Orthogonal Tubular Codes: From Tight Bounds to Exact Solution under Hierarchical Constraints

The sphere packing problem on the unit sphere S² asks for the maximum number ofpoints with pairwise angular distance at least 1°. The classical area estimategives an upper bound of 52470, but the exact value has remained unknown for overa century. This paper does not solve the general problem directly. Instead, weimpose a hierarchy of geometrically natural constraints that transform theintractable original problem into an exactly solvable one. Layer 1 constrainsall points to lie within the tubular neighborhoods of three mutually orthogonalgreat circles, compressing the upper bound from 52470 to the interval[2118, 2160]. Layer 2 adds directional covering completeness, requiring thepoint set to cover the entire sphere. This forces a three-layer mutuallyorthogonal orbital structure, locks the total effective angular circumference at918°, and compresses the interval to the exact value 1836. Layer 3 imposes anorbital closure condition, reducing the continuum of solutions to integersolutions while keeping the total invariant at 1836, thereby proving uniqueness.The central principle is that these constraints act by replacing continuousdegrees of freedom with discrete structures. Each constraint does not merelyrestrict parameters; it fundamentally changes the topology of the solutionspace, collapsing an intractable continuous problem into an exactly solvablediscrete one.