The Period Valley Bijection - Geometry Encodes Arithmetic

We establish a bijection between periods in a symbolic sequence and valleys in its coor-dination surface. The correspondence preserves quantitative structure: valley depth equalsnormalized period depth (d∗/N ), and curvature is determined by depth via K = 3(d∗/N )2. Thebijection respects the divisor lattice: primitive periods correspond to primitive valleys, and theM¨obius function governs the extraction of primitives from harmonics. This unifies the geometric,arithmetic, and topological aspects of the theory.