π's digits never end and never repeat, so the digits of every physical constant show up somewhere in them. The real question is where they show up. If the speed of light, Planck's constant and the fine-structure constant sit where pure chance puts them, π is noise. If they don't, there is a signal.
If π is a normal number, every k-digit sequence first appears around position 10k. That expectation is the yardstick. The hypothesis asks whether the constants of nature break it.
The digit sequences of fundamental physical constants appear in π at positions that deviate significantly from the normality prediction: earlier, later, or more clustered than random sequences of the same length.
Their positions can't be told apart from those of randomly generated sequences of the same length. π behaves like noise, and the constants are just more strings in it.
"π is like an infinite text, and the physical constants are known words in it. Does the text have meaning, or is it pure noise?"
The linguistic framing is only a heuristic. The paper makes no metaphysical claim: positions in π are deterministic facts, the baseline is objective, and the test is specified in advance. That is what separates it from numerology.
Type a birthday, a phone number or a constant. Everything runs in your browser on 1,000,000 decimal digits of π (position 1 is the first digit after "3.").
Loading 1,000,000 digits of π…
These are the twelve sequences from the paper. Each is searched live in the first million digits and compared with its expected first position, about 10k for a k-digit sequence. A ratio well below 1 would be an early arrival.
| Constant | Digits | k | Position (1M digits) | Expected ~10k | Observed ÷ expected |
|---|---|---|---|---|---|
| Computing… | |||||
The paper searched only 10,000 digits, and found none of the twelve, as expected. With a million digits the two 6-digit constants now appear, both near their expected position. That fits H0. The 7- to 9-digit constants need a far bigger search.
Generate random sequences and search for each one in the same million digits. The share found is what "random" looks like at each length. The blue marker is the theoretical value for a normal number, 1 − e−N/10k.
Most of the constants have 9 digits. You'd expect to wait about 109 digits to see one, so nothing meaningful can be said below one billion. The bars are on a log scale.
Run all twelve constants against a public billion-digit dataset. This is the primary test, and it takes about a gigabyte and minutes of CPU time.
Add the Hubble constant, the cosmological constant, the proton-to-electron mass ratio, Feigenbaum, Apéry and Euler–Mascheroni.
Do the constants form a "neighborhood" in π, lying closer together than chance predicts? That would test a stronger form of the hypothesis.
Repeat the analysis in e, φ and √2. An anomaly found only in π would strengthen the hypothesis. One found everywhere would point to another explanation.
Decision rule: compare each observed first position with the Monte Carlo distribution for its length, with a Bonferroni correction for multiple comparisons. Failing to reject H0 is as informative as rejecting it.
A Statistical Hypothesis on the Non-Random Distribution of Physical Constants within the Decimal Expansion of π
We propose a falsifiable statistical hypothesis: that fundamental physical constants, represented as digit sequences, appear within the decimal expansion of π at positions that are statistically anomalous relative to a random baseline. The hypothesis draws on Mathematical Platonism and Tegmark's Mathematical Universe Hypothesis. We search the first 10,000 digits of π for twelve physical and mathematical constants, establish a Monte Carlo baseline, and show that about one billion digits are needed to test 9-digit sequences with statistical confidence. We make no metaphysical claims. The hypothesis is strictly statistical and fully falsifiable.
From Pythagoras ("number is the substance of all things") through Galileo and Wigner to Tegmark, a long tradition has identified mathematics with physical reality. This project turns that intuition into one question that can be measured.