A Complete Conditional Proof that P ≠ NP via 1RSB Condensation, the Black Hole Trilemma, and the Reduction Conjecture
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2026
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| author | Derscariu, Radu-Daniel |
| author_facet | Derscariu, Radu-Daniel |
| contents | <p>We present a complete conditional proof that P ̸= NP by reducing the problem to a single, well-defined Reduction Conjecture. First, we establish that canonical random 3-SAT at clause density α = 4.2 belongs to the one-step replica symmetry breaking (1RSB) universality class, extending the Nam–Sly–Sohn framework. The 1RSB condensation implies the Overlap Gap Property (OGP), persistent topological voids (β2 > 0), and macroscopic free-energy barriers. Second, we prove the Black Hole Trilemma: any polynomial-time algorithm for this problem must be either global (blocked by the OGP), systematic (blocked by exponential backtracking from β2 > 0), or local (blocked by exponentially slow mixing). Third, we formulate the Reduction Conjecture: any algorithm that uses global information about the solution space can be simulated by a Statistical Query (SQ) algorithm, and therefore fails when the Franz–Parisi potential is strictly positive. We provide strong theoretical justification for the conjecture via the transfer principle of Jones et al. and the SQ/low-degree equivalence of Brennan et al., as well as direct experimental evidence showing that low-degree predictability collapses to zero at the critical density. Under the Reduction Conjecture, the Black Hole Trilemma covers all remaining cases, and P ̸= NP follows unconditionally. All other components are rigorously proved either in the literature or in the author’s prior work.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_20103979 |
| institution | Zenodo |
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| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | A Complete Conditional Proof that P ≠ NP via 1RSB Condensation, the Black Hole Trilemma, and the Reduction Conjecture Derscariu, Radu-Daniel P vs NP 1RSB condensation Overlap gap property Black Hole Trilemma Reduction Conjecture Statistical Query <p>We present a complete conditional proof that P ̸= NP by reducing the problem to a single, well-defined Reduction Conjecture. First, we establish that canonical random 3-SAT at clause density α = 4.2 belongs to the one-step replica symmetry breaking (1RSB) universality class, extending the Nam–Sly–Sohn framework. The 1RSB condensation implies the Overlap Gap Property (OGP), persistent topological voids (β2 > 0), and macroscopic free-energy barriers. Second, we prove the Black Hole Trilemma: any polynomial-time algorithm for this problem must be either global (blocked by the OGP), systematic (blocked by exponential backtracking from β2 > 0), or local (blocked by exponentially slow mixing). Third, we formulate the Reduction Conjecture: any algorithm that uses global information about the solution space can be simulated by a Statistical Query (SQ) algorithm, and therefore fails when the Franz–Parisi potential is strictly positive. We provide strong theoretical justification for the conjecture via the transfer principle of Jones et al. and the SQ/low-degree equivalence of Brennan et al., as well as direct experimental evidence showing that low-degree predictability collapses to zero at the critical density. Under the Reduction Conjecture, the Black Hole Trilemma covers all remaining cases, and P ̸= NP follows unconditionally. All other components are rigorously proved either in the literature or in the author’s prior work.</p> |
| title | A Complete Conditional Proof that P ≠ NP via 1RSB Condensation, the Black Hole Trilemma, and the Reduction Conjecture |
| topic | P vs NP 1RSB condensation Overlap gap property Black Hole Trilemma Reduction Conjecture Statistical Query |
| url | https://doi.org/10.5281/zenodo.20103979 |