Statistical Structure of the Historical Orderings of the I Ching Hexagrams: Pair Rule, Family Gradient, and the Limits of Demonstrability

The King Wen sequence orders the 64 hexagrams of the I Ching in an arrangement that has resisted explanation for three millennia. We present a statistical comparison of the three historical orderings (King Wen, Mawangdui, Jing Fang) against the binary ordering and against one another. Our central findings are the following. (1) All three historical orderings sit at the random-expected Kendall distance from the binary order (1013, 1008 and 1008 inversions, where the expectation is n(n-1)/4 = 1008): none is rank-correlated with binary counting, and two of the three achieve Kendall tau exactly zero. (2) Yet King Wen and Mawangdui are significantly correlated with each other (759 inversions, z = -2.89, Monte Carlo p = 0.0034, Bonferroni-surviving), and we identify the mechanism: a shared gradient of upper-trigram families, explicit in Mawangdui's octets and implicit in King Wen. (3) Re-analyzing recent Monte Carlo results (Chan, 2026) under a pair-preserving null shows three of the four reported signatures to be corollaries of the pair rule; we also report a small numerical correction, triangulated against a Lean-verified theorem (Radisic, 2026). (4) For the open question of within-pair orientation, the strongest admissible criterion is the classical doctrine of correct centers (12/16, uncorrected one-tailed p = 0.038; two-sided 0.077), and we show that no orientation rule of this strength can reach significance over 28 fixed pairs: a measurable limit of demonstrability. Every numerical claim in this paper is asserted by an automated test suite in a public repository.