Could this be chance?
A coincidence budget for a theory with no knobs to turn
Any theory that claims to hit a dozen measured numbers invites one reflex, and it is the right reflex: with enough formulas, something is bound to match. Numerology is exactly this — a wide net of arbitrary expressions, quietly keeping the ones that land. Before spending time on the physics, a scientist is right to ask whether this framework is anything more than that.
This page answers with arithmetic, not adjectives. It is deliberately built to be losable: the null hypothesis is “these matches are luck,” and we give that hypothesis every advantage we honestly can. The scorecard has the physics; this page has the accounting. You can rerun every number in it — audit/coincidence_budget.py.
The one principle that matters
The look-elsewhere penalty — the thing that turns a match into noise — attaches to free parameters you searched over, not to matches you got. This is the whole game, and it is worth being blunt about it:
- A model with 7 adjustable knobs that fits 7 numbers has explained nothing. It had exactly enough freedom to draw a line through the points. Its successes are guaranteed, so they carry no information.
- A model with zero knobs that fits 7 numbers has made 7 chances to be wrong and taken none of them. Each match is a coin it could have lost and didn’t.
This framework’s constraint summary is explicit that its spine carries no curve-fit knobs: the lattice size \xi, the geometry f=4\pi/(K\sqrt2), and the coupling \alpha_{mf}=\tan^2\theta_W are each fixed once, from a measured constant, and then reused everywhere with nothing new added. So the honest question is not “how many matches?” but “how improbable is each match, and how many formulas did it really take to find it?”
Both halves of that question have numbers. Here they are.
The coincidence budget
For each zero-parameter prediction we know the fractional error. Under the null (“this formula is just an arbitrary combination that happened to land near the data”) the chance of a match this good is the fraction of the plausible range that sits within that error of the target. For a dimensionless number we take the range to be a factor of e each way — i.e. “we already knew the answer was of order one to within about 2.7\times.” That is a generous prior: it makes coincidences look as likely as we can defend, so the odds below are a floor, not a ceiling.
| Prediction | Error | Odds against chance | Honest caveat |
|---|---|---|---|
| Packing fraction f=4\pi/(K\sqrt2) | 0.20\% | \sim\!510:1 | connects a Bessel-zero geometry to the cosmic dark-matter density — unrelated inputs |
| MOND scale a_0=c\sqrt{G\rho_\text{DM}} | 3.3\% | \sim\!700:1 | dimensionful; range taken as 20 decades of astrophysical accelerations |
| Koide relation Q=2/3 | 9 ppm | \sim\!10^5:1 | ⚠️ 2/3 is a natural centroid value — a skeptic can call it an attractor |
| Cosmic coincidence a_0/cH_0 | 0.7\% | \sim\!140:1 | ⚠️ a known GR combination — also a possible attractor |
The two flagged rows land on “special” numbers (2/3; a standard \sqrt{3\Omega/8\pi}). A determined skeptic can argue those values were reverse-engineered toward a pretty target, so we throw them out of the count below. We keep only the rows whose hit value is not special — where landing near the data buys you nothing aesthetic.
The number that can’t be fished
Strip the framework to a single fact and it is this. The lattice size \xi can be computed two completely separate ways:
\underbrace{\xi=\left(\frac{\hbar}{\rho_\text{DM}\,c}\right)^{1/4}=111.8\,\mu\text{m}}_{\text{cosmology — input is the dark-matter density}} \qquad \underbrace{\xi=\left(\frac{K\hbar\,\alpha_{mf}}{2m_e c}\right)^{1/3}=96.9\,\mu\text{m}}_{\text{electroweak — input is }\sin^2\theta_W,\ m_e}
These two routes share no dimensionful input except \hbar and c. One is fed a cosmological density measured from the cosmic microwave background; the other is fed the Weinberg angle and the electron mass from particle colliders. A length could a priori come out anywhere from the Planck length to the size of the visible universe — about 61 orders of magnitude. They land within 13\% of each other.
The chance of that agreement by luck is about 2\times10^{-3} — call it 560 : 1 — and that is the conservative reading at the raw 13\% level. Read through the packing fraction (which compares the fourth powers of the two lengths, so 0.2\% not 13\%), it is far sharper. You cannot get this from formula-fishing, because fishing works by adjusting an O(1) coefficient until a number matches — and here there is no number to match to until both independent routes have already been run. It is the framework predicting the dark-matter density from the electroweak scale, or the reverse, with no dial in between.
Now stack the deck against it
Take the two non-attractor spine rows (packing fraction, a_0) and the \xi convergence, and grant the null every advantage. Suppose — with no evidence, purely to be fair — that the author privately tried T different formulas of comparable simplicity for each row before one matched, and quietly discarded the failures. That multiplies each row’s odds down by T. Here is the combined “odds against chance” as the trials penalty grows:
| Formulas secretly tried per row | Combined odds against chance |
|---|---|
| 1 (no fishing) | \sim\!2\times10^{8}:1 |
| 3 | \sim\!8\times10^{6}:1 |
| 10 | \sim\!2\times10^{5}:1 |
| 30 | \sim\!7500:1 |
| 100 | \sim\!200:1 |
Even under the frankly absurd assumption that a hundred throwaway formulas were tried for every surviving one, the pure spine plus the convergence stay at a couple hundred to one against luck — and this is after discarding the two most eye-catching matches (Koide, the cosmic coincidence) for being too pretty to trust. That is the honest floor. The physics case is stronger; the accounting alone already clears numerology.
Where this argument stops — and what actually settles it
Honesty cuts both ways, so here is what the budget does not buy:
- The wider net does not count. Once you leave the spine — into chemistry, geology, biology — the framework is selecting which phenomena to point at, and that selection is an unbounded trials factor no one can score. Those chapters are suggestive pattern-matches, and the paper labels them as exactly that. None of them are in the numbers above.
- The trials count on the spine is bounded, not zero. We cannot prove how many formula-forms were tried per row; we can only show the result survives a punishing assumed penalty. That residual is real and stated openly.
- Retrodiction impresses; only prediction convinces. Every number on this page was measured before it was fit. That is worth less than one number measured after.
Which is why the framework’s weight ultimately rests not here but on its falsifiable, pre-registered predictions — the live rows that name a value no one has measured yet:
- Dark energy is transient (C=1): the DESI crust fit has already moved toward it; DR2 and beyond will confirm or kill it.
- A \sim\!300 keV \gamma-ray shoulder in terrestrial gamma flashes and solar flares, from an electron torn at the substrate’s own rotation speed.
- A refractive, chromatic signature at cosmic-web walls that mass-lensing cannot fake.
Any one of these coming back wrong does damage no coincidence budget can repair. That is the point: a theory earns belief by putting itself at risk, and the budget on this page only shows it has already survived the risk it took by having no knobs at all.
Every figure here comes out of audit/coincidence_budget.py, which reads the same stated inputs as the Tier-1 audit (\sin^2\theta_W=0.2312, \rho_\text{DM}=2.254\times10^{-27} kg/m³, j_{11}=3.8317) and CODATA-2018 constants. Change the null-model assumptions in the header and watch the odds move — the argument is meant to be poked at, not admired.