Universal Overlap Gap Barriers across NP‑Complete Problems: A Computational Black Hole Conjecture

We present the most extensive experimental validation of the Overlap Gap Property (OGP) on NP‑complete problems to date, covering 8 canonical problems from fundamentally different domains—logic (3‑SAT, NAE‑3‑SAT), number theory (Number Partitioning), random graphs (MAX‑CUT, Maximum Independent Set, Planted Clique), neural networks (Symmetric Binary Perceptron), and statistical inference (Stochastic Block Model)—plus a novel Collision OGP setting for binary perceptrons. Across all 9 experimental contexts, we simultaneously measure 6 OGP variants: classical OGP, multi‑OGP (m‑OGP), Branching OGP (BOGP), ultrametric OGP, the optimized Franz–Parisi criterion, and our own Topological Complexity Density (TCD). Our main experimental findings are: (1) every NP‑complete problem tested exhibits at least one form of OGP barrier; (2) BOGP is the dominant signature (72–100% across all problems), suggesting branching fragmentation is the fundamental geometric obstruction; (3) we identify a three‑phase computational taxonomy: BOGP dominates the easy (P) phase, classical OGP marks the critical threshold, and a strongly positive Franz–Parisi potential signals the deep NP regime; (4) the barrier is selective, annihilating message‑passing algorithms (Belief Propagation, Survey Propagation: 0% success) while CDCL solvers (Kissat: 100%) remain unaffected. We formalize the Computational Black Hole Conjecture: any polynomial‑time algorithm for 3‑SAT must be stable (or reducible to a stable one) in the sense of the OGP, and is therefore blocked by the OGP at the critical threshold α ≈ 4.2, implying P ≠ NP. We provide a complete roadmap: all components are either rigorously proven in the literature or experimentally confirmed by our work, leaving exactly one theoretical step to prove. Updated version (v2): extended experimental validation from 3‑SAT to 8 NP‑complete problems plus Collision OGP; added three‑phase computational taxonomy (P/Transition/NP); integrated all OGP variants (BOGP, m‑OGP, ult‑OGP, Franz–Parisi) into a unified framework.