Exact Solution to the Strongly Constrained Sphere Packing-Covering Joint Problem

The sphere packing problem asks for the maximum number of points that can be placed on the unit sphere S² with a pairwise spherical distance of at least a given value. The sphere covering problem asks for the minimum number of points to cover the entire sphere. These two problems are usually studied independently. This paper unites them, seeking the minimum number of points that simultaneously satisfies the following three constraints: (i) any two points have a spherical distance ≥ 1°, (ii) the point set covers the entire sphere—for any direction, there exists a point within its 1° neighborhood, and (iii) the points lie on closed orbits and are uniformly arranged. We prove that under this strong constraint, the point set necessarily forms a three-layer mutually orthogonal closed orbital structure, with a total effective angular circumference of 918° and a total of 1836 points. The three orbital angular circumferences form an arithmetic progression (215.5°, 306°, 397.5°), and the total number of points is 2 × 918 = 1836. The orbital closure condition reduces the continuous solution to integer solutions (215, 306, 397) and (216, 306, 396), while the total point count remains unchanged at 1836. 1836 is the minimum even number satisfying all constraints. This paper does not claim to solve the general sphere packing problem, but rather solves the strongly constrained variant of the packing-covering joint problem, yielding the exact solution 1836.