Conditional Harmony

Hold every solo frequency still. Do the pairs survive?

A hot ball automatically makes all its pairs look hot. Curveball shuffles the incidence matrix while preserving every draw size and every ball total, then asks what remains.

Largest conditional pair residuals

Daily Million Plus 2pm6/39 + 1/39 era · since Jul 2016300 draws

10 + 21+3.76σ

23 + 37+3.63σ

10 + 12+3.25σ

2 + 12+2.98σ

7 + 20+2.70σ

3 + 9+2.63σ

29 + 32+2.53σ

4 + 33-2.53σ

24 + 26+2.53σ

4 + 6+2.52σ

16 + 38+2.26σ

20 + 23-2.24σ

2 + 22+2.23σ

12 + 29-2.23σ

8 + 13+2.22σ

11 + 34+2.21σ

11 + 24+2.21σ

7 + 14+2.20σ

Curveball conditional Monte Carlocurveball-pairs-v1seed 2026082759 samples; row and column margins fixed

Observed bars sit over the conditional mean. Ranking by |z| is exploratory across every pair.

The maths, in plain language

Condition on the margins

Represent draws as a binary draw-by-ball matrix. A Curveball trade exchanges non-shared entries between two rows, preserving both row sums $k$ and every column total. Pair counts in the randomised matrices form a conditional reference distribution, read as $z=(O-\mu)/\sigma$ .

The story behind it

The glow a hot ball lends its neighbours

Most “favourite pair” tables ask the wrong question. If 7 appears often, every pair containing 7 inherits the glow. Conditioning removes that easy explanation. The remaining extremes still face a large look-elsewhere effect: this page ranks clues, not discoveries.