In 2022 a Portuguese mathematician named Fabio Chalub published a paper solving the neutral Kimura equation called Solution of the Neutral Kimura equation with two integral constraints. It is not a critique of neutral theory. Chalub gives no sign that he is even aware that anyone is contesting neutral theory. He is a specialist in degenerate partial differential equations doing what specialists do, which is testing the assumption that a well-known equation actually has the solution everyone has assumed it has.
Chalub opens by pointing out something that seventy years of science textbooks have failed to notice. The classical solution to the neutral Kimura equation — the one Kimura wrote down in 1955, the one reproduced in Crow and Kimura and in every graduate course since — decays to zero. All of the probability drains out of the interior of the interval and nothing is left. As a description of a population, it is nonsense: the alleles have to go somewhere, which is either fixation or extinction. The standard solution does not track them. It simply stops accounting for them. To fix this, Chalub adds two point masses at the boundaries, one representing the fraction of populations in which the allele has fixed and one representing the fraction in which it has been lost. Fine. Now comes the interesting part.
Nothing in the Kimura equation determines how the escaping probability splits between those two boundaries. The equation degenerates at both endpoints — the diffusion term goes to zero there — and there is no condition inside the mathematics that says how much goes to fixation and how much goes to extinction. The equation upon which the entire molecular clock is built does not provide the one number that it needs. So Chalub supplies the number from outside, in the form of two constraints he imposes by hand before solving anything. The first is that total probability is conserved. The second is that the average allele frequency never changes, which is not a fact about diffusion but an inherited assumption from the Wright-Fisher model, the very assumption that has been successfully disputed by me and three scientists before me. Impose that assumption and the fixation probability comes out equal to the starting frequency. Of course it does. Chalub writes in the constraint before he solves anything, recovers it as a result two pages later, and he says plainly that the constraints are inherited from the discrete process.
He has no reason to be aware of what he has demonstrated. The diffusion machinery is invoked throughout the literature as independent confirmation that a neutral allele fixes with probability equal to its frequency. Chalub shows the machinery cannot confirm any such thing, because it was given the answer in advance. That is the circular reasoning. Chalub shows that the 1/(2N) is an assumption imported from the Wright-Fisher model. It is not a result produced by the mathematics. It was never tested against, or derived from, an actual population.
This changes the shape of the Neutral theory critique even further in my favor, not that I needed any more ammunition given the N/Ne switcheroo as well as Frankham’s empirical evidence that conclusively demonstrates the inapplicability of Kimura to sexually reproducing populations. But until now, the defense of Kimura’s cancellation still had one retreat available: the critic does not understand the diffusion approximation, the result is a theorem of the continuum limit, the mathematics is above his pay grade.
That retreat is now closed, and it was eliminated by a mathematician who was not even paying any attention to its manifold implications for evolution, neutral theory, or the molecular clock. The initial-frequency theorem is not a theorem of the diffusion. The diffusion cannot produce it. The diffusion has to be provided as an assumed condition that is supplied in order to solve the equation. Which means that when we say Kimura’s cancellation rests on an assumption about how populations reproduce rather than on the math, we are no longer making an assertion that requires the reader to side with us against the scientific consensus on the basis of the evidence and the logic. We are simply repeating an observation that has been independently established in the mathematical literature and can be confirmed by anyone with the math to do so.
Kimura’s assumption about populations were always the whole of his argument. Chalub proved this, published it, and no one in population genetics appears to have either read it or realized its implications.
And if you don’t grasp what this means, here is the short version: every molecular single clock has been eliminated. All of the scientific estimates about deep time are gone. Even my own attempts at recalibration are hopeless, irrelevant, and wrong. It’s not just the CHLCA estimate. It’s also the Cambrian “molecular dates predate the fossils” literature, which was used for two decades to argue the fossil record was systematically incomplete rather than that the dates were all systematically wrong. This includes the mammalian radiation and whether it preceded or followed the K-Pg. The out-of-Africa chronology. Every single eukaryotic node in TimeTree. And the dating of angiosperm origins that produced Darwin’s abominable mystery in its modern form.
All of these time frames were calculated with a rate constant that has no derivation and no non-circular calibration. And all of it is now worthless and mathematically invalid nonsense.
For those who still have no idea what I’m talking about, let me make it even more straightforward for you. And rest assured there isn’t an evolutionary biologist on the planet who can even begin to successfully argue against any of this.
Dear Biologists,
Evolution is mathematically impossible. Your historical time estimates are not only wrong, they cannot even be calculated on the basis of the information you have. And your most important equations from your best guys are mathematically nonsensical.
Love,
The Mathematicians
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