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Fourier over the cube: the Walsh-Hadamard transform

Fourier analysis over the group (Z/2)⁶

Walk the energy by interaction order and see in which line interactions the structure lives.

Original finding

This is Fourier analysis, but over the group (Z/2)^6 instead of the circle: the 64 Walsh characters of the cube are its frequencies, each one a parity pattern over a subset of the 6 lines. The fast transform (the butterfly) is exactly the tensor product of 6 Hadamard matrices, H⊗6, and agrees bit for bit with the direct definition. Applied to the King Wen sequence (to each hexagram its chapter number), it shows where its energy concentrates. It complements is the King Wen ordering random?: that one says whether it is random as a whole; this one, in which frequencies it departs.

50%

energy in order 2 (pairs of lines)

energy in order 1 (linear)

Parseval ✓

Σ F² = 64 Σ f²

WHT² = 64f

involution verified

The complete spectrum

Where the structure lives: energy by order

order 1 (linear)

4.1%

order 2 (pairs of lines)

50.3%

order 3 (3 lines)

14.3%

order 4 (4 lines)

27.1%

order 5 (5 lines)

4.1%

order 6 (6 lines)

0.0%

The structure of King Wen lives above all in order 2: 50.3% of its energy (not counting the mean) is in interactions between pairs of lines, not in any single line (the linear order barely reaches 4.1%). Order 6 (all the lines at once) is almost null. It fits the randomness test: globally it looks like chance, but the little signal it has consists of correlations two by two.

Added together, the even orders 2 and 4 gather 77.4% of the energy, against the 48% that a random ordering would spread. It is the spectral confirmation, by an independent route, of what is the King Wen ordering random? found by Monte Carlo: under the inversion statistic, the dominant structure of the traditional ordering is its pair rule, which appears here as even-order interactions between lines. (The dialogue with Chan (2026), on that same page, qualifies that under other statistics a marginal signal also remains.)

The most marked subsets of lines

lines 4·5

F = −410

lines 2·3

F = −406

lines 1·3

F = −350

lines 4·6

F = −346

lines 2·3·4·5

F = 304

lines 1·6

F = −256

lines 1·2·5·6

F = −216

lines 1·2·4·6

F = −194

The characters with the most weight are almost all pairs of lines (4·5, 2·3, 1·3…): the concrete combinations where the traditional ordering leaves its mark.