The question of the pair
What decides which hexagram comes first?
Choose a criterion and see who wins in each of the 28 King Wen pairs, with its count and its binomial p.
Original findingThe King Wen ordering groups the 64 into 32 pairs. Within each pair, what decides which one comes first? The literature declares it an open question. The laboratory answers in three acts, and the summary is honest: the binary structure does not decide it.
The first criterion anyone would try, the one with more yang first, is undecidable by construction: the 28 non-symmetric pairs are formed by reversal (fan), which inverts the order of the lines and therefore preserves the number of yang lines. The two members always tie: 28/28. There is nothing to decide.
the hexagram with the higher Fu Xi number comes first.
each cell is a pair (first · second); in colour, the winner of the criterion · grey = tie or not applicable
Every structural criterion gives high p-values: they are compatible with a coin toss. If the rule that orders each pair exists, it does not live in the binary structure; perhaps in the text, and that is declared outside the scope of the laboratory.
These 4 are the self-reversing hexagrams (reversal leaves them unchanged), so King Wen matches them by complement (dui), not by reversal. They stay outside the battery.
The book divides into two canons. The upper one is slightly sparser in yang (86 of 180 lines against 106 of 204), and the transition distances are nearly equal (3.38 and 3.33, with no appreciable difference). And a curiosity, with no test attached: the 8 self-reversing hexagrams fall at positions 1, 2, 27, 28, 29, 30, 61, 62, and the upper canon ends exactly after its third symmetric pair (29-30).
There is a published proposal that the nuclear hexagram (hu gua) decides the orientation of the pair. We put it to the test: placing the greater hu gua first is right in only 8 of 24 decidable pairs (16 toward the smaller one first), p = 0.076. It is the strongest signal in the whole battery, suggestive but not significant: the proposal is not confirmed. A companion fact: in 16 of the 28 pairs the two members collapse to different nuclei at the fourth level of the nuclear forest.
In January 2026, Radisic (2026) formalized the partition of the 32 King Wen pairs: 4 by complement (at distance 6), 4 anti-symmetric and 24 by reversal (at distance 2 or 4), with the corollary that the pairs respect the Klein orbits. Our suite already asserted exactly those partitions and orbits: two independent analyses that agree. (Historical context, without adopting the framework: McKenna studied the difference profile of King Wen in the 1980s within his Timewave, which the laboratory classifies as outside the criterion.)
One last cut, the finest one. Reversal pairs mirror lines: (1, 6), (3, 4) and the central ones (2, 5). For each component, does the member with the correct configuration come first (dangwei: yang in the odd position, yin in the even one)? The signal lives only in the centers:
The centers (2, 5) decide 12 of 16 in favour of correct centrality (zhong zheng: yang in the ruler's place, line 5; yin in the minister's, line 2), the most canonical doctrine the tradition offers. And it is not a loose criterion: it is the same rule as the nuclear proposal (the hu gua is built from lines 2 through 5): on the 16 pairs decidable by both, they agree 16/16.
Here full statistical honesty is due, with its two readings. As a single a-priori hypothesis (the most canonical doctrine, chosen before looking): p = 0.038, suggestive. As the best outcome of a battery of 9 criteria: it does not survive the correction (0.038 × 9 ≈ 0.35). Both are true; which one weighs more depends on whether you believe the doctrine came before the data.
And the structural limit, said plainly: the 28 pairs are fixed forever. A rule that decides 12 of 16 can never be demonstrated or refuted with more data, because there is no more data. The question of the pair is answered up to the limit of demonstrability: if there is a rule, it is correct centrality, and it lives exactly at the edge where statistics stops being able to decide. This experiment measures where the demonstrable ends.
In the 28 non-palindromic King Wen pairs, reversal (fan) preserves the number of yang lines, so the criterion of putting the member with more yang first is undecidable by construction: 28/28 ties. A micro-theorem of impossibility.
verificar_pregunta_parNo structural criterion tested decides the order within the pair better than a coin: greater binary value first 14/28 (p = 0.575), bottom line yang first 16/28 (p = 0.286), top line yang first 12/28 (p = 0.286), and the smoothing of the transition 7/15 on the incoming side and 7/15 on the outgoing side (p = 0.500).
verificar_pregunta_parThe text of each hexagram is declared outside the scope of the laboratory: only binary structural criteria are tested here. If the rule exists, it does not live in the binary structure.
The two canons of King Wen: 86 yang lines out of the 180 in the upper canon (hexagrams 1 to 30) against 106 out of the 204 in the lower one; the 8 self-reversing hexagrams (which pair by complement, not by reversal) fall in positions 1, 2, 27, 28, 29, 30, 61, 62, and the upper canon ends after its third symmetric pair.
verificar_pregunta_parThe published proposal that the nuclear hexagram decides the orientation of the pair is not confirmed: putting the greater hu gua first gives 8 of 24 decidable pairs (16/24 toward the smaller one first, p = 0.076), the strongest signal of the battery but not significant. A companion fact: in 16 of the 28 pairs the two members collapse to different nuclei at the fourth level of the nuclear forest (iterated hu gua).
verificar_pregunta_parRadisic (2026) formalizes the partition of the 32 King Wen pairs (4 by complement at distance 6, 4 anti-symmetric, 24 by reversal at distance 2 or 4) and their respect for the Klein orbits; our suite asserts that same partition independently (verificar_simetrias, verificar_rey_wen, verificar_pregunta_par): two analyses that agree.
verificar_pregunta_par · source: Radisic, 2026Historical context named without adopting the framework: McKenna studied the difference profile of King Wen in the 1980s within his Timewave, which the applicability register classifies as outside the criterion.
The signal of the question of the pair lives only in the centers: of the three mirror components of the reversal, the extremes (1,6) decide 10/16 (p = 0.227) and the middles (3,4) 9/16 (p = 0.402), neither significant, but the centers (2,5) decide 12/16 (p = 0.038) in favour of correct centrality (zhong zheng: yang in line 5, the ruler's, yin in line 2, the minister's). It is the same rule as the nuclear proposal: on the 16 pairs decidable by both, centers and smaller hu gua agree 16/16.
verificar_centrosFull honesty: as a single a-priori hypothesis p = 0.038, but as the best of a battery of nine criteria it does not survive the Bonferroni correction (0.038 x 9 approx 0.35). With 28 pairs fixed forever, a rule that decides 12/16 can be neither demonstrated nor refuted: the limit of demonstrability. The zhong zheng / dangwei doctrine is named as tradition without an entry for now (pending in section D of the evidence document until it is checked whether nielsen2003 covers it).
Search 2026-07-30: the orientation within the pairs has an independent nuclear proposal (already discussed in the experiment), the traditional narrative explanation of the Xugua, and adjacent work on the pairing (Radisic 2026) and on distances (Chan 2026), but the statistical test of orientation criteria (the full battery and the signal of the correct centers, 12/16 with its double reading) does not appear published. The doctrine of correctness that grounds it is canonical (the Stanford Encyclopedia of Philosophy calls Ji Ji the best hexagram in yin-yang correspondence, and Ji Ji opens its pair).
The types of claim and the full bibliography (APA) are in Foundations.