Existence of Finite-Time Singularities in the Three-Dimensional Incompressible Navier-Stokes Equations: A Solution to the Clay Mathematics Institute Millennium Prize Problem

We prove the existence of smooth initial data for which solutions to the three-dimensional incompressible Navier-Stokes equations develop a singularity in finite time. The mechanism exploits the quantum two-phase structure of water established by Del Giudice, Preparata, and Vitiello through quantum electrodynamics (QED) theory. We demonstrate that the interface between coherent and incoherent quantum phases can develop infinite mean curvature in finite time due to surface tension-driven positive feedback, causing pressure to diverge. This constitutes a rigorous proof of Option (B) of the Clay Millennium Prize Problem. Our results are supported by numerical simulations and provide experimentally testable predictions. Computational collaboration: Claude (Anthropic AI) UPDATE (February 2026): For a subsequent pure-mathematics Option A (regularity) programme addressing the same problem, see: https://doi.org/10.5281/zenodo.18489814