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

Mega Millions5/70 + 1/24 era · since Apr 2025153 draws

single-ball band: 1.057
pair band after singles: 0.982
all-degree set kernel: 0.004

Largest single-ball loadings

17+1.91σ

18+1.91σ

40+1.91σ

42+1.91σ

49+1.91σ

28-1.86σ

67-1.86σ

30+1.59σ

43+1.59σ

56+1.59σ

63+1.59σ

29-1.55σ

Largest pair loadings after singles

13 + 52+5.58σ

25 + 52+4.64σ

7 + 40+4.25σ

16 + 21+4.25σ

5 + 30+4.17σ

31 + 63+4.17σ

2 + 42+3.93σ

10 + 24+3.93σ

16 + 40+3.93σ

22 + 42+3.93σ

34 + 48+3.93σ

34 + 59+3.93σ

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.