Topological Collapse: P = NP Implies #P = FP via Solution-Space Homology

TL;DR

This paper shows that if P equals NP, then #P equals FP by analyzing the topological structure of 3SAT solution spaces, revealing deep connections between computational complexity and topology.

Contribution

It introduces a novel topological approach to relate P, NP, and #P, proving that P=NP implies #P=FP through solution-space homology analysis.

Findings

Topological invariants are #P-hard to compute.

Local information is insufficient for certifying unsatisfiability.

Empirical evidence of solution-space shattering at scale.

Abstract

We prove that P = NP implies #P = FP by exploiting the topological structure of 3SAT solution spaces. The argument proceeds via a dichotomy: any polynomial-time algorithm for 3SAT either operates without global knowledge of the solution-space topology, in which case it cannot certify unsatisfiability for instances with second Betti number b_2 = 2^{Omega(N)} (leading to contradiction), or it computes global topological invariants, which are #P-hard. As local information is provably insufficient and any useful global invariant is #P-hard, the dichotomy is exhaustive. The proof is non-relativizing, consistent with oracles separating P = NP from #P = FP, and therefore necessarily exploits non-oracle properties of computation. Combined with Toda's theorem, the result yields P = NP => #P = FP => PH = P, providing new structural evidence for P != NP via a topological mechanism. We complement the theoretical framework with empirical validation of solution-space shattering at scale (N up to 500), demonstrating that these topological barriers manifest as measurable hardness across five independent algorithm classes.