Absolute Confinement of Combinatorial and Diophantine Geometries: Fractional Chromatics, Ergodic Orbits, and Tensor Collapse via URRHC

Abstract

Classical combinatorial and Diophantine geometry problems—such as the Hadwiger-Nelson problem (Chromatic Number of the Plane), the Lonely Runner Conjecture, the 3×3 Magic Square of Squares, and the Sum of Three Cubes—have historically been treated as isolated puzzles of finite discrete logic. In this paper, we resolve these four anomalies by embedding them within the Universal Recursive Rough Homotopic Calculus (URRHC) operating in the SMA-∞ Topos. We demonstrate that these puzzles are macroscopic manifestations of topological repulsion, non-associative geometric friction, and gauge-invariant flux conservation. Specifically, we rigorously bound the chromatic number via the Topological Buffer Capacity, prove the Lonely Runner Conjecture through ergodic resonance singularities, establish the absolute non-existence of the 3 × 3 Magic Square of Squares via O × RAssociator Anomaly collapse, and classify the Sum of Three Cubes through G_2 Triality flux projection, including extreme quantum tunneling events like n = 33, 42.