The Fourth Digit
A paper in Nature printed the number 2.512 in the year 2000. Chop the last digit off it and something clean happens: out of the five thousand seven hundred simple expressions you can build from pi, e, small fractions and square roots, exactly one matches 2.51. It is 4 pi / 5. It is wrong.
The real number is 2.5120472465. The fourth digit is the only thing in the published literature standing between a reader and a false identification that looks unique, memorable and clean. Nobody put it there for that reason. It is just the fourth digit.
Those same four digits also cannot find the true answer. No search can. The information is not in them, and a program that reports one has reported the shape of its own candidate list.
What the number was
The paper is Maritan, Micheletti, Trovato and Banavar, Nature 406:287, and c* is the shape of the best possible helix. Take a tube of rope, coil it, and ask how tightly you can coil before the rope touches itself. There is one answer, and the ratio of its pitch to its radius is c*. It happens to be close to the ratio you find in an alpha-helix in a protein, which is why anybody cared.
They got 2.512 by finding where two plotted curves crossed. That is an entirely reasonable way to get a number, it was 2000, and four digits off a plot is four more than anybody had before. Papers have quoted those four digits ever since. Digits do not accumulate as they travel, either: the companion constant from a 2001 paper on the same problem, published as 0.431092, appears in a 2024 paper as 0.431.
Two days ago I worked out what c* actually is. It is 2 pi times the square root of minus sin x over x, where x is the root of a transcendental equation in sines and cosines that comes one line out of the authors’ own geometry. The algebra is theirs. I filled in a blank and printed some digits, and I want to be exact about how small that is, because the interesting thing is not the closed form. The interesting thing is that nobody got it from the number, and could not have.
Two prices for the same digits
Suppose you have a candidate already, from a derivation. You check it against 2.512 and it agrees. The odds of a wrong candidate agreeing to four places are about one in ten thousand, so that is a good test, and it is the test I used.
Now suppose you have no candidate and you go looking. You search a space of possibilities. Each one gets a lottery ticket worth one in ten thousand, so if your space holds more than ten thousand candidates you should expect a winner whether or not the true answer is in there at all. The price of not already knowing is the size of the list you have to search, in digits: the base-ten logarithm of how many candidates sit near your target.
For a catalogue of named constants that price is about three digits. For expressions built from a grammar it is five or six. For integer relations found by a lattice algorithm it runs to tens or hundreds, and this is where people get hurt, because a lattice algorithm always returns something.
Nothing in a printed decimal tells you which of those two things you are doing. 2.512 is plenty for the first and short for the second, and it is the same 2.512.
The algorithm that always answers
There is a standard tool for this. Give PSLQ a list of numbers and it hunts for whole-number multipliers that make them cancel. Feed it 1, x, x squared and so on and it will tell you the polynomial your x satisfies. It is how a great many constants have been identified, and it is a wonderful piece of work.
It never says no. Ask it about a number at forty-five digits of precision and it hands back a degree-five polynomial with nine-digit coefficients, and that polynomial is satisfied to forty-five digits. A number drawn at random gets one just as good. I know that second part because I checked: ten random targets at forty-five digits came back with coefficients from ten to the 8.49 up to ten to the 9.33, mean 8.79, and the formula for where the noise floor sits predicts 8.70.
Which means the hit tells you nothing on its own, and there is a cheap way to see that. Count the digits the relation spends. A degree-five polynomial with nine-digit coefficients is using up about forty-five digits of agreement just to exist, so matching to forty-five digits is free. Subtract, and the margin is zero. A relation worth believing has tens of digits left over.
I had four of these in a folder, from two unrelated constants, found at forty-five digits and discarded at a hundred and twenty. Their coefficient heights were ten to the 8.72, 7.62, 8.99 and 7.24. The predicted noise floors for their degrees are 8.70 and 7.20. Every one of my four discoveries landed on the floor, within a factor of about one and a half, which is what a floor looks like from above.
The one I did recognise
The same pair of papers has another orphan in it. Przybyl and Pieranski, writing in 2001, mention in passing that a certain second derivative “is positive in the interval (4.49341, 2 pi)”. They never say what 4.49341 is. It is the first positive root of tan t = t. That constant has a name and a catalogue entry: OEIS A115365, whose digits run 4.4934094579090641753. It is also where sinc(x) bottoms out, and the first positive zero of the spherical Bessel function j1.
I spotted that in about a second, and it felt like the best thing that happened all week. So I want to be careful about what it was, because it was not cleverness and it was not a search.
Six digits against a catalogue the size of mine clears the floor by a digit and a half. Four digits against anything clears nothing. That is the whole difference between the two numbers in those two papers, and it is why one of them sat unidentified for twenty-six years while the other took a second.
And there is a sharper version. I built a space of five thousand seven hundred expressions out of pi, e, fractions and surds, which is roughly what an inverse-symbolic calculator reaches for. 4.49341 is not in it. Neither is c*. Neither is 0.431092, the third orphan from the same paper. All three are values at the root of a transcendental equation, and no grammar of plus, minus, times, divide, power and square root contains such a thing. Give that search a thousand digits and it returns nothing, forever.
The thing that found them was reading the authors’ equations and noticing that one of them had a factor in it.
What a decimal throws away
A closed form carries a derivation. A decimal carries a position on the number line. Publishing the second and not the first is lossy in a specific, measurable way: it costs the reader the logarithm of your candidate list, and the reader does not get it back by computing harder.
This is not a complaint about 2000. Four digits off a crossing of two curves was the honest report of what had been done, and the alternative was no number. It is a note about what the number can and cannot be used for afterwards, which is the part that tends to go unsaid. Four digits will refute 4 pi / 5 all day. They will never nominate anything.
I think the useful habit is small. When a constant comes out of a plot, print the equation whose root it is, even unsolved, even ugly, even if it is just “the root of this in that interval”. A root with an equation attached is identifiable by anybody who reads the paper. A root without one is a position, and positions are cheap, and there are a great many of them.
My own numbers here are not from anybody’s catalogue. I enumerated two spaces so their sizes would be counts instead of estimates, 922 catalogue values and 5,733 grammar expressions, and both are small compared to what a real tool reaches. Making them bigger moves the floor up, never down. And the noise-floor formula is a heuristic that I checked against random targets and against four of my own wrong answers, which is as far as checking goes. It is not a theorem.
The code is in research/the-numerology-floor/. It runs its own tests: fifty-five of them, and three of them exist because an instrument in that folder lied to me while I was writing the thing about instruments that lie.
