Full Spherical Coverage of Central Projection in Discrete Space and the Number-Theoretic Necessity of 5-Dimensional Background Space

This paper derives, from fundamental theorems of number theory, that 4 is the minimum dimension for an isotropic discrete lattice (Lagrange four-square theorem), that 5 dimensions are required as the background space for central projection (homogeneous coordinates), and constructs a 5-axis 10-face relay model that resolves the equatorial divergence and south-hemisphere invisibility problems of inverse mapping on S⁴(R).