This paper provides a formal proof of the P is not NP conjecture by introducing Structural Manifold Compression (SMC). By mapping Boolean circuits to multilinear manifolds, we define "Metric Tension" as a global spectral invariant derived from the Hessian permanent. We demonstrate that polynomial-sized circuits are restricted to a sub-factorial curvature envelope, while the Permanent function requires a factorial interaction entropy. This geometric separation bypasses the Natural Proofs barrier, establishing that NP-hard functions are fundamentally incompressible into polynomial-time architectures.