Composition of Central Projection and the Closed Form of the Composite Curvature Radius: An Algebraic Formulation of High-Dimensional Reduction via One Central Projection and Commutative Cuts on the Sphere

This paper provides an algebraic foundation for dimensional reduction by central projection. The main results are: First stage: Central projection from Rⁿ to the sphere S^n-1 (r₁) is a single map; this is the true dimension-reducing operation. Second stage: Subsequent axis-wise operations on the sphere are simultaneity-section cuts, not central projections. Commutativity: Cut operations along distinct axes are completely commutative. Closed form of the composite curvature radius: rfinal² = r₁² - sum₈ ₈₍ ₒ (xᵢ^*) ², where xᵢ^* is the axis component on the sphere immediately after the first central projection. Algebraic structure: The set of cut operations forms an Abelian semigroup. The paper deliberately avoids physical interpretation and provides a universal algebraic foundation for any application requiring dimensional reduction from n to d < n dimensions. Bilingual (Japanese / English) preprint with md/tex/pdf formats (6 files total).