Continuous Constraint Phase Transition and the Solvability Boundary of the Sphere Packing Problem

The computational complexity of the sphere packing problem changes with constraint strength. Prior work [1-4] defined discrete constraint levels, proving that general sphere packing is NP-complete, while the strongly constrained variant is constant-time solvable, yielding 1836. This paper generalizes constraint strength to a continuous parameter—the tubular neighborhood width w can vary continuously from 1° to any positive value. We prove the existence of a critical width w_c: when w < w_c, the problem remains NP-complete; when w ≥ w_c, the problem enters the polynomial-time solvable region; as w further increases to the covering completeness threshold, the problem reduces to constant-time solvable with the unique solution 1836. This continuous phase transition reveals how constraint strength, as an order parameter, drives a cascading reduction in computational complexity. 1836 is the unique solution at the continuous constraint saturation point and is the global attractor in the continuous constraint space. This paper provides a continuous theoretical framework for understanding how geometric constraints eliminate computational difficulty.