Boolean Cube Cutting Reveals the Geometric Origin of the 0.72 Threshold in NP-hard Problems

The critical threshold Γc ≈ 0.72 appears universally across NP-hard problems including 3-SAT, TSP, VRPTW, and Max-Cut, yet its theoretical origin has remained unknown. Here we establish that this value is a mathematical necessity arising from the geometry of Boolean cube cutting. By modeling constraints as subcube deletions from the ndimensional hypercube {0, 1}n, we prove that the complexity phase transition occurs exactly at Γc = H(0.2) = −0.2 log2 0.2 − 0.8 log2 0.8 ≈ 0.7219. The key term log2 5 emerges naturally from the geometric ratio 0.2 = 1/5 imposed by critical constraint density. This derivation requires no empirical fitting, curve-crossing arguments, or external models—only the combinatorial geometry of vertex deletion on the Boolean cube. Our results establish computational complexity phase transitions as geometric percolation phenomena, providing the first predictive theory for the Easy-Hard transition in combinatorial optimization.