Organized Contingency: Structure and Across-History Contingency as Two Coordinates of Complexity

Complexity measures, from the López-Ruiz–Mancini–Calbet (LMC) measure to computational mechanics, are usually estimated from one realization, assuming ergodicity, and cannot tell whether an organization was inevitable or a frozen accident, a property of the across-history law (defined on that law, excess entropy and statistical complexity can). We propose reporting two coordinates, a structure degree δ_C and a contingency degree μ_path, the normalized variance across histories of which organization forms (Page's equilibrium dependence, not when the outcome was fixed or whether it stays fixed), summarized by their product Ω. No relabeling-invariant single-realization measure determines μ_path. On a two-knob reinforced urn the coordinates are weakly related (Spearman −0.29), Ω is nearly unrelated to the LMC measure (−0.06), and pairs indistinguishable on it have Ω of 0.000 against 0.991 and 0.014 against 0.496. Davey's incoherence rests on a similar pair, within-trial and pooled entropy, and tracks Ω on that urn (Pearson 0.95), not on a sign-free sweep (Spearman −0.04); ours is a variance-based version of his pair. Ensemble measures tell a fair lottery from a weak tilt (area under the ROC curve 0.934 to 0.935), relabeling-invariant single-realization measures barely (0.423 to 0.569), and the tilted color's share better (0.778; averaged over histories, 0.965). On trained networks the order parameter is usually a capability score, whose mean caps its seed variance: three capabilities of nine Pythia-160M seeds reach 0.0062 to 0.1087 of that cap, each μ_path 72 to 281 times its measurement null. In an exploratory reading, which of four adjacent layers hosts the leading induction head varies by seed (mean margin 0.053; categorical μ_path = 0.91, scalar depth 0.050). No real system here reaches the frozen-accident corner; capability scores cannot. The evidence comes from constructed families, a simulated spin glass, engineered networks and a descriptive reading of an artificial cultural market, not cells or codes.