Orthogonal Tubular Codes: An Exactly Solvable Constrained Variant of the Sphere Packing Problem

The sphere packing problem seeks the maximum number of points that can be placed on the unit sphere S² such that the spherical angular distance between any two points is no less than a given value. For a separation of 1°, the classical area estimate gives an upper bound of 52470, while the exact value remains unknown. This paper does not directly solve the general M(1°). Instead, we introduce a specific geometric constraint—the Orthogonal Tubular Code—and rigorously solve the packing problem under this constraint. The constraint requires all points to lie within the union of the tubular neighborhoods of three mutually orthogonal great circles. This constraint reduces the two-dimensional spherical packing problem to a one-dimensional orbital arrangement problem. We strictly prove that the exact capacity of a single tubular neighborhood is 720, the global upper bound is 2160, and the global lower bound is 2118, leaving a gap of only about 2%. The small gap between the upper and lower bounds arises from boundary effects near the six intersection points of the great circles. Closing this gap is not the goal of this paper; rather, we precisely locate its origin and provide a clear target for future refinement. This result is far smaller than the classical area estimate upper bound of 52470, demonstrating the significant reduction in problem complexity achieved by strong geometric constraints. The Orthogonal Tubular Code is a novel exactly solvable model in sphere packing theory.