Terence Tao

Two densities, and only one of them can see

Before 2019 the standing partial result on Collatz was Korec's: for any $\theta > \log 3/\log 4 \approx 0.7925$, the orbit of $N$ descends below $N^{\theta}$ for almost all $N$, in the sense of natural density. Note what that leaves. A bound that is still a power of $N$ says an orbit starting at a trillion need only come down to about ten billion. It is very far from saying it reaches 1.

Tao's 2019 paper replaces that with: for any function $f$ with $f(N) \to \infty$ — as slowly as you like, $\log\log\log N$ if you want — $\mathrm{Col}_{\min}(N) \le f(N)$ for almost all $N$, in the sense of logarithmic density. The bound improved from a power of $N$ to an arbitrarily slow function. And the measure changed. The obvious suspicion is that the second bought the first: that this is a weakening dressed as a strengthening.

So the two densities are not a strong claim and a weak one. They are different instruments, and Tao's contribution — the part that is method rather than technique — was recognising which instrument the statement wanted. The proof itself runs on an approximate transport property for a first-passage random variable and an estimate on the characteristic function of a skew random walk on a $3$-adic cyclic group, which is real machinery and not a trick. But the machinery only became applicable once the statement was rewritten in a measure that could hold it.

The sentence in Book 3, and the test it has never had

Book 3 · Ch.H opens with this:

That is a strong claim and it is the same claim Tao's paper demonstrates: a gap of exactly that shape, closed by changing the language rather than by finding new truth. Which makes his work the best available evidence for the sentence — and his own writing the sharpest warning attached to it. From There's more to mathematics than rigour and proofs:

Twenty-two million dictionaries, all machine-checked

In September 2024 Tao opened the Equational Theories Project: take every equational law of magmas expressible in at most four operations — there are 4,694 of them — and settle, for every ordered pair, whether the first implies the second. That is $4694 \times 4693 = 22{,}028{,}942$ questions, each one proved or refuted, each proof checked in Lean, by a crowd rather than a author.

Volume XIII, Chapter 9 asks what a dictionary between two theories transports, and answers that an interpretation carries “only what the dictionary was written to cover, and in one direction.” The Equational Theories Project is that question asked twenty-two million times and answered mechanically. A volume trying to work out what its own model carries has, sitting there, a completed worked atlas of the general case.

Place in the Series

The register for a Tao citation here is methodological. He is not evidence for dm³, does not work on it, and nothing in this series bears on anything of his. What his work supplies is a demonstration, at full rigour, of the move this corpus keeps reaching for and has not yet made: find the language in which the thing you can see becomes sayable. Every failure recorded in this series over the past week has been a translation failure and not a mathematical one — a band-width prediction that died on a sign convention, a control variable that smuggled the calendar back in, a constant that was an asymptote and read as a value. Tao's Collatz paper is what the successful version looks like, and it is worth knowing that the successful version is also just a change of measure, arrived at by someone who had first tried it the other way.

Grothendieck built the object the statement needed. Weil found the column of the analogy where the statement was already true. Tao changed the measure the statement was made in. Three ways to close a gap that is not a gap in truth, and this corpus has attempted a fourth — a change of geometry — without yet having written down what its dictionary transports. That is Chapter 9, and it is open.

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