The family tree of the orderings
Who resembles whom, measured in inversions
Walk the edges of the tree and compare the inversions of each pair of orderings with the distance chance would give.
Original findingThe experiment on King Wen as a permutation measured that the traditional ordering has 1013 inversions against the binary one (Fu Xi), nearly tied with Mawangdui and Jing Fang (1008). Here is what that really means: between two random orderings of 64 elements, the expected number of inversions is exactly n(n−1)/4 = 1008, with standard deviation 86.3. The three historical orderings fall right on that expectation: against the binary ordering, they sit at the distance chance would give, uncorrelated with it.
dotted lines: each ordering sits at the distance chance would give (≈1008) from the binary one · solid lines: inversions between the historical orderings
Is the kinship inherited through the pairs? No. The 32 King Wen pairs end up completely separated in Mawangdui: none is adjacent, none falls in the same octet, and the mean distance between the members of a pair (24.4 positions) is greater than what chance would give (21.7). The reason is structural: reversal changes the upper trigram almost always, and Mawangdui organizes precisely by upper trigram, so it separates the pairs by construction. The kinship is real, but its mechanism is not the pairs: it is something else, resolved below.
Where does the kinship come from, then? We decompose the 759 inversions according to whether the pair of hexagrams falls within a single Mawangdui octet (the same upper-trigram family) or between octets. The deficit against chance lives between families:
mean King Wen position of each family (Mawangdui octet)
| Qian | father | 18.0 |
| Gen | youngest son | 26.6 |
| Kan | middle son | 31.9 |
| Zhen | eldest son | 43.0 |
| Kun | mother | 20.0 |
| Dui | youngest daughter | 39.8 |
| Li | middle daughter | 38.5 |
| Xun | eldest daughter | 42.3 |
The order is familial: father and sons (Qian 18.0 · Gen 26.6 · Kan 31.9 · Zhen 43.0) rise monotonically, and the mother early (Kun 20.0) with the daughters late (38 to 42). Mawangdui makes it explicit through its octets (its manuscript, c. 168 BCE, organized by upper trigram, is edited in Shaughnessy, 2014); King Wen carries it as a statistical tendency. The family doctrine of the trigrams is documented Han tradition (Nielsen, 2003); the gradient is our own computation.
Resolved: the kinship between the two oldest orderings is, almost entirely, a shared gradient of upper-trigram families. It is neither copying nor chance: it is the same cosmological doctrine, explicit in Mawangdui and implicit in King Wen.
Among themselves, the three are not equidistant. Only the King Wen to Mawangdui pair (759 inversions, z = −2.89) departs from chance: its Monte Carlo p is 0.0034, which survives the Bonferroni correction for the 3 comparisons. The other two pairs (King Wen to Jing Fang, z = −1.15; Mawangdui to Jing Fang, z = −1.58) do not. The reading: none of the three resembles the binary ordering, but the two oldest resemble each other, and Jing Fang is the solitary one. Computed with 20,000 random orderings and a fixed seed, verified by the suite.
Between two random orderings of 64 elements, the expected number of inversions is exactly n(n-1)/4 = 1008, with standard deviation sqrt(n(n-1)(2n+5)/72) = 86.3 (verified against Monte Carlo). The three historical orderings measure 1013, 1008 and 1008 inversions against Fu Xi: they sit at the distance chance would give from the binary ordering, uncorrelated with it.
verificar_hermandadThe inversions between the three historical orderings are 759 (King Wen to Mawangdui, z = -2.89), 909 (King Wen to Jing Fang, z = -1.15) and 872 (Mawangdui to Jing Fang, z = -1.58). Only the King Wen to Mawangdui pair departs from chance (Monte Carlo p = 0.003, surviving the Bonferroni correction for 3 comparisons): the two oldest orderings resemble each other, and Jing Fang is the solitary one.
verificar_hermandadThe mechanism of the kinship, in the negative: the 32 King Wen pairs end up 0/32 adjacent in Mawangdui and 0/32 within the same octet, with a mean position distance of 24.4 (chance 21.7). Reversal changes the upper trigram almost always, so Mawangdui's organization by octets separates the pairs by construction; the kinship is not inherited through the pairs, its origin is elsewhere.
verificar_hermandadThe mechanism, resolved: decomposing the 759 inversions between King Wen and Mawangdui, within octets there are 95 discordant pairs out of 224 (expected 112) and between octets 664 out of 1792 (expected 896): the deficit lives between upper-trigram families. The mean King Wen position of each octet follows the family order (father and sons monotone: Qian 18, Gen 27, Kan 32, Zhen 43; mother 20 early, daughters late). Conditional nulls: fixing the families, the residual is not significant (z = -1.5).
verificar_mecanismoThe family doctrine of the trigrams (Qian the father and Zhen, Kan, Gen the sons; Kun the mother and Xun, Li, Dui the daughters) is documented Han tradition; the family gradient that explains the kinship between King Wen and Mawangdui is our own computation, not the doctrine.
The Mawangdui ordering (c. 168 BCE), organized by upper trigram, is documented in the scholarly edition of the recently unearthed manuscripts (Shaughnessy, 2014); the standard view attributes no mathematical meaning to it, and the family gradient that explains its kinship with King Wen is our own computation, not a traditional reading.
Search 2026-07-27: there is comparative research on the Mawangdui sequence against King Wen (Han Zhongmin, in Chinese), analysis of King Wen against binary (Chan 2026; iching-math with cycle type [52, 10, 2]) and qualitative comparisons (Yijing Dao), but no permutation statistics BETWEEN the historical orderings themselves; the 759 of KW-Mawangdui (z = −2.89) and the full picture of the kinship do not appear published. Search 2026-07-30 (mechanism): the existing comparative KW-Mawangdui scholarship is qualitative (Han Zhongmin; Shaughnessy, Unearthing the Changes, 2014) and the standard view denies mathematical meaning to Mawangdui (Yijing Dao, citing Rutt); the quantitative explanation of the kinship as a shared gradient of trigram families does not appear published.
The types of claim and the full bibliography (APA) are in Foundations.