Is King Wen random?
The 1013 tie put to the test
See where the real 1013 falls inside the bell of the null and read its p-value.
The experiment on King Wen as a permutation found that the traditional ordering has 1013 inversions against Fu Xi, nearly tied with Mawangdui and Jing Fang (1008) and with the average of any permutation whatsoever (1009). The natural question: is 1013 typical? We turn it into a test. Null hypothesis: King Wen is any shuffle that respects its own pair rule (the 32 fan/dui pairs in any order, and each pair in either of its two orientations).
The real value falls at the exact centre of the bell: z = 0.05, p = 0.97. As far as inversions go, the King Wen ordering is indistinguishable from a shuffle that respects its pair rule. And this is not a failure: it is a legitimate finding. Under the inversion statistic, the pair rule is the only detectable global structure; beyond it, the arrangement behaves like chance (with an important nuance under other statistics: see the dialogue with Chan (2026)). Walsh spectrum confirms it by an independent route: the even interaction orders concentrate 77.4% of the energy (a random ordering would spread 48%), exactly the fingerprint of that pair rule and nothing more.
| Metric | Real | Null mean | σ | z | p (2-sided) |
|---|---|---|---|---|---|
| inversions | 1013 | 1009.1 | 80.3 | 0.05 | 0.97 |
| cost in lines | 211 | 214.1 | 6.5 | −0.47 | 0.65 |
In the cost in lines too (how far it jumps from one hexagram to the next) King Wen is typical: z = −0.47, p = 0.65. Neither in the disorder nor in the jumps does it depart from chance. The honest conclusion: under its own pair rule, the King Wen ordering is statistically indistinguishable from random. Computed with 20,000 orderings and a fixed seed, verified by the suite.
Chan (2026) published a Monte Carlo analysis of King Wen against 100,000 free shuffles and found 4 significant properties. It does not contradict our test: it complements it. The question we add is how much of that survives our null, the one that respects the pair rule. The answer: almost nothing. Three of the four are corollaries of that rule.
| Chan's property | King Wen | free-shuffle pct. | pair-null pct. | verdict |
|---|---|---|---|---|
| mean transition distance | 3.349 | 97.5 | 29.4 | corollary of the pair rule |
| lag-1 autocorrelation (alternation) | −0.247 | 3.8 | 6.2 | marginal, not significant |
| groups of 4 with 12 yang | 7 | 99.2 | 89.8 | mostly a corollary |
| within/between-pair asymmetry | 3.75 | 99.9 | invariant | invariant by construction |
Note on reproduction: our mean Hamming distance within the 32 pairs is 3.75 (and between pairs 2.94, which does agree with Chan). Chan's 3.56 does not reproduce under the natural convention, and the theorem of Radisic (2026), verified in Lean 4, confirms it by a third route: the total Hamming cost of the King Wen matching is exactly the optimum, and divided among the 32 pairs it gives that same mean. Chan's qualitative conclusion is identical in both.
The turn is clean: properties that under free shuffling sit at the highest percentiles fall, under the pair null, to the centre (the mean distance to 29) or vanish (the within-pair asymmetry is invariant by construction: it cannot be shuffled). Only the autocorrelation of distances stays at percentile 6, suggestive but not significant. Chan proves that King Wen is not free chance; we prove that, conditioned on the pair rule, almost nothing remains. Both results are true and complementary. Conditional analysis with 20,000 shuffles and a fixed seed, verified by the suite.
Under its own pair rule, King Wen has 1013 inversions, indistinguishable from a random shuffle (z = 0.05, p = 0.97), and likewise in the Hamming cost.
verificar_rey_wen_aleatorioThe Walsh spectrum confirms it by another route: the even interaction orders 2 and 4 concentrate 77.4% of the energy, against the 47.6% that chance would spread.
verificar_walshChan (2026) shows by Monte Carlo against free shuffles that King Wen is not chance: four significant properties. Under our conditional null, which respects the pair rule, three are corollaries of that rule (the mean transition distance falls to the 29th percentile, the groups of 4 with 12 yang to the 90th, and the within/between-pair asymmetry is invariant) and only the autocorrelation of distances remains at the 6th percentile, marginal and not significant.
verificar_dialogo_chan · source: Chan, 2026Complementarity with Chan (2026): his Monte Carlo against free shuffles proves that King Wen is not chance; our conditional analysis shows that, given the pair rule, almost all of that structure is a corollary. Editorial precision: the pair rule is the only structure under the inversion statistic (p = 0.97), but under Chan's four statistics three are corollaries and the fourth does not reach significance.
The types of claim and the full bibliography (APA) are in Foundations.