Scheme-theoretic Approach to Computational Complexity I. The Separation of P and NP

We lay the foundations of a new theory for algorithms and computational complexity by parameterizing the instances of a computational problem as a moduli scheme. Considering the geometry of the scheme associated to 3-SAT, we separate P and NP. In particular, we show that no deterministic algorithm can solve $\mathsf{\text{3-SAT}}$ in time less than ${1.296839}^n$ in the worst case.