The Exact Solution to the Strongly Constrained Sphere Packing Problem: A Hierarchical Constraint Approach from Bounds to Uniqueness

The sphere packing problem seeks the maximum number of points on the unit sphere S² with pairwise angular distance at least a given value. For a 1° separation, the classical upper bound is 52470, but the exact value remains unknown. This paper does not solve the general problem directly. Instead, we impose three layers of geometrically natural constraints that transform it into an exactly solvable variant. Layer 1 constrains all points to lie within the tubular neighborhoods of three mutually orthogonal great circles, compressing the upper bound from 52470 to the interval [2118, 2160]. Layer 2 adds directional covering completeness, requiring the point set to cover the entire sphere. This forces a three-layer mutually orthogonal orbital structure, locks the total effective angular circumference at 918°, and compresses the interval to the exact value 1836. Layer 3 imposes an orbital closure condition, reducing the continuum of solutions to integer solutions while keeping the total invariant at 1836, thereby proving uniqueness. This hierarchical constraint approach offers a new pathway for tackling intractable problems: rather than solving the original problem, one solves its strongest constrained variant and proves its uniqueness.