Topological Collapse: P = NP Implies #P = FP via Solution-Space Homology
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866910066229116928 |
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| author | Alasli, M. |
| author_facet | Alasli, M. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_22211 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Topological Collapse: P = NP Implies #P = FP via Solution-Space Homology Alasli, M. Computational Complexity 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. |
| title | Topological Collapse: P = NP Implies #P = FP via Solution-Space Homology |
| topic | Computational Complexity |
| url | https://arxiv.org/abs/2603.22211 |