The Irreducible Overhead Theorem: Why Exponential Cost Cannot Be Perfectly Conserved

We examine the tradeoff between time and parallelism in exact algorithms for NP-complete problems. While it is folklore that exponential parallelism can simulate nondeterministic search in polynomial time, this observation is often interpreted as permitting a perfect conservation of exponential cost. We show that such an exact tradeoff is impossible. For any exact algorithm deciding an NP-complete language, the product of time and parallelism must exceed the size of the nondeterministic search space by a constant factor. This irreducible loss arises not from physical constraints but from structural properties of computation: non-injectivity of verification, information erasure, and limits on compressibility. We formalize this loss via Kolmogorov complexity and prefix-free coding and prove the Irreducible Overhead theorem: exponential computational cost cannot be conserved exactly across time and parallelism.