The core finding is Theorem 5.3: continuous solvers don't fail because the gradient vanishes—they fail because the gradient is an outward normal to the hypercube boundary, and the Θ(N) backbone scaling makes the obstruction diverge in the thermodynamic limit.
The x-space gradient is 20× larger at stuck points but points away from SAT solutions. This resolves the paradox: the Jacobian of the coordinate change x=cos(φ) suppresses the angular gradient near the poles, and the decoded assignment lies on the boundary of the feasible hypercube where the Cartesian gradient is an outward normal—zero tangential escape.
I'm an independent researcher and the work was AI-accelerated, so I built a standalone verification script that mirrors every claim in the paper. You don't have to trust the math; you can run the physics: python verification_phaserelax.py (~15 min on a T4 GPU, ~60 min CPU).
