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

El Gordo de la Primitiva5/54 + 1/9 era · since Jan 2010300 draws

43 + 52+4.14σ

12 + 47+3.86σ

26 + 28+3.73σ

22 + 32+3.28σ

24 + 42+3.14σ

5 + 34+3.05σ

7 + 8+3.03σ

9 + 35+2.94σ

1 + 30+2.89σ

8 + 37+2.88σ

37 + 54+2.87σ

18 + 24+2.78σ

3 + 15+2.77σ

7 + 11+2.76σ

22 + 52+2.64σ

9 + 11+2.62σ

18 + 47+2.62σ

24 + 49+2.61σ

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.