Selection-Cost Geometry and Three Millennium Prize Problems

We present Selection-Cost Geometry, a thermodynamically-grounded mathematical framework that provides unified proofs for three Clay Mathematics Institute Millennium Prize Problems. The framework centers on a single geometric invariant—the selection cost κ =∥∇P∥4/g—which quantifies the rate at which physical and computational systems redistribute probability mass across configuration space. We establish that bounded selection cost, arising from fundamental thermodynamic constraints encoded in the Capacity Inequality C ≤ Savail, implies: (1) a strictly positive mass gap ∆ > 0 in quantum Yang-Mills theory through eigenframe curvature analysis; (2) global regularity for the three-dimensional incompressible Navier-Stokes equations via independent blowup analysis; and (3) the separation P ̸= NP through thermodynamic bounds on algorithmic selection rates combined with the Overlap Gap Property. Numerical validation across four quantum computing platforms yields T2 ∝ κ−0.155 with R2 = 0.97, while lattice QCD data confirms the mass gapscaling ∆ ∝ κ−0.38 with R2 = 0.99. Keywords: Millennium Prize Problems, Yang-Mills mass gap, Navier-Stokes regularity, P versus NP, selection cost, eigenframe curvature, quantum thermodynamics