symmetry, and critical behavior. It formalizes decomposition even in cases without explicit error terms, such as the triangle inequality or Brunn–Minkowski, and benchmarks falsifiability through subconvexity bounds and structural rigidity. The paper includes canonical examples, counterexamples, admissibility criteria, and a general theorem. It concludes with a propagation workflow and scaffolds for applying the decomposition to new problems across mathematical domains.