Johnson Spectrum

If the sequence bends, where does the bend live?

Fixed-size draws live on a mathematical slice, not in an ordinary grid. This spectrum separates single-ball imbalance, pair structure left after singles, and structure needing whole draws.

Three registered structural bands

Classic Lotto 476/47 era · since Apr 2010250 draws

single-ball band: 1.299
pair band after singles: 0.980
all-degree set kernel: 0.003

Largest single-ball loadings

2+3.43σ

33+3.24σ

21+2.48σ

9+1.91σ

45+1.91σ

46-1.88σ

39-1.50σ

47-1.31σ

37+1.15σ

1-1.12σ

17-1.12σ

18-1.12σ

Largest pair loadings after singles

8 + 21+3.06σ

10 + 22+2.99σ

15 + 20+2.99σ

11 + 38+2.92σ

6 + 47+2.86σ

14 + 19+2.72σ

28 + 29+2.72σ

29 + 41+2.72σ

1 + 3+2.65σ

25 + 41+2.65σ

36 + 41+2.65σ

16 + 30+2.59σ

Johnson-slice harmonic spectrumjohnson-spectrum-v3seed 2026082799 fair-machine histories; three-band Bonferroni

The maths, in plain language

Harmonics on fixed-size subsets

A $k$ -of- $N$ draw is a vertex of the Johnson graph: two vertices are close when their sets overlap. Degree 1 measures centred single-ball indicators. Degree 2 starts with pair indicators and removes the part already explained by their two single-ball coordinates. The all-degree band uses an overlap kernel.

Fresh fair histories calibrate all three statistics. If $p_1,p_2,p_\ast$ are their raw Monte Carlo p-values, each displayed adjusted value is .

The story behind it

A prism for pattern claims

“Something is unusual” is not yet an explanation. A heavy first band points toward marginal ball frequencies. A heavy second band says some pairing remains after those margins are removed. A kernel-only signal suggests structure that neither summary captures alone.

The separation makes follow-up work narrower and more falsifiable. It does not make an unusual band predictive.