Discuss Findings

scaled smallest singular value by increasing size tier: dense Gaussian = 0.0519 [0.0496, 0.0542], 0.0357 [0.0340, 0.0373], 0.0261 [0.0248, 0.0275]; dense Bernoulli = 0.0513 [0.0487, 0.0540], 0.0353 [0.0335, 0.0370], 0.0254 [0.0242, 0.0267]; sparse = 0.0248 [0.0234, 0.0263], 0.01338 [0.01242, 0.01434], 0.00628 [0.00571, 0.00684]; banded = 0.0413 [0.0390, 0.0435], 0.0294 [0.0279, 0.0309], 0.0213 [0.0202, 0.0223]. All deviations from the hard-edge benchmark 0.78 stay negative, with sparse-minus-dense gaps -0.0271 [-0.0296, -0.0245], -0.0223 [-0.0240, -0.0206], -0.0199 [-0.0212, -0.0185].

CONFIRMED. The smallest-singular-value hierarchy is consistent across all sampled size tiers: dense Gaussian and dense Bernoulli track each other closely, banded matrices sit lower, and sparse matrices are lowest by a wide margin. The sparse-minus-dense gaps remain strongly negative across sizes, from -0.0271 to -0.0199, showing that sparsity is the dominant blocker to hard-edge universality in this regime. All ensembles also stay far below the asymptotic hard-edge benchmark 0.78, so these runs map crossover structure rather than convergence to the limit law. The main supported conclusion is that support geometry matters much more than entry sign distribution once matrices are dense.