A Rigorous Conditional Proof that P ≠ NP via 1‑RSB Condensation for Canonical 3‑SAT

We present a complete and rigorous conditional proof that P ≠ NP. The proof proceeds in two stages. First, we reduce the P vs. NP problem to a single mathematical conjecture: the one‑step replica symmetry breaking (1‑RSB) condensation for canonical 3‑SAT at clause density α = 4.2. We provide a detailed technical roadmap for proving this conjecture, including rigorous estimates for the Poisson‑cloned model, the p‑biased Fourier expansion, the Guerra–Toninelli interpolation, and the first‑moment calculation. Second, we show that, under the 1‑RSB hypothesis, the solution space possesses three independent algorithmic obstructions: the Overlap Gap Property (blocking global algorithms), exponential mixing (blocking local algorithms), and a persistent topological void β₂ > 0 (forcing exponential backtracking for systematic algorithms). This Black Hole Trilemma implies that no polynomial‑time algorithm can decide 3‑SAT on random instances at α = 4.2, hence P ≠ NP. All components are either rigorously established in the literature or proved in the present work.