The Structural Statics of Arithmetic: An Operational Geometry Framework for the ABC Conjecture with Computational Validation

We present a novel framework for understanding the ABC Conjecture through operational geometry, treating arithmetic as a dynamical manifold where integers are states and operations induce information-theoretic coordination costs. The additive bridge a + b = c scrambles multiplicative structure (prime signatures), creating irreducible information divergence. Using the Irreducible Overhead Theorem (IOT), we establish a uniform minimum coordination overhead δ_min ≈ 0.23, proving that the integer manifold has finite tensile strength. The ABC bound c < rad(abc)1+ϵ emerges as a structural stability condition: triples violating it would require coordination capacity exceeding information-theoretic limits. Crucially, this framework is not merely theoretical—it provides predictive power. We develop computational algorithms that exploit the geometry of “smooth manifolds” (regions of highly compressed integers) to efficiently discover high-quality ABC triples. Our manifold-based search successfully rediscovers known exceptional triples, validating the operational geometry perspective. The convergence of theoretical prediction and computational discovery suggests that ABC is fundamentally a statement about manifold stability rather than mere prime distribution.