Hodge–Toric Geometry Beyond CY20

This article develops a higher-dimensional research programme for Calabi–Yau geometry beyond the classical threefold setting and through the explicit CY₂₀ horizon. It integrates hypersurface and toric constructions, Hodge statistics, mirror laws, special-holonomy constraints, conditional SYZ lifting, and dimensional-saturation models into a single framework for studying the growth and organization of Calabi–Yau landscapes. The manuscript distinguishes proved construction results from computational evidence and asymptotic conjectures, while incorporating a corrected treatment of toric intersection products, including repeated divisors and self-intersections. The resulting programme provides a systematic mathematical and computational architecture for testing higher-dimensional regularities, mirror behaviour, and possible limiting laws in Calabi–Yau geometry.