We prove that computational complexity classes correspond to topological invariants of semantic manifolds.

Polynomial-time decidability requires genus ≤ 1, while NP-complete problems induce genus ≥ 2 via self-referential structure.

This genus gap creates an uncrossable operational velocity boundary, yielding P ≠ NP as a corollary.

The same topological constraint explains Gödel incompleteness and democratic impossibility.

Using the Coulon complex visualization above, we demonstrate how semantic chains (axioms → theorems → proofs) form higher-dimensional structures whose genus determines computational tractability.

The framework unifies complexity theory, metamathematics, and political philosophy through the common language of algebraic topology.