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

Cash Pop1/15 era · since Sep 2022250 draws

single-ball band: 1.129
pair band after singles: 0.000
all-degree set kernel: 0.003

Largest single-ball loadings

15+1.86σ

4-1.69σ

5+1.61σ

7-1.18σ

11-1.18σ

14+1.10σ

6-0.93σ

8+0.85σ

10+0.85σ

12+0.85σ

9-0.68σ

2-0.42σ

Largest pair loadings after singles

1 + 2+0.00σ

1 + 3+0.00σ

1 + 4+0.00σ

1 + 5+0.00σ

1 + 6+0.00σ

1 + 7+0.00σ

1 + 8+0.00σ

1 + 9+0.00σ

1 + 10+0.00σ

1 + 11+0.00σ

1 + 12+0.00σ

1 + 13+0.00σ

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.