A Prime Distribution Formula Valid from N_x to Infinity: Domain of Validity, Geometric Foundation, and the Inversion of Predictive Difficulty in Prime Number Theory

A prime distribution formula is presented with an explicit domain of validity: from a threshold prime N_x to infinity. The formula is validated on M_82589933 = 2^82589933 - 1, the largest known prime (24,862,048 decimal digits), with a residual of 0.001% — 4.5 times smaller than the training residual, directly confirming the arithmetic mass principle. The cylindrical foot of the vase corresponds to the inner face of the toroidal electron in Genesis-1. The threshold N_x is analogous to the Bohr radius. Interactive visualizations at genesis1.net. Fourth in the Genesis-1 prime geometry series.