Seed Universality for Reflexive Polytopes in Dimension Four: Generation of the Kreuzer—Skarke Calabi—Yau Landscape via Toric Seed Orbits

We prove Convex Seed Universality for the Kreuzer—Skarke classification of four-dimensional reflexive polytopes. Every reflexive polytope in the Kreuzer—Skarke dataset arises from a primitive convex seed through a finite sequence of four toric operations: unimodular transformations, stellar subdivisions, polar duality, and lattice translations. Seed orbits coincide with connected components of the GKZ secondary fan, and the Hodge numbers of the associated Calabi—Yau hypersurfaces remain constant on each orbit. The seed invariant matrix is identified with the GLSM charge matrix, providing a natural toric-geometric interpretation of the construction. Four structural theorems: Seed Completeness, Orbit Connectivity, Hodge Invariance, and Exhaustiveness, together establish seed universality for the entire Kreuzer—Skarke dataset.