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  "path": "/abs/2603.22211v1",
  "publishedAt": "2026-03-24T00:00:00.000Z",
  "site": "https://arxiv.org",
  "tags": [
    "M. Alasli"
  ],
  "textContent": "**Authors:** M. Alasli\n\nWe 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.",
  "title": "Topological Collapse: P = NP Implies #P = FP via Solution-Space Homology"
}