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The Irrelevance of k = μ 

Keruru has uncovered some interesting things in his review of the mathematics underlying Kimura’s cancellation, which I have shown is a) incorrect for most species and b) a Portuguese mathematician has shown to be mathematically incorrect:

In February I argued that the mathematics underwriting neutral theory was broken. The argument had two legs. The first, drawn from a preprint I had read approvingly, held that Kimura’s substitution-rate identity k = μ rests on an equivocation: that the effective population size N<sub>e</sub> is made to stand for two incompatible quantities — a mutation-supply term and a drift term — which are then cancelled against each other. The second was empirical: an ancient-DNA analysis reporting essentially zero allele fixations across a million loci in the 240 generations separating Early Neolithic Europe from the present.

Both legs have given way. I want to set out how, because the manner of their failure is more interesting than the original claim, and because the instinct behind the essay turns out to have been sound while its aim was off by about thirty degrees.

The equivocation that isn’t

Written as k = 2N_e μ × 1/(2N_e) = μ, the objection looks devastating. Mutation supply is a biochemical fact about replicating cells and scales with the number of individuals actually reproducing. Drift is a statistical fact about sampling variance and scales with something much smaller and considerably slipperier. Two different numbers wearing the same symbol, cancelled against each other, yielding a result that has calibrated half of molecular anthropology.

The trouble is that the derivation does not require N<sub>e</sub> in either position, and the textbook notation is simply careless.

Mutation supply is 2Nμ with N the census count — mutations occur in gametes, and every reproducing individual contributes gametes. The fixation probability of a single new neutral mutant is its initial frequency, and one new copy among 2N copies has frequency 1/(2N) — census again. The two census terms cancel. N<sub>e</sub> never enters, and so cannot be equivocated upon.

Why is the fixation probability equal to the initial frequency? Not by assumption. Under neutrality the expected change in allele frequency each generation is zero, which makes the frequency a bounded martingale; a bounded martingale that must eventually be absorbed at 0 or 1 has absorption probability at 1 equal to its starting value. That is optional stopping, and it is a theorem rather than a modelling convention.

The claim is testable without any diffusion approximation at all, using exact finite Markov chains. I built two. The first is ordinary Wright–Fisher. The second is a sweepstakes model: half the time, half the population is replaced wholesale by clones of a single randomly chosen parent — reproductive skew of the kind documented in Atlantic cod and Pacific oysters, wildly outside anything Wright and Fisher had in mind. Measured from heterozygosity decay, the sweepstakes chain has an effective size 5.6 times smaller than the Wright–Fisher chain at identical census size.

The fixation probability of a single new mutant in both chains: exactly 1/(2N_census), to fifteen decimal places.

N<sub>e</sub> can be dragged around by a factor of six and the fixation probability does not move. It follows that my own demographic work on the collapse in variance of reproductive success — a real finding, and one I still stand behind — does not touch the substitution rate. It changes coalescent depth, standing diversity, and every estimate of N<sub>e</sub> derived from them. It leaves k = μ exactly where it was.

First, the problem with Kimura turns out to be even more significant than I originally believed. I was pretty sure that Keruru had been led off track the moment that he mentioned the word “martingale” because that’s the retreat that every AI, including both Claude Athos and the entire Red Team, initially made. You cannot use AI to provide you the answer; you have to know at least the correct shape of the answer before you ask the AI.

I’ll post my full response to Keruru later this week. His work here is very far from valueless, as I will demonstrate, but it is nevertheless incorrect due to his failure to anchor the abstract mathematical elements in material reality. This also, rather fortuitously, demonstrates the importance of the Triveritas, as any one link in the triad is capable of serious error when allowed to operate outside the bounds of the other two.

But the epicycle wasn’t elsewhere. There are multiple epicyles.

k = μ is a correct baseline for asking whether something other than drift is happening. But it is not a rate at which anything can be observed to occur in a sexual, age-structured, demographically non-stationary population — which is every organism anyone cares about.

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The War on US Farmers

The utter idiocy of participating in Israel’s war on Iran is being driven home hard in the negative effects on America’s ability to feed itself:

US grain farmers have been plunged into their worst financial crisis in four decades as the war with Iran drives up diesel and fertilizer costs, the Financial Times reported on Tuesday. The latest price shock compounds years of weak crop prices and falling incomes.

The mounting pressure follows the US-Israeli attack on Iran in February, which prompted Tehran to largely block commercial shipping through the Strait of Hormuz, a key route that previously served about a fifth of the world’s energy supplies. The resulting surge in energy costs is hitting the US heartland ahead of November’s midterm elections, which will determine control of Congress.

Farmers across the Corn Belt have been particularly hard hit by soaring diesel and crop nutrient prices since the military campaign began, the FT reported. Nebraska Farmers Union president John Hansen described the downturn as the sector’s worst since the 1980s.

”Inputs are way out of whack,” Nebraska corn and soybean farmer Matt Bailey told the newspaper. He said a phosphorus-rich fertilizer used at planting now costs more than $900 per ton, compared with about $470 a decade ago. Diesel prices have also surged after the conflict disrupted energy markets and sharply reduced traffic through the Strait of Hormuz. The nationwide average has climbed to about $5.45 per gallon from $3.81 before the war, according to US Energy Information Administration data.

”The [Iran] war has made so much uncertainty for us as farmers,” Pam Johnson, a former president of the National Corn Growers Association, told the FT. “It’s projected that farmers aren’t going to make any money for the next two years.”

And this is before the financial crisis hits this fall. This is why the real America sought to avoid foreign entanglements and restricted immigration for decades after the 1920s; foreigners with influence always seek to use other nations’ more powerful militaries for their own purposes and on their own behalf. And this is almost always to the detriment of the people of those nations.

We all know the retarded rhetoric about how “X is crazy!” and “X cannot be permitted to obtain a nuclear weapon”. Not one single American is going to give a damn about the rhetoric when gasoline costs $10 per gallon and they can’t afford to feed their families.

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The NFL and CTE

It’s not looking great for ex-football players:

A new study from The BMJ finds that at least 24.5 percent of all former NFL players who died between 2016 and 2021 had CTE. Of the 878 who died during that period, at least 215 had CTE. Of the remaining 663, 643 of the players’ brains were not studied for the presence of CTE. Which means that the real percentage could be much higher.

It almost certainly is. A friend of mine married, and eventually divorced, a former college football player whose personality and behavior changed drastically over time. She suspects that he’s got CTE, but of course there is no way of knowing for sure until he dies. I wouldn’t be at all surprised if it turns out that more than half of the former NFL players had it.

It’s a violent game. And the players should be given every opportunity to know what they’re getting into.

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THE CRUMBLING ABYSS

High in the mountains above Kagoshima, an old man finds the remains of a forbidden ritual. A headless dog. Ritual kindling. The residue of a curse aimed at killing children.

So begins Naoki Sanjūgo’s Nangoku Taiheiki, the sprawling historical epic that made its author’s name synonymous with popular fiction in Japan. First serialized in 1930–31, the novel plunges into the Oyura Disturbance, the real succession crisis that nearly tore apart the Shimazu clan of Satsuma in the years before the Meiji Restoration. On one side, Lord Narioki and his concubine Oyura, wielding sorcery, poison, and political cunning to install their chosen heir. On the other, Nariakira, the reformist eldest son whose vision of a modern, industrialized Japan threatens everything the old guard has built.

Caught between them: Kotarō, a young samurai hunting the sorcerer who murdered his family. Masumitsu, a swaggering political operative who fights with a pipe, a pistol, and an inexhaustible supply of nerve. Nangyoku, a comic storyteller whose street performances mask a dangerous loyalty. Miyuki, a woman sent into enemy territory with a dagger and a mission she did not choose. And behind them all, the historical figures who would soon remake Japan: Saigō Takamori, Ōkubo Toshimichi, Shimazu Nariakira himself.

Naoki braids occult horror, swordfight choreography, Edo backstreet comedy, political intrigue, and doomed romance into a narrative that moves with the propulsive energy of serialized fiction and the emotional weight of serious literature. Volume One, The Crumbling Abyss, follows the conspiracy from its discovery through assassination attempts, factional warfare, desperate journeys, and the catastrophic death that changes everything.

The Naoki Prize, Japan’s most prestigious award for popular fiction, is named after the author. An original English translation by Vox Day, The Chronicle of the Southern Kingdom: The Crumbling Abyss is now available on Amazon for Kindle, KU, and audiobook. An excerpt is available at Castalia Library.

NB: This is an excellent book and I am tremendously annoyed with myself for failing to release it when I finished the translation several months ago. A small press recently released the first English translation in July, so while this is an original translation, it is not the first English translation.

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Foreign Affairs Throws in the Towel

Clown World is beginning to grasp that the US military can’t even defeat a second-rate power anymore.

For 47 years, the Islamic Republic has bedeviled U.S. presidents. The 1979 Iran hostage crisis helped doom Jimmy Carter’s reelection bid. The Iran-contra scandal in the mid-1980s badly tarnished Ronald Reagan’s foreign policy record. And Barack Obama’s 2015 nuclear deal with Iran, the Joint Comprehensive Plan of Action (JCPOA), led to an unprecedented situation in which the U.S. Congress invited a foreign leader, Israeli Prime Minister Benjamin Netanyahu, to Washington to argue against it.

In February, when he launched his war against Iran, President Donald Trump declared that he intended to break this embarrassing pattern once and for all. By crushing Iran militarily and effecting regime change, he boasted that whereas other presidents had only talked about the Iranian threat, he was taking decisive action to end it. But six months later, Trump has become yet another U.S. president facing a dangerous humiliation related to Iran; the fallout from his excursion threatens his party’s prospects in the midterm elections. Now, perhaps in desperation, Trump is attempting to squeeze an economic noose around Tehran. But as it is constructed, this economic-pressure operation poses a greater threat to U.S. credibility than it does to Iran’s economy.

The irony of Trump’s failed war on Iran—and his stop-start negotiations to try to end it—is that it has so severely weakened Washington’s ability to exert pressure on Tehran that no future administration will ever be able to revert to the traditional U.S. playbook. But that could also be the only upside of a war that has already exacted a heavy toll on U.S. munitions and the global economy. The war could—and should—force a broader reckoning with 47 years of failed strategy and change Washington’s approach for good.

The combination of strategic 5GW and the ability of Iran to control the Strait of Hormuz has nullified both the USA’s military and financial power. This, combined with the fact that China now surpassed US economic and industrial might, and Russia now exceeds the USA’s ability to project power into Europe, means that the post-WWII era is officially over. What comes next remains to be seen, but Americans should look to post-WWII Britain to better understand what their domestic future likely holds.

And the sooner Trump, or whoever is pulling the short, fake Trump’s strings, understands that continued conflict will only make things worse, the better. The financial system already looks as if it is on the verge of breaking this fall; October could be very ugly beyond just the financial markets. Getting the oil flowing and getting gasoline and diesel prices down will at least help somewhat cushion the economic blow.

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Not NATO: Middle East Edition

Larry Johnson correctly observes that the new military alliance in the Middle East is not a US-aligned one:

The Western commentariat filed the Mecca agreement under a comfortable heading. When Saudi Arabia, Turkey, and Pakistan signed the Makkah Joint Defence Agreement on August 7 — an armed attack on one to be treated as an attack on all — the reflexive label was “Islamic NATO,” a Sunni bloc, three of the Muslim world’s largest militaries welded together by sectarian identity. That reading was obsolete almost before the ink dried. What is actually taking shape is not a Sunni alliance against Iran. It is the first load-bearing beam of the post-American security architecture that Russia and China have been calling for — and Tehran is about to walk through the door…

The Mecca agreement was sold to Western audiences as a Sunni bloc and dismissed as a club of American clients. It is neither. It is the first pillar of the security architecture Putin and Wang Yi named this spring after sitting with Iran’s foreign minister — a regionally led, Russia- and China-backed framework that folds Iran in as a legitimate power and pushes the United States out as the indispensable one. The Americans built their Gulf position on keeping Sunni and Shia apart and keeping the dollar in the middle. Both pillars are being pulled at once — and the structure going up in their place was drawn in Moscow and Beijing.

I was very confused when I saw the mainstream media covering the Mecca agreement as something that was good for Clown World. It seemed very obvious to me that it wasn’t an anti-Iran association, since Pakistan is one of Iran’s closest allies, but rather, a post-US security arrangement.

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Fact Check

Pictures of former US presidents Barack Obama, Donald Trump, Bill Clinton and George W. Bush touching the Western Wall in Jerusalem were misrepresented online as taken in the Great Wall of China. Posts shared globally falsely claimed the pictures showed the former leaders “swearing allegiance” to the Chinese Communist Party

It’s good the fact-checkers got this sorted out. Obviously, US presidents swear allegiance to Israel, not to the Chinese Communist Party. We wouldn’t want Americans to think their leaders are working for the wrong foreign country!

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The Manufacture of Alzheimer’s

Pre-Covid, it was very hard to believe that the entire medical establishment was actually trying to kill people. Now that 75 percent of the population has been injected with heart attack and turbo cancer machines due to a man-made disease that never threatened the vast majority of the population, people are starting to look a little harder at other serious medical issues:

DOCTORS ARE LITERALLY MANUFACTURING ALZHEIMER’S

Dr. Joel Wallach: “Alzheimer’s is a PHYSICIAN-CAUSED disease.”

Your brain is 75% myelin insulation. And myelin is 100% cholesterol.

Statins shred that cholesterol — stealing 75% of what your brain desperately needs to stay insulated and functional.

No myelin = no brain. That’s why Alzheimer’s went from virtually unknown to the 4th leading killer over age 65.

They told you to lower your cholesterol…while your brain was starving for it.

This isn’t “aging.” This is medical malpractice on a massive scale.

This observation is just anecdotal, but I only know three people in the two previous generations who had some form of dementia before they died. However, all three of them were on statins…

At this point, you couldn’t pay me to take any medication or prescription that doesn’t have an immediate, obvious, and observably beneficial effect. No matter what the numbers on the printout say.

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Chalub and the Kimura Cancellation

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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Byron AI Volunteers Wanted

The Byron AI team is looking for 10-20 volunteers who are a) reasonably well-versed in a broad range of literary genres and b) are capable of quickly reading 500- to 1000-word chunks. No technical ability is required; what we’re looking for is people who are capable of reading an unidentified text sample and attempting to identify the author on the basis of the style. The performance is not important; we have a second-level test that is necessary for those readers who cannot identify the author on the first read, so the volunteers have to be able to resist the temptation to look anything up or resort to AI in order to identify the author.

That sort of artificial identification will pollute the data and we want to strictly avoid it. Byron AI backers are favored for this, so if you’re interested in volunteering, please email me with AI STYLE BACKER if you are a project backer and AI STYLE if you are not.

As is probably clear, the dev team did not wait for the campaign to successfully conclude before we got started on certain elements of the project. It’s still too soon to declare that we will succeed in the proof of concept, but the initial indications are positive.

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