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Coherence Amplitude Framework for Mathematical Problem Analysis by John Gavel Abstract The Coherence Amplitude Framework provides a unified mathematical approach for analyzing complexity bottlenecks across diverse problem domains. By quantifying the interplay between structural constraints, combinatorial resources, and local irregularities, the framework predicts critical points where problems transition from tractable to intractable. I. Foundational Definitions Definition 1.1: Problem Instance Space Let 𝒫 be a mathematical problem domain with parameter space Ω ⊂ ℝᵈ . For each ω ∈ Ω , denote P(ω) as a specific problem instance. Definition 1.2: Basic Structural Components For any problem instance P(ω) , define: Basic Units 𝒰(ω) : The fundamental combinatorial or algebraic objects that compose solutions. Configuration Space 𝒞(ω) : The set of all possible arrangements of basic units. Constraint Set 𝒦(ω) : The conditions that valid solutions must sa...