Riemannian Geometry of the Local Class Transition Tensor

This supplement provides rigorous mathematical derivations of the Riemannian structure induced by the transition tensor Tcc′j. It includes:
(1) the complete derivation of the Christoffel symbols, (2) the sectional curvature and its properties, (3) the Bianchi identities with proofs, (4) the Gauss-Bonnet theorem for the simplex, (5) the formal comparison with Fisher-Rao geometry, and (6) the existence and uniqueness theorems. The aim is to provide a foundation that meets the most rigorous standards of mathematical review.