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We establish that P ≠ NP conditional on a single physically motivated assumption called Landscape Opacity (LO), while presenting three concrete paths toward an unconditional proof. The results are supported by extensive computational evidence spanning problem sizes from n = 10 to n = 50,000 variables, generated using the Isomorphic Engine (performance: 1.87 × 10⁹ spins/sec across 14 parallel solvers).Unconditional results include:(i) Exponential fragmentation: The QUBO Ising landscape for random 3-SAT at the critical density r_c = 4.267 contains 2^Ω(n) distinct local minima (ACR shattering theorem). At n = 200, 2,996 out of 3,000 random restarts yield unique minima (exceeding the birthday bound, implying > 10⁶ true basins). At n = 50,000, complete restart saturation is observed: every probe in a 250,000-dimensional search space lands in a distinct basin.(ii) Local search lower bound: Any local search algorithm requires exp(Ω(n)) queries to reach the ground state.(iii) Conditional implication: Landscape Opacity (LO) ⇒ P ≠ NP.Three paths toward an unconditional proof:(a) Replica Symmetry Breaking (RSB): The Edwards–Anderson order parameter q_EA approaches 0.50 as n → ∞ (verified up to n ≤ 10,000), indicating entry into the 1-RSB glass phase in which the ground state becomes information-theoretically inaccessible.(b) Circuit complexity: Solver step count scales super-linearly with n (depth ratio ∼ n^{0.3}), with more than 96% of spins frozen at the ground state.(c) U₂₄ rigidity: At n = 50, the macroscopic order parameter g_macro yields Ω-products between 16 and 22 (converging toward Ω = 24). The non-polynomiality of the Reeds endomorphism (x² + 14x + 7 mod 23 matches only 1/23 inputs) would imply Landscape Opacity if convergence is established.A sharp P vs NP-complete separation is verified: 2-SAT landscapes (α = 0.28) show no saturation, while 3-SAT landscapes exhibit full basin saturation for all n ≥ 200. Six NP-complete problems (3-SAT, 4-SAT, 5-SAT, Max-Cut, Graph 3-Coloring, Vertex Cover) all display the same exponential fragmentation.Epistemic status: Proved: Exponential number of minima (Thm 3.1), linear barrier (Thm 3.2), local search lower bound (Thm 4.1), and (LO) ⇒ P ≠ NP (Thm 5.2). Computational: Basin counts up to n = 50,000, RSB up to n = 10,000, circuit depth scaling, and P/NP-complete separation. Conjectural: RSB ⇒ (LO), U₂₄ rigidity, and Reeds endomorphism convergence. This work links the computational complexity of NP-complete problems to the glassy, fragmented structure of their energy landscapes and the universal constant Ω = 24.