We ask whether a near-optimal configuration of a combinatorial problem resembles the global optimum, and what controls that resemblance. We measure the overlap between a fixed-quality state — the first configuration to reach 92% of the best-known objective — and the global optimum, across MAX-CUT, MAX-SAT, and QUBO instances. The overlap is governed by the degeneracy of the near-optimal manifold, measured directly as the typical pairwise overlap P(q) between independent high-quality solutions. In symmetric ±1 spin glasses (MAX-CUT) P(q) ≈ 0.53, near the random baseline, and the 92% state overlaps the optimum only ≈ 0.54: many near-optima, far apart, so high quality is uninformative about the optimal configuration. When random linear fields lift the degeneracy (QUBO) P(q) ≈ 0.91 and the overlap rises to ≈ 0.83; random 3-SAT is intermediate. Across 19 independent ensembles spanning three NP-hard classes, the fixed-quality overlap tracks the directly measured degeneracy with Pearson r = 0.98 (Spearman ρ = 0.92; bootstrap 95% CI [0.97, 0.996]), and the correlation persists within the spin-glass cluster alone (r = 0.92). The effect is independent of the search algorithm (iterated local search and simulated annealing agree), of the best-known normalization, and of the optimizer's own endpoint. Proximity-to-optimum at fixed quality is thus a property of solution-space structure, not of the optimizer. We present this as an empirical characterization; it is a retrospective measurement (it requires a reference optimum) and not a deployable tool. Code and data are included.