Foundations and sources

The 64 hexagrams correspond bijectively to the integers 0 to 63, with the bottom line as the most significant bit.

The reflected Gray code walks the 64 states changing a single line per step: a Hamiltonian cycle on the Q6 hypercube of 192 edges.

The binary ordering (Fu Xi) is traditionally associated with Shao Yong; the identity with binary counting was recognized by Leibniz, not by the culture that produced it.

Jing Fang's eight houses partition the 64 hexagrams without repeating any, generated by successive line changes from the eight pure hexagrams.

The system of the Eight Palaces (bagong) is attributed to Jing Fang, with the order Qian, Zhen, Kan, Gen, Kun, Xun, Li, Dui and the generations pure, first to fifth world, wandering soul (you hun) and returning soul (gui hun).

A consultation is an XOR of the moving lines; the original hexagram and the resulting one are exactly as far apart as the number of lines that changed (Hamming distance in Q6).

With coins P(yang) = 1/2 on every line; with yarrow the line probabilities are 1/16, 3/16, 5/16 and 7/16. Both methods give P(yang) = 1/2, but a different proportion of old lines.

The Klein group generated by the flip (fan) and the opposite (dui) splits the 64 into 20 orbits; the 8 palindromes are the special King Wen pairs; the nuclear map falls into 3 attractors.

A person's history of consultations is a path through the 64 states; the distance between two readings is the Hamming distance in the hypercube Q6.

The King Wen ordering groups the 64 into 32 pairs: 28 by reversal (fan) and 4 by complement (dui). Reversal preserves the number of yang lines; complement inverts it.

The 64 cells of the 8x8 square are exactly 0 to 63; the 8 geometric symmetries of the square are 8 trigram operations.

Shao Yong's diagram reached Europe through Bouvet: his letter is dated 4 November 1701, Leibniz received it on 1 April 1703 and published the Explication that same year.

The Mawangdui ordering arranges the 64 by upper trigram with a fixed rule for the lower trigrams; it is a scholarly reconstruction of the 1973 manuscript, not a direct datum.

As a permutation of Fu Xi, King Wen has 1013 inversions (order 260), Mawangdui 1008 (order 600) and Jing Fang 1008; the Hamming cost in lines is 211, 141 and 93.

The exact enumeration of the 4^3 = 64 cases of the ritual of the 49 yarrow stalks gives 3/16, 5/16, 7/16 and 1/16, identical to the table of the probabilities experiment.

Going from the Earlier Heaven bagua to the Later Heaven one is a permutation of two 4-cycles; in the Earlier arrangement the 4 axes join binary complements, in the Later one only the axis of Li and Kan.

The three projections of the hexeract (12-vertex Petrie polygon, cube of cubes, and yang levels) preserve the 192 edges of the hypercube.

The dominance order (submask) forms a lattice of 7 levels with the 192 edges oriented upward; from Kun to Qian there are 720 = 6! ascending paths.

The leaves of the yin and yang branching tree, from left to right, are the binary ordering 0 to 63; the path from the root to a leaf matches the lines of the hexagram bit by bit.

The tree reconstructs the genesis of the Earlier Heaven sequence attributed to Shao Yong.

The hu gua map collapses the images 64, 16 and 4 down to 3 attractors: two fixed points and a cycle of 2, with basins 16, 16 and 32.

Hu gua is linear over F2: there is a 6x6 matrix with ranks 6, 4 and 2, and M^4 = M^2, which explains the single cycle (Ji Ji with Wei Ji).

There is a ring of 64 bits whose 64 windows of 6 consecutive bits are the 64 hexagrams, each exactly once; there are 2^26 such rings.

The canonical construction used is that of Fredricksen and Maiorana: concatenating the Lyndon words in lexicographic order gives the minimal De Bruijn sequence.

With XOR the 64 form the group (Z/2)^6; the 8 pure hexagrams are a subgroup whose 8 cosets partition the set, and the dominance matrix is Pascal mod 2 (Lucas), the Sierpinski triangle.

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.

The 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.

Chan (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.

Complementarity 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 Markov chain of consultations has a uniform stationary distribution with coins and one skewed to yin with yarrow (Kun is 729 = 3^6 times more likely than Qian); both mix at the same speed, with a simple eigenvalue 1 and second modulus 0.5.

Three casting methods with their exact distributions: P(mutates) = 1/4 in all three; the equivalent modern method of 16 tokens (3, 7, 5, 1) reproduces the yarrow by construction, whereas the coins are different.

The 16-token method is named as an equivalent modern method; its authorship remains to be verified (section D of the evidence document).

The adjusted Rand index measures the similarity between two partitions of the 64; palaces and cosets give ARI = -0.125, further apart than chance, because the 8 generating masks do not form a subgroup.

The Walsh transform of King Wen concentrates its energy in order 2 (50.3%) and in the even orders 2 and 4 (77.4%), against the 47.6% of chance: its only structure is correlations between lines. The transform is Fourier analysis over the group (Z/2)^6: the matrix is the tensor product of six Hadamards (H tensor 6).

The counts of the cube: 46080 automorphisms (2^6 times 6!), 2^26 De Bruijn sequences, the distances C(6,k) = 1, 6, 15, 20, 15, 6, 1 and 720 chains from Kun to Qian.

The number of cyclic Gray codes of Q6 is cited to OEIS without reproducing its digits; Knuth is the general reference for Gray codes and De Bruijn sequences.

Figure cited, not reproduced.

The random walk on Q6 has a uniform stationary distribution; the expected return time to the origin is exactly 64 and covering the 64 states takes around 360 steps.

The stationary distribution of the walker's number of yang is exactly the binomial C(6,k)/64; with 200000 steps and a fixed seed the maximum deviation between the frequencies and the binomial stays below 0.003 (0.0008 with seed 99): the bell emerges from the throws.

The Chinese did not practise binary arithmetic: Shao Yong was after cosmology, and it was Leibniz, with his binary already invented, who recognized the structural identity.

Documented chronology: Bouvet communicated the analogy in a letter of 4 November 1701, received by Leibniz on 1 April 1703; the Explication de l'arithmetique binaire appeared in 1703.

The genetic code has 64 = 4^3 = 2^6 codons, like the hexagrams, but the correspondence is a combinatorial isomorphism, not a biological fact: it depends on one of the 24 base-to-bit encodings, none of them canonical.

Analogy with a disclaimer: the coincidence in number is real as form and arbitrary in the details; the standard code is cited in the experiment itself (NCBI), which is not part of this bibliography.

The 12 bi gua satisfy four properties: they form a closed Gray cycle, the changing line advances 2, 3, 4, 5, 6, 1, opposite months are dui complements, and they are exactly the 12 monotone hexagrams of Q6.

The 12 bi gua (xiaoxi gua) were associated with the 12 lunar months in the Han tradition, linked to Meng Xi and to the gua qi system.

The 12 sovereign hexagrams (the monotone ones of Q6) are exactly the first six generations of the palace of Qian and the first six of Kun (six and six; zero in the other six palaces): a bridge between the sovereigns and Jing Fang's palaces.

The qualitative observation is in Yijing Dao; here it is proved and asserted.

The hexagrams with no two consecutive yin (and, by symmetry, no two yang) are exactly 21 = F(8); counting by number of lines, the ladder is 2, 3, 5, 8, 13, 21 = F(n+2).

In the circular version (line 6 a neighbour of line 1) the survivors are 18 = L(6), the Lucas number.

The intersection of both rules (perfect alternation) is exactly Ji Ji and Wei Ji, and the breakdown of the 21 by number of yin is 1, 6, 10, 4 = C(7-k, k), the Fibonacci identity in Pascal’s triangle.

A counting theorem, not a hidden code: the numerology of Fibonacci and the golden ratio in the I Ching is left out as undemonstrable.

The same count generalises with a 2x2 transfer matrix: the no-two-yin rule is one case and its dominant eigenvalue is phi.

With the transfer matrix T (beta = 0.7, J = 1), the open-chain partition function is 1ᵀT⁵5 = 199.384322 and the ring one Tr(T⁶) = 262.456561; with beta at zero, Z = 64 and the distribution is uniform.

At low temperature with positive J, Qian and Kun dominate 50/50, and with negative J, Ji Ji and Wei Ji; the hard constraint (no adjacent yin-yin) reproduces F(8) on the chain and L(6) on the ring, with dominant eigenvalue phi, the same transfer matrix as the design-your-rule experiment.

The model, formulated by Lenz in 1920, was solved by Ising in 1925 in one dimension, proving that in 1D there is no phase transition: the chain orders gradually on cooling.

The connection with the I Ching is a mathematical identity of structure, not a physical claim about the oracle.

A uniform hexagram is exactly 6 bits (the maximum for 64 states); a line of coins has 1.8113 bits and a yarrow one 1.7490, and the difference of 0.0623 lives entirely in the movement, because the yin/yang value is 1 bit in both.

The stationary distribution of the yarrow chain has 4.8677 = 6 H(1/4) bits, against the 6 of the uniform one of the coins.

The optimal Huffman code of one line gives an expected length of 30/16 = 1.875 bits with coins and 29/16 = 1.8125 with yarrow: the ancient method has less entropy and also compresses better, satisfying H <= L < H+1.

Every adjacency rule between lines is a 2x2 matrix; its powers give the counts by number of lines, its trace the cyclic ones and its dominant eigenvalue the growth ratio (phi for the Fibonacci rules, 2 for the free one, 1 for alternation).

The balanced hexagrams where yang never falls behind yin are exactly C₃ = 5, the values 42, 44, 50, 52 and 56 (Catalan numbers): it is the z-transform of the count, not Laplace’s, because there is no continuous time.

The eigenvalues of the adjacency of Q6 are 6−2k with multiplicity C(6,k): the multiplicities are the yang levels of the lattice B6, and the spectrum of the simple walk is this one divided by 6, which fixes its mixing speed.

The DFT over Z/64 (the Shao Yong circle) decomposes the King Wen sequence into cyclic harmonics and satisfies Parseval; the DFT of a constant is a delta and the round trip recovers the signal.

The dominant harmonic is k = 8, the period of the eight trigrams; it is the same signal as the Walsh experiment in another geometry: the circle Z/64 instead of the cube (Z/2)^6.

For the no-two-adjacent-yin rule, the influences by line are 10, 22, 18, 18, 22, 10 out of 64 (largest at lines 2 and 5); for yang parity, the influence of every line is exactly 1.

The total influence of a property coincides with the weighted spectral sum of Walsh (the sum of popcount(w) times the coefficient squared): the theorem that links this experiment with Fourier over the cube.

The sphere-packing bound gives at most 9 words at distance 3 in Q6 (7 does not divide 64: there is no perfect code); the real maximum is 8, proved by exhaustive search, and a code of 8 corrects any single-line error.

The 32 hexagrams with an even number of yang lines and the 32 with an odd number form a bipartition: the 192 edges all cross from one half to the other, so Q6 is bipartite and there are no odd cycles of mutations.

Under rotation of the 6 lines there are exactly 14 necklaces (Pólya's formula = direct enumeration) and 13 bracelets once the reflection is added.

Sanskrit prosody counted the metres of short (1) and long (2) syllables, and the count satisfies C(n) = C(n-1) + C(n-2): 1, 2, 3, 5, 8, 13, 21, the Fibonacci numbers, formulated by Virahanka, Gopala and Hemachandra centuries before Fibonacci.

The 21 figures of 6 lines with no two adjacent yin lines are in bijection with the 21 metres of duration 7: a sentinel yang is added and each yin is paired with the yang above it as a long syllable. Two civilizations counted the same thing with different symbols.

In late 1950 Cage received the I Ching, built charts of 64 values indexed by the hexagrams and selected from them with coin tosses; Music of Changes (1951) is the founding work of composition by chance.

The method is documented, not the work: the demonstration uses a chart of our own and the coin engine of experiment 21, without reproducing any score or fragment by Cage.

The selection method (six coin tosses, P(yang) = 1/2 per line) chooses uniformly one of the 64 hexagrams, each with probability 1/64; our demonstration chart assigns a value to each hexagram, as Cage did with his charts.

The Haar basis is the other classical orthogonal basis of 64 points, sister of Walsh: its 64 rows are orthogonal (the Gram matrix is diagonal) though of different norms, the reconstruction is exact and Parseval holds with those norms.

Applied to the King Wen sequence, the Haar coefficients localise where the book changes; the largest live in the scale of the trigram blocks (the stretches 0 to 16 and 48 to 64) and are frozen as fixed values. It is the same signal as Walsh's and as the Fourier of the ring, on the line 0..63.

A hexagram is a state of the computational basis of 6 qubits, and the site's Walsh transform is the Hadamard gate H tensor 6 (asserted in the Fourier-over-the-cube experiment); applied to |Kun> = |000000> it gives the uniform superposition of the 64 hexagrams, with amplitude 1/8 and probability 1/64 each.

The normalized transformation is unitary (H·H^T = I): it preserves norms, as a quantum evolution must. It is a formal identity between transforms and mathematical states; nothing quantum is claimed about the oracle.

A mathematical identity, with a reinforced disclaimer: the commercial quantum I Ching is in the applicability register as rejected.

The faces of the 6-cube are the partial hexagrams (words over yin, yang and undetermined): the number of faces of dimension k is f_k = C(6,k) 2^(6-k), verified by formula and by direct enumeration of {0,1,*}^6; the f-vector is 64, 192, 240, 160, 60, 12, 1.

The faces add up to 3^6 = 729 (three options per line, the generating function (2+x)^6), the same number as the Kun/Qian ratio of the Markov chain; and the Euler characteristic is 0 on the boundary (a 5-sphere) and 1 on the whole solid (contractible).

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.

The 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.

The 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.

The 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).

The 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.

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.

No 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).

The 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.

The 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).

Radisic (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.

Historical 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.

Full 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).

The spectral signature of the five orderings (common signal position -> binary value, centred, different from the King Wen signal of Fourier over the cube): Fu Xi is a pure tone (100% of the Walsh energy in order 1) and Gray almost linear (75% order 1, 25% order 2), the calibration rods; Jing Fang is the anti-linear one, with zero order-1 energy and 74.5% in four-line interactions, the strangest signature, consistent with being the loner of the brotherhood.

The dominant DFT harmonic of King Wen is k = 16 (period 4), in resonance with the groups of four hexagrams that Chan (2026) detects by another route: two independent methods pointing to the same scale. The dominant harmonics of the five orderings are 1, 1, 24, 16 and 7; the dominant Haar waves have widths 64, 64, 4, 4 and 32.

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.

Independent 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.

The 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.

Both 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.