The Wave That Wasn't: Fitting a Calculus-Style Formula to the Powerball

A Late-Night Calculus Video

I was watching a YouTube video the other night called The Language of Calculus I Wish I Learned First. About halfway through, the host put up an equation that stopped me in my tracks:

√(1 + sin x) + √(1 - sin x) = √3

What hit me wasn't the algebra. It was the picture. The left side of that equation, plotted as a function, traces a deterministic wavy curve that crosses the horizontal line at √3 over and over again. Crests, troughs, crests, troughs — and a clean horizontal line cutting through the middle.

It's worth being a little careful here, because the equation is doing something subtle. The left side isn't equal to √3 as a constant — it's a genuine periodic function whose value bounces between √2 and 2 (this is easier to see if you square both sides: f²(x) = 2 + 2|cos x|). The "= √3" is the puzzle: at which values of x does the wavy curve cross that horizontal line? The "wave bouncing across a constant" picture is itself a story your eye is telling you about an average — √3 ≈ 1.73 is just where the curve spends most of its time. The constant isn't really a constant. It's the mean of the wave.

That's worth holding onto, because I'd seen that exact picture before.

The Shape on Our Homepage

Pull up the Recent Drawing Trends chart on Balliqa. The thin line is each draw's data; the bold line is a 20-draw rolling average. The bold line is almost flat — it sits near the long-run mean — but it has a gentle wiggle. A wave. Bouncing around a constant.

Same shape. Different domain.

I sat there for a minute and thought: if there's a closed-form formula for the calculus version, is there one for the lottery version? Could the rolling-average wiggle be expressed as a sum of well-defined sine waves, the way Fourier showed any periodic signal can be? And if so — could you extrapolate the formula forward? Could you predict?

Spoiler: nothing was there. But the way nothing was there is the actual story — and the journey to that nothing teaches the same lesson the calculus identity teaches once you look closely. Both pictures are stories your eye is telling about a constant that isn't really a constant.

Theories are cheap. The only thing that separates a real theory from a pretty one is whether you're willing to write down what would falsify it before you run the test. So before I built anything, I wrote down what I was actually claiming.

The Theory, Stated Properly

Here's what I was actually claiming, made falsifiable:

1. A formula exists. The Powerball rolling average could be expressed as a Fourier series — a sum of sine waves at specific frequencies — with a small handful of dominant terms standing out clearly above the background.
2. The formula is robust to smoothing. Whether I smoothed over 5 draws or 100 draws, the same dominant frequencies should keep showing up. If anything, wider smoothing should clarify the underlying signal because high-frequency noise gets killed.
3. The formula has predictive power. The fitted curve, evaluated at the next draw, would beat the dumbest possible prediction (always guess the mean, μ ≈ 13.5).

If even one of those held, the lottery would have a structural property worth investigating. If all three held, that would be extraordinary.

How We Tested It

The new experiment is live at /experiments/fourier-fit. Most of the pipeline is mechanical — take draws, smooth them, run an FFT, fit the top peaks. The only step that actually matters is the third one, so let me state it on its own:

A rolling average is a low-pass filter. Even pure random noise, run through a rolling average, produces a non-flat power spectrum with what look like peaks. The only honest test isn't whether real peaks exist — they always do. It's whether real peaks beat what noise alone produces at the same smoothing level.

That single idea is what separates this from a bad statistics paper. To pin down "what noise alone produces," we ran 200 Monte Carlo simulations: pure Gaussian noise of the same length and σ as real Powerball draws, fed through the same smoothing-and-FFT pipeline, with the loudest accidental sine wave recorded per trial. The 95th percentile of those random-noise maxima became the threshold any real peak would have to clear — corrected for the fact that we were testing hundreds of frequency bins simultaneously. We did this independently for each of five smoothing windows, because a wider window is a more aggressive low-pass filter and therefore demands its own threshold.

With that bar set, the rest of the pipeline is just bookkeeping:

1. The series. Every Powerball draw since the 2015 matrix change — 1,343 of them, each an integer 1–26.
2. The smoothing. Compute the rolling average at five window sizes: 5, 10, 20, 50, 100 draws.
3. The Fourier transform. At each window, decompose the de-meaned smoothed signal into a power spectrum — every sine wave from one full cycle across history down to the Nyquist limit (period of two draws).
4. The threshold. Apply the Monte Carlo noise floor described above.
5. The fit. Take the top 10 highest-power frequencies, write them as a Fourier series, plot it on top of the rolling average. See whether anything sticks out.

The Result

Zero significant peaks at any of the five smoothing windows.

Not at W=5. Not at W=10, or 20, or 50, or 100. Every window's "best" sine wave came in under the noise floor — the loudest Fourier component the real Powerball produced was no louder than what random noise alone produces at the same smoothing level.

The fitted formula at W=50, all ten harmonics included:

f̂(t) = 13.59
     + 0.54 · cos(2π·t / 143.8 − 1.22)
     + 0.44 · cos(2π·t / 1294.0 − 0.76)
     + 0.43 · cos(2π·t / 258.8 − 0.02)
     + 0.43 · cos(2π·t / 431.3 + 2.42)
     + 0.39 · cos(2π·t / 99.5  + 0.42)
     + 0.37 · cos(2π·t / 184.9 − 2.98)
     + 0.36 · cos(2π·t / 161.8 + 1.97)
     + 0.32 · cos(2π·t / 86.3  + 2.05)
     + 0.32 · cos(2π·t / 215.7 − 0.17)
     + 0.24 · cos(2π·t / 129.4 − 1.65)

Look at it. Every fingerprint of randomness is in there:

• Tiny amplitudes. The largest is 0.54 — half a Powerball value, on a 1–26 scale. None of them moves the prediction by even a single integer step.
• No clustering. A real periodic signal puts most of its power at one fundamental frequency and its harmonics. These ten periods (143.8, 1294, 258.8, 431.3, 99.5, 184.9, 161.8, 86.3, 215.7, 129.4 draws) are unrelated — they're just the top 10 of about 640 nearly-equal bins.
• Scrambled phases. The phase angles span the full range from −π to +π with no structure. A real signal phase-locks; noise doesn't.
• Slow amplitude drop-off. 0.54 → 0.44 → 0.43 → 0.43 → 0.39 → 0.37 → 0.36 → 0.32 → 0.32 → 0.24. A real signal has one big amplitude with a sharp fall-off after; this is a gentle slope characteristic of order statistics from a flat random spectrum.

What I Already Knew

Here's the part I want to be honest about: I knew before I built this what the result was going to be.

If the rolling average of Powerball draws contained a real Fourier signal, somebody would have found it forty years ago. The lottery would be broken. The matrix change in 2015 would have triggered a public reckoning. Instead, every randomness test ever run on the modern Powerball has come back empty — and we've already run a lot of them on this site.

But I like running these tests anyway. Two reasons:

The first is that "the lottery is random" stated as an opinion is worth almost nothing. Stated as the conclusion of a Fourier analysis at five smoothing windows, with a Monte Carlo null hypothesis correcting for multiple comparisons — that's a different kind of statement. It's a falsifiable claim that the data backed up. The fairness of the lottery isn't an article of faith on this site; it's a stack of experiments, each of which could have found something and didn't.

The second is that the moment of pattern recognition itself — looking at a calculus video, then a lottery chart, and going "wait, that's the same picture" — is worth taking seriously. Most of the time the pattern won't be real. But sometimes it will be, and the only way to know which is which is to do the test.

The wave on our chart isn't real. The wave in the calculus video is. The two pictures look the same because your eye is built to recognize wave shapes — including in places where the underlying generator is just smoothed random noise. The Fourier-fit experiment is, more than anything, a way of asking the chart: "are you a wave, or are you my brain?"

In this case, the chart was very clear about which one it was.

See It For Yourself

The full interactive experiment is at /experiments/fourier-fit. You can click through all five smoothing windows, drag the K slider to add or remove harmonics from the fit, change how many recent draws are displayed, and read the fitted formula update in real time. Every number on the page is recomputed from the live Powerball data — there's no canned result.

If a future draw, or a future format change, ever produces a Fourier peak above the noise floor, the page will show it. I'm not holding my breath. But that's the whole point of building it as a test, not as an opinion.