Axioms of Curvature: Revealing and Correcting Euclid's Hidden Assumption

Axioms of Curvature establishes a corrected geometric foundation by identifying and repairing a hidden assumption in Euclid’s curvature treatment. From this correction, the paper derives the Tier‑0 curvature metric, a dimensional structural ladder, and a unified operator framework that stabilizes curvature inheritance across tiers. The theory explains triangular curvature amplification, tier‑mismatch collapse, and the emergence of superposition and tensor behavior at stabilized tiers. This work provides a coherent axiomatic basis for curvature dynamics with implications for geometry, mathematical physics, and curvature‑based modeling.