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Gap Uniformizer

Every lawful wait, translated onto one ruler.

A five-draw wait for one game and a twelve-draw wait for another are not directly comparable. The geometric null transforms every completed wait to 0–1, where fair residuals fill the ruler evenly.

Geometric residual ruler

Powerball5/69 + 1/26 era · since Oct 2015600 draws

Completed gaps
2931
Uniformity D
0.015
adjusted p 0.660
Serial correlation
0.020
adjusted p 0.610
Whole-lab p
0.610
0–0.1274
0.1–0.2316
0.2–0.3325
0.3–0.4289
0.4–0.5273
0.5–0.6287
0.6–0.7297
0.7–0.8280
0.8–0.9286
0.9–1304

Largest per-ball residual shifts

64-2.55σ · 60 gaps
22+2.10σ · 30 gaps
10+2.01σ · 29 gaps
52-2.01σ · 50 gaps
23-2.00σ · 47 gaps
28-1.97σ · 51 gaps
41+1.74σ · 30 gaps
49+1.71σ · 35 gaps
27-1.70σ · 52 gaps
20+1.65σ · 34 gaps
13+1.50σ · 30 gaps
21-1.45σ · 53 gaps

Randomised geometric PIT + complete-draw simulationgeometric-gap-rpit-v1seed 20260827199 fair histories; two-test Bonferroni

Expected bars are flat. Complete-draw simulations preserve the dependence between balls chosen in the same draw.
Analysis ledger
Draw stream
main
Usable evidence
600 recent / 1418 draws
Trust filter
all canonical draws
Source omitted
none
Null law
geometric with p = 5/69
Correction
uniformity and serial-dependence readings adjusted together by Bonferroni
∑The maths, in plain language

Turning a geometric wait into a uniform residual

For one ball, a fair completed gap GGG is geometric with appearance probability p=k/Np=k/Np=k/N:

P(G=g)=(1−p)g−1p.P(G=g)=(1-p)^{g-1}p.P(G=g)=(1−p)g−1p.

Because GGG is discrete, its randomised probability-integral transform is

U=F(G−1)+V P(G),V∼Uniform⁡(0,1).U=F(G-1)+V\,P(G),\qquad V\sim\operatorname{Uniform}(0,1).U=F(G−1)+VP(G),V∼Uniform(0,1).

Under the null, UUU is uniform. The laboratory checks the pooled uniform distribution and lag-one residual correlation. It does not pretend residuals from different balls are independent: every null run simulates complete kkk-of-NNN draws, recomputes both statistics, and corrects their minimum p-value.

The final open wait is censored. For every gap that does finish, the transform therefore uses the geometric law conditional on completion before the recorded history ends. Long unfinished waits are not silently discarded from an untruncated reference law.

✦The story behind it

“Long” depends on the machine

A wait only feels meaningful relative to its opportunity to end. Uniformisation removes that changing scale: every correctly modelled wait becomes another point on the same ruler.

The ruler is a diagnostic, not a due-number engine. A residual near one says a completed wait was long under the model; the geometric law remains memoryless before the next draw. Discrete forecast residuals follow the model-checking approach described by Czado, Gneiting, and Held.