I introduce Structural Manifold Compression (SMC), a framework for analyzingcircuit complexity through differential-geometric invariants. I define the MetricTension (τ )—the integrated magnitude of a polynomial’s Hessian determinant—asa measure of structural complexity. By establishing a bound on the localized curva-ture capacity of polynomial-size circuits, I demonstrate an asymptotic gap betweenthe tension required by #P-complete manifolds and the capacity of P-class circuitgenerators. This framework provides a novel, non-natural pathway for establish-ing lower bounds in algebraic complexity by treating computational ”hardness” asgeometric incompressibility.