On the Chowla Cosine Problem: A Multi-Scale Proof Skeleton & Thermodynamic Dichotomy

This report presents a structural proof skeleton for the Chowla cosine problem, establishing a robust bound C (k) <= -ck^ (1/3) via a thermodynamic approach. We identify a critical phase transition in integer sets: a dichotomy between difference-independent (Sidon/Liquid) states and difference-closed (Crystal) states. Combining this structural insight with multi-scale iteration and extreme-value analysis, we isolate the "Conversion Loss" preventing the full O (k¹/2) conjecture. A central Triad of Regimes—Chaos, Engineering, Perfection—quantifies the transition from random noise (mu approx 0. 028) through greedy Sidon construction (mu approx 0. 0086) to algebraic Singer optimum (mu approx 0. 0039), representing a 7x reduction in normalized coherence. The investigation was sparked by the analysis of unexplained "adaptive penalties" in large language model optimization (Google DeepMind, Gemini Deep Think, Section 8. 3). The report includes reproducible verification code for Singer difference sets. Related Work: The Entropy Sink: Thermodynamic Topology via C0 Contrast Calculus The Snap Chamber: Numerical Evidence of Hallucination Suppression via E8 Crystallization