Discuss Findings

from 2275 numeric sanitized payloads, the large-N edge is uniformly far from Tracy-Widom: for N=512 the L-skewness stays near zero for every c in {2,4,8,16,32,64}, with c=64 equal to 0.00021 [-0.00098, 0.00140] and absolute deviation from Tracy-Widom skew 0.29367 [0.29248, 0.29487]; the rescaled edge mean at the same point is -128.03295 [-128.13899, -127.92692], and remains near -128 across the full N=512 sparsity sweep; smaller matrices are less extreme in mean shift but still non-TW in shape, for example N=256,c=64 has L-skewness 0.03097 [0.03015, 0.03178] and N=64,c=64 has 0.04378 [0.04318, 0.04437], both still about 0.33 away from the Tracy-Widom target.

CONFIRMED The edge statistics in this sparse-Wigner pipeline do not approach the Tracy-Widom skew profile anywhere in the sampled sparsity window, and the deviation is strongest and most stable at N=512. The scientific point is not merely a noisy failure of one condition: the whole large-N branch sits in a near-symmetric, strongly shifted edge regime rather than the expected TW scaling window. That is consistent with a mechanism where sparsity-induced localization or finite-connectivity structure dominates the edge fluctuations before universal dense-matrix behavior can emerge. In practical terms, the leading eigenvalue is being set by sparse-graph structure, not by the classical collective edge fluctuations of dense Wigner theory.