The ladder of nulls
A hierarchy of alibis, and what is left to explain at each rung
Tap a rung of the map to see what that null grants and how the four King Wen signatures stand under it.
Saying that an ordering "is not chance" depends on which chance. Here we climb a ladder of conditional nulls, from the free shuffle to the most constrained one, and at each rung we re-evaluate the four known signatures of King Wen. Each null is an alibi granted: it fixes part of the structure and asks what is left to explain.
| transition | alternation | quartets | asymmetry | |
|---|---|---|---|---|
| 97.5 | 3.8 | 99.2 | 99.9 | |
| 29.4 | 6.2 | 89.8 | const. | |
| 31.9 | 6.5 | 87.7 | const. | |
| 33.2 | 10.8 | 90.4 | const. | |
| 29.9 | 3.3 | const. | const. | |
| 17.6 | 3.5 | const. | const. |
colour = how far the percentile departs from the centre (50) · marked cells are constant: the statistic does not vary under that null
Grants the pair rule: the 32 pairs are permuted as blocks and each is flipped or not. It is the null of the paper.
observed value: s1 3.3492 · s2 −0.2469 · s3 7 · s4 3.75
Canons and anchors add no explanation. Between P1 and P3 the percentiles barely move: once the pair rule is granted, also knowing that the canons do not mix, or that the four anchors are nailed down, does not change what is left to explain. The pair rule already carries almost everything.
Preserving the quartets preserves the yang sum of each block of four: permuting whole blocks and flipping within a pair does not change how many yang lines the block holds. Since in King Wen 7 of the 16 blocks add up to 12 yang, under P4 and P5 the signature equals 7 in EVERY sample. Its percentile of 0.0 is not an extreme signal: it is the arithmetic of a constant. The quartet signature is a corollary of quartet membership.
At P5, with the order of the pairs already fixed and only the flips drawn at random, the alternation sits at percentile 3.5, more extreme than the 6.2 of P1: the received orientations alternate more than orientations drawn by lot. This is the residue the ladder does not manage to explain, and it lives in the last degree of freedom left.
Both readings, as the house requires: read on its own, that residue is suggestive; read as the best outcome of a battery of signatures and rungs, it does not survive correction. And it connects with the question of the pair: the same degree of freedom that decides which hexagram opens each pair takes part in the anti-habituation of the distances.
A note on the anchors, which is a lesson about reversal and complement: the four are the pairs whose members are palindromes, which pair by complement out of necessity, because reversing them leaves them unchanged. The shortcut of looking for "a XOR b = 63" catches 8 pairs, not 4: it is also satisfied by four pairs in which reversal and complement coincide (11-12, 17-18, 53-54, 63-64), and those are reversal pairs. Nailing eight instead of four moves rung P3.
20,000 samples per rung, seed 20260722. Percentile = 100 × P(statistic < observed), the house convention. The definitions of the four signatures are the same as in the dialogue with Chan, reused rather than rewritten.
No seal: the originality search for this method is pending. The ladder is a candidate, not a finding.
The ladder of nulls re-evaluates Chan's four signatures under six nested randomizations (20,000 samples per rung, seed 20260722, percentile = 100 x P(statistic < observed)). Once the pair rule is granted, adding the canons or nailing the four anchors does not change what is left to explain: the mean transition distance stays between the 29th and 34th percentiles in P1, P2 and P3.
verificar_escaleraIndependent cross-verification with a different seed: the largest difference was 0.4 percentile points.
Micro-theorem of invariance: permuting blocks of two pairs and flipping within a pair preserves the yang sum of each block of four positions, so the signature of the groups with 12 yang equals 7 in EVERY sample of rungs P4 and P5. Its percentile of 0.0 is the arithmetic of a constant, not an extreme signal: the signature is a corollary of quartet membership.
verificar_escaleraThe residue at the top: with the order of the pairs fixed and only the 32 flips drawn at random (P5), the lag-1 autocorrelation of the distances sits at percentile 3.5, more extreme than the 6.2 of the pair null. The received orientations alternate more than orientations drawn by lot.
verificar_escaleraBoth readings: on its own it is suggestive; as the best outcome of a battery of signatures and rungs it does not survive correction. No seal is granted: the originality search for the method is pending.
The types of claim and the full bibliography (APA) are in Foundations.