Structural Manifold Compression: A Geometric Theory of Computational Limits

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