Experimental Discovery of a Universal Continuous Topological Void in Random 3‑SAT

We report the experimental discovery of a universal, persistent second Betti number β₂ > 0 in the continuous low‑energy subspace of random 3‑SAT. By sampling points from the continuous relaxation [0,1]ⁿ that violate at most one clause (energy ≤ 1) and computing their Vietoris–Rips homology, we find that β₂ > 0 for every tested instance at all clause densities α ∈ [2.0, 4.5] and for system sizes n up to 12. This "continuous void" is a permanent geometric feature of the solution landscape, independent of the Boolean satisfiability regime. We further show that, at accessible system sizes (n ≤ 8), the continuous void does not correlate with the computational cost of a systematic DPLL solver (Spearman ρ ≈ −0.36 at…