Orthogonal Tubular Codes and the Exact Solution to the Strongly Constrained Sphere Packing-Covering Joint Problem

The sphere packing problem asks for the maximum number of points that can beplaced on the unit sphere S² with pairwise angular distance at least 1°. Theexact value has remained unknown for over a century, with the classical areaestimate giving an upper bound of 52470. This paper introduces a newgeometric constraint—the Orthogonal Tubular Code—and solves the stronglyconstrained sphere packing-covering joint problem exactly. The constraintrequires all points to lie within the union of the tubular neighborhoods ofthree mutually orthogonal great circles. Under this constraint, the two-dimensional packing problem is reduced to a one-dimensional orbitalarrangement. The exact capacity of a single tubular neighborhood is provedto be 720, yielding global bounds 2118 ≤ M_OTC(1°) ≤ 2160. Upon furtherimposing directional covering completeness and the orbital closurecondition, the point set necessarily forms a three-layer mutually orthogonalclosed orbital structure. The total effective angular circumference isrigorously proved to be 918°, yielding exactly 1836 points. The orbitalclosure condition reduces the continuous solution to integer solutions whilepreserving the total count of 1836. A proof by contradiction demonstratesthat no other even number can satisfy all constraints. The number 1836 isproved to be the minimum even number satisfying all constraints, and isuniquely determined as the exact solution to the strongly constrainedvariant of the sphere packing problem. The variational derivation of thecommon difference δ = 91° is provided in complete detail, and thetransition from the continuous optimum to integer solutions is rigorouslyestablished.