Vox Popoli

The Word of the Lord of Barad-dûr

“The Witch-king proposes an assault,” said Khamûl. “A direct strike on Imladris. The full strength of the Nine, supported by two battalions from Dol Guldur and whatever host Saruman can muster from the south. The Witch-king believes that Rivendell’s defenses, while formidable, have not been tested by true military force in —”

“No.”

The word was quiet, but it filled the room the way darkness fills a cellar, completely and without effort. Khamûl fell silent.

Sauron rose from his chair and walked to the fire. He stood with his back to the room, looking into the flames, and the flames, one might have noticed, were not reflected in his eyes. Something older burned there.

“An assault on Rivendell is precisely what Elrond would want,” he said. “It is what Gandalf expects. They are not fools, Khamûl. Rivendell is a fortress of the spirit as much as of stone. The valley itself resists the Shadow — the Bruinen obeys Elrond, the passes are warded with arts that predate the founding of Mordor. An assault would cost us thousands of Orcs, at least three Nazgûl, and months of preparation, and even then success would not be certain. And while we spent our strength against those waterfalls and singing stones, Gondor would have time to marshal, Rohan would consolidate, and every petty lord from Dol Amroth to the Iron Hills would take heart. No. We will not assault Rivendell.”

“Then what does the Dark Lord command?”

Sauron turned from the fire. “The crebain. Saruman’s crows answer to him, but they will answer to me as well, and in any case the birds will do as they are told. I want crows over the Misty Mountains, crows over every pass and path between Rivendell and the south. I want to know what moves in and out of that valley — every rider, every company, every cursed halfing with a walking stick. Nothing leaves Imladris without my knowledge.”

“Anything else?”

“Two of the Nine, on their winged mounts, from a rotation of six. High patrol; they should not stoop low enough to provoke Elrond into a response, but close enough to see who travels in and out of Rivendell. They are to observe. They are not to engage. If they see the Ring-bearer moving, they report. They shall not attack. Not yet.”

“The Witch-king will find this response to be… restrained.”

“The Witch-king will find this wise, once he has thought about it for more than the three seconds he typically devotes to reflection. You may tell him the Lord of Barad-dûr said so.”

Khamûl bowed deeply and departed. The door closed behind him with a sound like a coffin lid settling uneasily into position.

Sauron stood by the fire a moment longer, then returned to his chair. Lúthiel had moved her knight while he was away from the table. He noticed this but said nothing. She cheated only when the position was already lost, and he found the habit endearing in a way he suspected said something unflattering about his character.

“You are concerned,” she said. It was not a question.

“I am thinking.”

He looked at the chessboard, but his eyes were not on the pieces. They were somewhere far to the west, in a valley of green and gold where his enemies were, at this very moment, deciding how to move against him.

“The Ring is in Rivendell,” he said. “And in Rivendell there are gathered, if Khamûl’s report is accurate — and it is at least partially accurate, which for Khamûl is exceptional — the Halfling who carried the Ring from the Shire, Gandalf, Elrond, and almost certainly representatives of the Dwarves and the men of the North. A council. They will be debating what to do with it.”

“And what will they do with it?”

“That is the question.” He moved a pawn, absently. “They cannot hide it. The Ring calls to me; wherever it rests, I will find it in time. They know this. They cannot unmake it by any ordinary means — Elrond knows this better than anyone, having watched Isildur refuse to cast it into the fire when he had the chance. So they must either wield it or destroy it.”

“And you think they will wield it.”

“I think they will be tempted.” He leaned back and pressed his fingertips together — nine fingers forming an incomplete arch. “The question is: who? Who among them has both the power to use the Ring effectively and the arrogance to believe they can control it?”

He was quiet for a time. The fire crackled. Lúthiel waited. She was, among her many virtues, an exceptional waiter.

“Elrond will not take it,” said Sauron. “He is too cautious. He remembers what happened with Isildur, and he has spent three thousand years being cautious as a form of penance for not having physically shoved Isildur into the fire when he had the chance, which, between us, he probably should have done. Elrond will counsel destruction. He will be right, and he will be ignored.”

“The Dwarf lords?”

“Dwarves are resistant to the Ring’s deeper corruptions — their minds are stone, slow to turn. But for the same reason, they cannot wield it with the subtlety it requires. A Dwarf with the One Ring would simply become a more stubborn Dwarf, which is a terrifying concept in its own right but not a strategic threat. No. Not the Dwarves.”

“The Halfling.”

“A carrier. A postman. The Ring chose him for proximity, not for power. He is no more capable of wielding the Ring against me than a sparrow is of wielding a siege engine. The Ring would eat him alive within a week.”

“Then who?”

Sauron’s eyes narrowed. “There is a Man in the north — Aragorn, they call him. Isildur’s heir. The last of the Númenórean line. He has power in his blood, old power, and the Ring would know it. The Ring would sing to him of kingship, of the throne of Gondor restored, of the Reunited Kingdom. He is dangerous.” He paused. “But he is also a Ranger. He has spent his life in the wild, deliberately avoiding power. A man who has refused the throne for sixty years is unlikely to suddenly decided to seize it through a weapon of the Enemy. Aragorn is not the threat.”

He fell silent, and the silence lengthened, and Lúthiel watched him arrive at the answer she suspected he had known since Khamûl opened his mouth.

“Gandalf,” said Sauron.

He said the name the way one says the name of an old colleague who has made a career of being underestimated and whose modesty one has never for a moment believed.

“Gandalf,” he repeated. “He is a Maia. My equal in nature, if not in craft. He has walked Middle-earth for two thousand years in the shape of an old man, pretending to be less than he is, playing the advisor, the wanderer, the friend of Hobbits and the lighter of fireworks. But he is a spirit of fire, and the Ring would amplify that fire a hundredfold. With the One Ring, Gandalf could challenge me directly. He could raise the Free Peoples not merely as a counselor but as a commander, and transform them into a power to rival this Dark Tower itself.”

“And his eagle,” said Lúthiel.

Sauron made a dismissive gesture. “Couriers and carriers. The eagles are proud creatures. They do favors for Gandalf out of old debts and older vanity, but they will not commit to a war on his behalf. Manwë’s birds have not intervened in the affairs of Middle-earth in any sustained fashion since the War of Wrath, and that was under direct instruction from the Valar. No. Gwaihir carried Gandalf out of Orthanc because it cost him nothing and flattered his self-regard. He cannot carry an army.”

“Would Gandalf take it?”

Sauron considered this with the seriousness it deserved. “He would tell himself he was taking it reluctantly. He would tell himself it was necessary — especially in light of Saruman coming over to me —that no one else could bear the burden, that he alone had the wisdom to wield it without being corrupted. He would be wrong, of course. They are always wrong. But he would believe it, and that belief is all the Ring needs.”

He moved his queen. “That is the danger. Not a warrior riding to my gates with the Ring on his finger. Gandalf. Working quietly, building alliances, using the Ring’s power to unite and strengthen and inspire, until one day I look west and find not a scattered collection of failing kingdoms but a single, coordinated force led by a Maia with the power of the One Ring and the submission of every once-free creature in Middle-earth.”

He studied the board. Lúthiel’s position was, despite her clandestine knight maneuver, quite hopeless. He could see checkmate in eleven moves.

“That,” said Sauron, “is what I must prevent.”

Lúthiel moved her rook. It was the wrong move, but she made it with great confidence, which he admired.

“And the crows and the Nazgûl patrol?” she said. “That will be enough?”

“For now. Gandalf is patient, but he is not infinitely patient. He will move soon, most likely within weeks, not months. And when he does, when he leaves that valley with the Ring, my servants will see him. And that is when I will strike.”

He took her rook with his bishop. “Your position is untenable, incidentally.”

She looked at the board, looked at him, and tipped over her king with one pale finger.

“Again?” she said.

“Again.”

She began resetting the pieces, bone-white and volcanic glass, and Sauron the Great, Lord of Mordor, Enemy of the Free Peoples of Middle-earth, settled into his chair and permitted himself a small, private smile. He had been outmaneuvered before. He had been defeated before. He did not intend to let it happen again. The Ring was once more in play, and the game — the true game, the one that mattered — was only beginning.

Outside the tower, far below, the plains of Gorgoroth stretched away under a sky of smoke and ember, and somewhere in the darkness, a large flock of crebain turned their black eyes westward and began to fly.

DISCUSS ON SG

The Real Rate of Molecular Evolution

Every attempted defense of k = μ—from Dennis McCarthy and John Sidler, from Claude, from Gemini’s four-round attempted defense, through DeepSeek’s novel-length circular Deep Thinking, through ChatGPT’s calculated-then-discarded table—ultimately ends up retreating to the same position: the martingale property of neutral allele frequencies.

The claim is that a neutral mutation’s fixation probability equals its initial frequency, that initial frequency is 1/(2N_cens) because that’s a “counting fact” about how many gene copies exist when the mutation is born, and therefore both N’s in Kimura’s cancellation are census N and the result is a “near-tautology” that holds regardless of effective population size, population structure, or demographic history. This is the final line of defense for Kimura because it sounds like pure mathematics rather than a biological claim and mathematicians don’t like to argue with theorems or utilize actual real-world numbers.

So here’s a new heuristic. Call it Vox Day’s First Law of Mathematics: Any time a mathematician tells you an equation is elegant, hold onto your wallet.

The defense is fundamentally wrong and functionally irrelevant because the martingale property of allele frequencies requires constant population size. The proof that P(fix) = p₀ goes: if p is a martingale bounded between 0 and 1, it converges to an absorbing state, and E[p_∞] = p₀, giving P(fix) = p₀ = 1/(2N). But frequency is defined as copies divided by total gene copies. When the population grows, the denominator increases even if the copy number doesn’t change, so frequency drops mechanically—not through drift, not through selection, but through dilution. A mutation that was 1 copy in 5 billion gene copies in 1950 is 1 copy in 16.4 billion gene copies in 2025. Its frequency fell by 70% with no evolutionary process acting on it.

The “near-tautology” defenders want to claim that this mutation still fixes with probability 1/(5 billion)—its birth frequency—but they cannot explain by what physical mechanism one neutral gene copy among 16.4 billion has a 3.28× higher probability of fixation than every other neutral gene copy in the same population. Under neutrality, all copies are equivalent. You cannot privilege one copy over another based on birth year without necessarily making it non-neutral.

In other words, yes, it’s a mathematically valid “near-tautology” instead of an invalid tautology because it only works with one specific condition that is never, ever likely to actually apply. Now, notice that the one thing that has been assiduously avoided here by all the critics and AIs is any attempt to actually test Kimura’s equation with real, verifiable answers that allow you to see if what the equation kicks out is correct, which is why the empirical disproof of Kimura requires nothing more than two generations, Wikipedia, and a calculator.

Here we’ll simply look at the actual human population from 1950 to 2025. If Kimura holds, then k = μ. And if I’m right, k != μ.

Kimura’s neutral substitution rate formula is k = 2Nμ × 1/(2N) = μ. Using real human census population numbers:
Generation 0 (1950): N = 2,500,000,000 Generation 1 (1975): N = 4,000,000,000 Generation 2 (2000): N = 6,100,000,000 Generation 3 (2025): N = 8,200,000,000
Of the 8.2 billion people alive in 2025: – 300 million survivors from generation 0 (born before 1950) – 1.2 billion survivors from generation 1 (born 1950-1975) – 2.7 billion survivors from generation 2 (born 1975-2000) – 4.0 billion born in generation 3 (born 2000-2025)
Use the standard per-site per-generation mutation rate for humans.
For each generation, calculate: 1. How many new mutations arose (supply = 2Nμ) 2. Each new mutation’s frequency at the time it arose (1/2N) 3. Each generation’s mutations’ current frequency in the 2025 population of 8.2 billion 4. k for each generation’s cohort of mutations as of 2025
What is k for the human population in 2025?

The application of Kimura is impeccable. The answer is straightforward. Everything is handed to you. The survival rates are right there. The four steps are explicit. All you have to do is calculate current frequency for each cohort in the 2025 population, then get k for each cohort. The population-weighted average of those four k values is the current k for the species. Kimura states that k will necessarily and always equal μ.

k = 0.743μ.

Now, even the average retard can grasp that x != 0.743x. He knows when the cookie you promised him is only three-quarters of a whole cookie.

Can you?

Deepseek can’t. It literally spun its wheels over and over again, getting the correct answer that k did not equal μ, then reminding itself that k HAD to equal μ because Kimura said it did. ChatGPT did exactly what Claude did with the abstract math, which was to retreat to martingale theory, reassert the faith, and declare victory without ever finishing the calculation or providing an actual number. Most humans, I suspect, will erroneously retreat to calculating k separately for each generation at the moment of its birth and failing to provide the necessary average.

Kimura’s equation is wrong, wrong, wrong. It is inevitably and always wrong. It is, in fact, a category error. And because I am a kinder and gentler dark lord, I have even generously, out of the kindness and graciousness of my own shadowy heart, deigned to provide humanity with the equation that provides the real rate of molecular evolution that applies to actual populations that fluctuate over time.

Quod erat fucking demonstrandum!

DISCUSS ON SG

Veriphysics: The Treatise 006

VII. The Scientific Failures

Science was the Enlightenment’s proudest achievement. Here, at last, was a method that worked: systematic observation, controlled experiment, mathematical formalization, rigorous testing. The results were undeniable. Physics, chemistry, medicine, engineering—the sciences transformed human life and demonstrated the power of disciplined reason applied to nature.

The prestige of science was not unearned. But the Enlightenment made a subtle and consequential error: it confused the success of scientific method within its proper domain with the sufficiency of scientific method for all domains. If physics could explain the motions of the planets, perhaps it could also explain the motions of the soul. If chemistry could analyze the composition of matter, perhaps it could also analyze the composition of morality. The success of science in one area became an argument for its supremacy in all areas.

This confidence has not aged well.

The institution of science, as distinct from the method, has proven vulnerable to precisely the corruptions that the Enlightenment imagined it would transcend. The guild structure of modern academia—tenure, peer review, grant funding, journal publication—was designed to ensure quality and independence. In practice, it has produced conformity and capture. The young scientist who wishes to advance must please senior scientists who control hiring, funding, and publication. Heterodox views are not refuted; they are simply not funded, not published, not hired. The revolutionary who challenges the paradigm does not receive a hearing and a refutation; he receives silence and exclusion.

The replication crisis has revealed the extent of the rot. Study after study, published in prestigious journals, approved by peer review, celebrated in the press, has proven impossible to replicate. The effect sizes shrink, the p-values evaporate, the findings dissolve upon examination. In psychology, in medicine, in nutrition science, in economics, the literature is contaminated with results that are not results at all but artifacts of bad statistics, selective reporting, and the relentless pressure to publish something—anything—novel and significant.

Peer review, that supposed guarantor of quality, has been exposed as inadequate to its function. The peers are competitors; the reviews are cursory; the incentives favor approval over scrutiny. Fraud, when it is detected, is detected years or decades after the damage is done. The process filters for conformity to existing paradigms, not for truth. The Enlightenment imagined science as a self-correcting enterprise; the corrections, it turns out, are slow, partial, and fiercely resisted by those whose careers depend on the errors.

It is in biology that the Enlightenment’s scientific project reaches its apex—and its most consequential failure.

Charles Darwin’s On the Origin of Species, published in 1859, proposed to explain the diversity of life through purely natural mechanisms: random variation and natural selection, operating over vast stretches of time, producing all the complexity we observe. No designer, no purpose, no direction—only the blind filter of differential reproduction. The theory was not merely scientific; it was the completion of the Enlightenment’s program to explain the world without recourse to anything beyond material causation.

Darwin’s idea, as Daniel Dennett observed, was “universal acid”—it ate through every traditional concept. If man is merely the product of blind variation and selection, then there is no soul, no purpose, no inherent dignity. Ethics becomes an evolved adaptation; consciousness becomes an epiphenomenon; free will becomes an illusion; man becomes a clever animal, nothing more. The stakes could not be higher. If Darwin was right, then the Enlightenment had completed its work: the world was fully explained in material terms, and everything else—meaning, value, purpose—was either reducible to matter or mere sentiment.

The scientific establishment embraced Darwin not merely as a hypothesis but as a foundation. To question evolution by natural selection was to mark oneself as a rube, a fundamentalist, an enemy of reason. The theory became unfalsifiable in practice—not because it was so well-confirmed, but because no alternative could be entertained within respectable discourse. The question was settled, and to reopen it was professional suicide.

But the question was never settled. It was merely avoided.

The mathematical problems with the theory were identified almost immediately. In 1867, Fleeming Jenkin raised an objection that Darwin never adequately answered: blending inheritance would dilute favorable variations before selection could act on them. The discovery of Mendelian genetics resolved this particular difficulty, but it raised others. The “Modern Synthesis” of the 1930s and 1940s combined Darwinian selection with Mendelian genetics and mathematical population genetics, creating the Neo-Darwinian framework that remains official orthodoxy today, even though it is honored mostly in the breach.

In 1966, mathematicians and engineers gathered at the Wistar Institute in Philadelphia to examine the mathematical foundations of the Modern Synthesis. Their verdict was devastating. The rates of mutation, the population sizes, the timescales available—the numbers did not work. The probability of generating the observed complexity through random mutation and natural selection was effectively zero.

The biologists were unimpressed. They did not engage with the mathematics; they simply noted that the mathematicians were not biologists, and continued as before. The pattern established in 1966 has held ever since: mathematically literate outsiders raise objections; biologically credentialed insiders ignore them; the textbooks remain unchanged.

The mapping of the human and chimpanzee genomes in the early 2000s provided the data necessary to test the theory quantitatively. The genetic difference between the species requires approximately forty million mutations to have become fixed in the relevant lineages since the hypothesized divergence from a common ancestor. Using the fastest fixation rate ever observed in any organism—bacteria under intense selection in laboratory conditions—and the most generous timescales proposed in the literature, the mathematics permits fewer than three hundred fixations.

The theory requires forty million. The math allows three hundred. The gap is not a matter of uncertainty or approximation; it is a difference of five orders of magnitude. No adjustment of parameters, no refinement of models, no appeal to undiscovered mechanisms can bridge such a chasm. The theory of evolution by natural selection, as an explanation for the origin of species, is mathematically impossible.

This is not a controversial claim among those who can do the arithmetic. It is simply not discussed by those whose careers depend on not discussing it. The Enlightenment’s greatest scientific achievement—the explanation of life itself through material causes alone—is empirically false. And the institution of science, that much-hallowed engine of supposed self-correction, has proven incapable of acknowledging the mathematical falsification for sixty years.

DISCUSS ON SG

Trying to Salvage Kimura

A commenter at Dennis McCarthy’s site, John Sidles, attempts to refute my demonstration that Mooto Kimura made a fatal mistake in his neutral-fixation equation “k=μ”.

VOX DAY asserts confidently, but wrongly, in a comment:
“Kimura made a mistake in the algebra in his derivation of the fixation equation by assigning two separate values to the same variable.”
It is instructive to work carefully through the details of Kimura’s derivation of the neutral-fixation equation “k=μ”, as given in Kimura’s graduate-level textbook “The Neutral Theory of Molecular Evolution” (1987), specifically in Chapter 3 “The neutral mutation-random drift hypothesis as an evolutionary paradigm”.
The derivation given in Kimura’s textbook in turn references, and summarizes, a series of thirteen articles written during 1969–1979, jointly by Kimura with his colleague Tomoko Ohta. It is striking that every article was published in a high-profile, carefully-reviewed journal. The editors of these high-profile journals, along with Kimura and Ohta themselves, evidently appreciated that the assumptions and the mathematics of these articles would be carefully, thoroughly, and critically checked by a large, intensely interested community of population geneticists.
Even in the face of this sustained critical review of neutral-fixation theory, no significant “algebraic errors” in Kimura’s theory were discovered. Perhaps one reason, is that the mathematical derivations in the Kimura-Ohta articles (and in Kimura’s textbook) are NOT ALGEBRAIC … but rather are grounded in the theory of ordinary differential equations (ODE’s) and stochastic processes (along with the theory of functional limits from elementary calculus).
Notable too, at the beginning of Chapter 3 of Kimura’s textbook, is the appearance of numerical simulations of genetic evolution … numerical simulations that serve both to illustrated and to validate the key elements of Kimura’s theoretical calculations.
As it became clear in the 1970s that Kimura’s theories were sound (both mathematically and biologically), the initial skepticism of population geneticists eolved into widespread appreciation, such that in the last decades of his life, Kimura received (deservedly IMHO) pretty much ALL the major awards of research in population genetics … with the sole exception of the Nobel Prizes in Chemistry or Medicine.

Claim 1: My claim that Kimura made a mistake in the algebra in his derivation of the fixation equation by assigning two separate values to the same variable” is “confidently, but wrongly” asserted.

No, my claim is observably correct. The k = μ derivation proceeds in three steps:

Step 1 (mutation supply): In a diploid population of size N, there are 2N gene copies, so 2Nμ new mutations arise per generation. Here N is the census population—individuals replicating DNA. Kimura’s own 1983 monograph makes this explicit: “Since each individual has two sets of chromosomes, there are 2N chromosome sets in a population of N individuals, and therefore 2Nv new, distinct mutants will be introduced into the population each generation” (p. 44). This is a physical count of bodies making DNA copies.

Step 2 (fixation probability): Each neutral mutation has fixation probability 1/(2N). This result derives from diffusion theory under Wright-Fisher model assumptions, where N is the effective population size—the size of an idealized Wright-Fisher population experiencing the same rate of genetic drift. Kimura himself uses N_e notation for drift-dependent quantities elsewhere in the same work: “S = 4N_e s, where N_e is the effective population size” (p. 44).

Step 3 (the “cancellation”): k = 2Nμ × 1/(2N) = μ.
The cancellation requires the N in Step 1 and the N in Step 2 to be the same number. They are not. Census N counts replicating individuals. Effective N_e is a theoretical parameter from an idealized model. In mammals, census N exceeds diversity-derived N_e by ratios of 10× to 46× (Frankham 1995; Yu et al. 2003, 2004; Hoelzel et al. 2002). If the two N’s are not equal, the correct formulation is:
k = 2Nμ × 1/(2N_e) = (N/N_e)μ

This is not a philosophical quibble. It is arithmetic. If you write X × (1/X) = 1, but the first X is 1,000,000 and the second X is 21,700, you have not performed a valid cancellation. You have performed an algebraic error. The fact that the two quantities could be equal in an idealized Wright-Fisher population with constant size, random mating, Poisson-distributed offspring, and discrete non-overlapping generations does not save the algebra when applied to any natural population, because no natural population satisfies these conditions.

Claim 2: The derivation references thirteen articles published in “high-profile, carefully-reviewed journals” and was subjected to “sustained critical review” by “a large, intensely interested community of population geneticists.”

This is true and it is irrelevant. The error was not caught because the notation obscures it. When you write 2Nμ × 1/(2N), the cancellation looks automatic—it appears to be a trivial identity. You have to stop and ask: “Is the N counting replicating bodies the same quantity as the N governing drift dynamics in a Wright-Fisher idealization?” The answer is no, but the question is invisible unless you distinguish between census N and effective N_e within the derivation itself.

Fifty years of peer review did not catch this because the reviewers were working within the same notational framework that obscures the distinction. This is not unusual in the history of science. Errors embedded in foundational notation persist precisely because every subsequent worker inherits the notation and its implicit assumptions. The longevity of the error is not evidence of its absence; it is evidence of how effectively notation can conceal an equivocation.

John Sidles treats peer review as a guarantee of mathematical correctness. It is not, and the population genetics community itself has acknowledged this in other contexts. The reproducibility crisis affects theoretical as well as empirical work. Appeals to the number and prestige of journals substitute sociological authority for mathematical argument.

Claim 3: “No significant ‘algebraic errors’ in Kimura’s theory were discovered.”

This is an argument from previous absence, which is ridiculous because I DISCOVERED THE ERROR. No one discovered the equivocation because no one looked for it. The k = μ result was celebrated as an elegant proof of population-size independence. It became a foundational assumption of neutral theory, molecular clock calculations, and coalescent inference. Questioning it would have required questioning the framework that built careers and departments for half a century.
Moreover, the claim that no errors were discovered is now empirically falsified. I demonstrated that the standard Kimura model, which implicitly assumes discrete non-overlapping generations and N = N_e, systematically overpredicts allele frequencies when tested against ancient DNA time series. The model overshoots observed trajectories at three independent loci (LCT, SLC45A2, TYR) under documented selection, and a corrected model reduces prediction error by 69% across all three. A separate analysis of 1,211,499 loci comparing Early Neolithic Europeans with modern Europeans found zero fixations over seven thousand years—against a prediction of dozens to hundreds under neutral theory’s substitution rate.
The error has now been discovered. The fact that it was not discovered sooner reflects the fundamental flaws of the field, not the soundness of the mathematics.

Claim 4: The mathematical derivations “are NOT ALGEBRAIC… but rather are grounded in the theory of ordinary differential equations (ODE’s) and stochastic processes.”

This is true of Kimura’s fixation probability formula, P_fix = (1 − e^(−2s)) / (1 − e^(−4N_e s)), which derives from solving the Kolmogorov backward equation—a genuine boundary-value problem for an ODE arising from the diffusion approximation to the Wright-Fisher process. The commenter is correct that this piece of Kimura’s mathematical apparatus is grounded in sophisticated mathematics and is INTERNALLY consistent.

But it is not externally consistent and the k = μ result does not come from the ODE machinery anyhow. It comes from the counting argument: 2Nμ mutations per generation × 1/(2N) fixation probability = μ. This is multiplication. The equivocation is in the multiplication, not in the diffusion theory. Invoking the sophistication of Kimura’s ODE work to defend a three-line counting argument is a red herring. Mr. Sidles is defending Kimura on ground where Kimura is correct (diffusion theory) while the error sits on ground where the math is elementary (the cancellation of two N terms that represent different quantities).

The distinction between census N and effective N_e is not a subtlety of diffusion theory. It is visible to anyone who simply asks what the symbols mean. Mr. Sidles’s invocation of ODEs and stochastic processes does not address the actual error.

Claim 5: Numerical simulations “serve both to illustrate and to validate the key elements of Kimura’s theoretical calculations.”

Numerical simulations of the Wright-Fisher model validate Kimura’s results within the Wright-Fisher model. This is unsurprising—if you simulate a constant-size population with discrete generations, random mating, and Poisson reproduction, you will recover k = μ, because the simulation satisfies the assumptions under which the result holds.

The question is not whether Kimura’s math is internally consistent within its model. It is. The question is whether the model’s assumptions map onto biological reality. They observably do not. No natural population has constant size. No natural population of a long-lived vertebrate has discrete, non-overlapping generations. Census population systematically exceeds effective population size in every mammalian species studied.

Simulations that assume the very conditions under which the cancellation holds cannot validate the cancellation’s applicability to populations that violate those conditions. This is circular reasoning: the model is validated by simulations of the model.

Ancient DNA provides a non-circular test. When the standard model’s predictions are compared to directly observed allele frequency trajectories over thousands of years, the model fails systematically, overpredicting the rate of change by orders of magnitude. This empirical failure cannot be explained by simulation results that assume the model is correct.

Summary: Mr. Sidles’s defense reduces to three arguments: (1) many smart people reviewed the work, (2) the math uses sophisticated techniques, and (3) simulations confirm the theory. None of these address the actual error.

The error is simple: the k = μ derivation uses a single symbol for two different quantities—census population size and effective population size—and cancels them as if they were identical. They are not identical in any natural population. The cancellation fails.

The result that substitution rate is independent of population size holds only in an idealized Wright-Fisher population with constant size, and is not a general law of evolution.

Kimura’s diffusion theory is internally consistent within the Wright-Fisher framework and only within that framework. His fixation probability formula follows validly from its premises—premises that no natural population satisfies, since N_e is not constant, generations are not discrete, and census N ≠ N_e in every species studied. His contributions to population genetics are substantial.

None of this changes the fact that the k = μ derivation contains an algebraic error that has propagated through nearly sixty years of molecular evolutionary analysis.

In spite of this, Mr. Sidles took another crack at it:

Vox, your explanation is so clear and simple, that your mistake is easily recognized and corrected.
THE MISTAKE: “Step 2 (fixation probability): Each neutral mutation has fixation probability 1/(2N).”
THE CORRECTION: “Step 2 (fixation probability): Each neutral mutation IN A REPRODUCING INDIVIDUAL (emphasis mine) has fixation probability 1/(2Ne). Each neutral mutation in a non-reproducing individual has fixation probability zero (not 1/N, as Vox’s “algebraic error” analysis wrongly assumes).”
Kimura’s celebrated result “k=μ” (albeit solely for neutral mutations) now follows immediately.
For historical context, two (relatively recent) survey articles by Masatoshi Nei and colleagues are highly recommended: “Selectionism and neutralism in molecular evolution” (2005), and “The Neutral Theory of molecular evolution in the Genomic Era” (2010). In a nutshell, Kimura’s Neutral Theory raises many new questions — questions that a present are far from answered — and as Nei’s lively articles remind us:
“The longstanding controversy over selectionism versus neutralism indicates that understanding of the mechanism of evolution is fundamental in biology and that the resolution of the problem is extremely complicated. However, some of the controversies were caused by misconceptions of the problems, misinterpretations of empirical observations, faulty statistical analysis, and others.”
Nowadays “AI-amplified delusional belief-systems” should perhaps be added to Nei’s list of controversy-causes, as a fresh modern-day challenge to the reconciliation of the (traditional) Humanistic Enlightenment with (evolving) scientific understanding.

Another strikeout. He removed the non-reproducers twice, because he doesn’t understand the equation well enough to recognize that Ne already incorporates their non-reproduction, so he can’t eliminate them a second time. This is the sort of error that someone who knows the equation well enough to use it, but doesn’t actually understand what the various symbols mean is usually going to make.

Kimura remains unsalvaged. Both natural selection and neutral theory remain dead.

DISCUSS ON SG

In the Tower of Barad-dûr

The library of Barad-dûr was not what most people would have expected, had most people been in a position to expect anything about it at all, which they were not, on account of being either dead or very far away and deeply committed to remaining so. It occupied the forty-third level of the Dark Tower, well below the great Eye’s chamber but far above the barracks and forges and pits where the common business of Mordor was conducted, and it was — there was no other word for it — comfortable.

The walls were black stone, naturally, but they had been hung with tapestries of deep crimson and charcoal grey, woven by captive artisans of considerable skill who had been treated quite well during their employment and then released to the interior settlements of Nurn, where they now ran a moderately successful textile cooperative. The shelves rose from floor to ceiling, carved from the dark wood of trees that grew in the sheltered vales south of the Ephel Dúath, and they held thousands of volumes — histories, treatises on metallurgy and linguistics, the collected philosophical works of the Second Age, several illustrated atlases of Middle Earth, and a modest but well-curated collection of erotic Sindarin poetry that Sauron would have denied owning if asked, which, of course, no one ever did.

A fire burned in a grate of black iron. Two chairs of dark leather faced each other across a table of polished obsidian, upon which a chess set had been arranged. The pieces were exquisite — one set carved from white bone, the other from volcanic glass — and the game was already underway.

Sauron sat in the chair to the left of the fire. He was not, at this moment, wearing the form of the great Eye, which he customarily maintained in the upper chamber by a combination of will and ancient sorcery and which he found, frankly, exhausting. In his library he preferred a more practical shape: tall, severe, dark-haired, with the handsome and slightly drawn features of a man who has been awake for several thousand years and finds the whole thing rather tedious. He wore a robe of black silk and no crown. His right hand, the one missing the finger where the Ring had once sat, rested on the arm of his chair. He had never bothered to restore it. He found that its absence tended to focus his mind rather helpfully when he found himself gravitating toward sloth.

Across from him, studying the board with an intensity that he found genuinely charming, sat Lúthiel.

She had been an Elf of Eregion, once. She was of Celebrimbor’s people, a seamstress of great talent who had worked alongside Sauron himself in his Annatar days, when he had walked among the Elves in a fairer form and taught them the craft of ring-making. She had seen through him long before the others. She had known perfectly well what he was. And she had, after a period of considerable internal deliberation that had lasted approximately three centuries, decided that it did not bother her. Indeed, much to the contrary, she found herself drawn to his darkness.

This was not, Sauron reflected, as uncommon as the histories of the Elves would have one believe. The Eldar presented themselves as uniformly noble, but immortality did strange things to those upon whom it was bestowed. Lúthiel had simply grown bored with virtue. She was not evil, at least, not by his standards. She had no interest in dominion or cruelty or the subjugation of peoples. She was merely done with the relentless earnestness of Elvish civilization, the repetitive songs about starlight, and the interminable councils about the proper stewardship of forests. She wanted to read interesting books and play chess with someone who could keep up with her, and Sauron, whatever his other failings might be, had always provided her with exceptional company.

She also liked wargs. A lot. To a degree that Sauron found almost disturbing. The orcs of Mordor knew her, and feared her, as Hiriel the Huntress, due to her habit of riding a very large pack leader at the head of a voracious pack and hunting orcs, goblins, and, Sauron suspected, the occasional Easterling.

She moved her bone-white bishop three squares. “Check,” she declared happily, looking pleased with herself.

Sauron looked at the board. She was right. His king was exposed along the diagonal, and her rook, previously blocked by the bishop, now commanded the entire file. He studied the position for a moment, then moved his king behind a pawn with a faint nod of acknowledgment.

“You are improving,” he said.

“I have been improving for nine hundred years. You might at least pretend to be threatened.”

“My king is in danger. That bishop maneuver was —”

There was a knock at the door. Three sharp raps, then silence. It was the knock of someone who had been taught precisely how to knock at this particular door and understood the negative consequences of improvisation.

“Enter,” said Sauron.

The door opened and a Nazgûl came in. It was not the Witch-king. From the figure entering emanated the pale, ephemeral menace of Khamûl the Easterling, the third of the Nine. He was still wearing his full kit of black robes, iron crown, and gauntlets, which meant he had come directly from the field and had not thought to change, which meant the news was either urgent or Khamûl had once again failed to grasp the concept of appropriate attire for different contexts. Sauron assumed the former.

“My lord,” said Khamûl. His voice came from the hood like wind through a keyhole. “I bring word from the western watches.”

“Speak.”

Khamûl glanced at Lúthiel. Sauron did not tell her to leave, and the Nazgûl had learned, over the centuries, that Lúthiel’s presence was a permanent feature of the library and that any question concerning her right to be there would be met with a decidedly negative response. Khamûl turned back to his master.

“Saruman’s prisoner has escaped. The wizard Gandalf — he was held atop Orthanc. He is gone.”

Sauron’s expression did not change, but his hand, the complete one, closed slowly around the arm of his chair. “How did this happen?”

“We are uncertain of the precise details. Saruman was reluctant to admit the wizard’s escape and his report was… incomplete. But our watchers in the mountains saw it clearly enough. An eagle, one of the skylords of the Misty Mountains, descended upon the tower of Orthanc and bore Gandalf away through the air.”

Sauron was quiet for a moment. “Which one?”

“Gwaihir, my lord. The Windlord. He carried Gandalf north and east at great speed. Our crebain tracked them as far as the upper vales of the Anduin before losing sight. It appears the eagle delivered him directly to Rivendell.”

“When?”

“Yesterday.”

“And you are telling me now?”

Khamûl shifted almost imperceptibly. The Nazgûl did not fidget — their bodies were too far past the threshold of ordinary physical response for anything so human — but there was something in the movement that served the same function. “As I said, Saruman was evasive, and so we wished to confirm the wizard’s escape rather than bring you false news, my lord. And there is more. The Halfling — the Ringbearer — we pursued him on horseback to the very borders of Rivendell but we were unable to cross the Bruinen.” The barest hint of resentment entered Khamûl’s voice. “The river rose against us.”

“Yes,” said Sauron. “It does that.”

He was quiet for a moment. Across the table, Lúthiel had returned her attention to the chessboard with the pointed discretion of someone who has lived long enough to know when to appear as if one has heard nothing.

“So,” said Sauron. “Gandalf is free. Gandalf has the Halfling. The Halfling has the Ring. And all three are in Rivendell, under the protection of Elrond.”

Saruman had failed. So, too, had the Nine. Sauron closed his eyes and repressed the urge to remove the iron crown from Khamûl’s head and force his entire body to pass through it in a very violent and painful manner.

He was not a happy dark lord.

The Word of the Lord of Barad-dûr

DISCUSS ON SG

Veriphysics: The Treatise 005

VI. The Usury Revolution

The failures of Enlightenment philosophy examined thus far—political, juridical, economic, scientific—share a common feature: they all represent the systematic failure of ideas. The social contract is a logical fiction. The law of supply and demand does not describe real markets. The theory of evolution by natural selection cannot survive the genetic arithmetic required. These are intellectual errors, and intellectual errors can, at least in principle, be corrected by the presentation of better arguments and more predictive models.

But the Enlightenment itself did not triumph through better arguments. It triumphed through rhetoric and institutional capture, and institutional capture requires resources. Ideas need patrons, publishers, platforms, and time. The philosophers needed salons; the salons needed hosts; the hosts needed wealth. The question of how the Enlightenment acquired the resources to propagate itself across centuries is not peripheral to its success; it is central. And the answer lies in a revolution that preceded and enabled all the others: the revolution in usury.

The Ancient Prohibition

The prohibition on usury is older than Christianity. It is older than Rome. The condemnation of lending at interest appears in the earliest legal codes of civilization and persists across cultures that had no contact with one another.

In Rome, the Twelve Tables—the foundation of Roman law, dating to approximately 450 BC—restricted interest rates and imposed severe penalties for usurious lending. The Lex Genucia of 342 BC banned interest entirely, though enforcement proved difficult. Cato the Elder, asked what he thought of lending at interest, replied: “What do you think of murder?” The Roman tradition understood usury as a form of theft—the extraction of wealth without the creation of value, the exploitation of necessity, the conversion of time itself into a commodity to be sold.

Long before Rome, the Greek philosophers concurred. Aristotle, in the Politics, condemned usury as the most unnatural form of wealth-acquisition. Money, he argued, is a medium of exchange, a measure of value, a tool for facilitating transactions. It is sterile; it does not breed. To charge for the use of money over time is to treat money as though it could generate offspring—to pretend that a tool has become a living thing. The unnaturalness of usury, for Aristotle, was not merely economic but metaphysical: it violated the nature of what money is.

The Jewish tradition prohibited usury among Israelites while permitting it in dealings with foreigners, a distinction that would later have significant historical consequences. The relevant passages in Exodus, Leviticus, and Deuteronomy are unambiguous: “If you lend money to one of my people among you who is needy, do not treat it like a business deal; charge no interest.” The prohibition was grounded in the covenantal relationship among the people of Israel and the recognition that interest charges exploit vulnerability.

Christianity universalized the prohibition. The Fathers of the Church—Clement of Alexandria, Basil the Great, Gregory of Nyssa, Ambrose, Augustine, Jerome—condemned usury without exception. The medieval canonists developed the prohibition into a sophisticated legal and theological framework. The Third Lateran Council (1179) declared that manifest usurers should be denied Christian burial. The Second Council of Lyon (1274) prohibited rulers from permitting usury in their territories. Thomas Aquinas, in the Summa Theologica, provided the definitive philosophical analysis: to charge for the use of money is to sell what does not exist, to charge twice for the same thing, to violate both justice and the nature of money itself.

This was not arbitrary religious scruple. The prohibition rested on reasoned analysis of what money is and what lending involves. It reflected practical observation of what usury does to communities: concentrating wealth, dispossessing debtors, converting productive economies into extractive ones, transferring resources from those who labor to those who lend. The ancient and medieval world understood what the modern world has forgotten: that unrestricted usury is a solvent that dissolves social bonds and a weapon that transfers power from the many to the few.

The Erosion

The prohibition held for over a millennium. But it eroded, gradually, under the pressure of commercial expansion and the ingenuity of those who wished to circumvent it.

The medieval casuists—the canon lawyers and moral theologians who applied general principles to particular cases—developed increasingly sophisticated distinctions. Certain forms of return on investment were permissible: the census, a contract to purchase future income from productive property; the societas, a partnership in which both profit and risk were shared; the triple contract, a complex arrangement that nominally converted a loan into an investment. The lender who forewent profitable opportunities by lending his money could claim lucrum cessans—compensation for the gain he had sacrificed. The lender who suffered loss because of the borrower’s default could claim damnum emergens—compensation for actual damage incurred.

These distinctions were not always sophistical. There is a genuine difference between a loan at interest and an investment in productive enterprise, between compensation for actual loss and extraction of gain from another’s necessity. But the distinctions multiplied, and as they multiplied, the exceptions threatened to swallow the rule. What had been a clear prohibition became a maze of qualifications that only specialists could navigate—and specialists could usually find a path to the desired destination.

The Reformation accelerated the erosion. Luther initially condemned usury in terms as strong as any Church Father, but Protestant practice soon diverged from Protestant rhetoric. Calvin, in a famous letter, argued that the blanket prohibition on interest could not be sustained from Scripture alone—that the Old Testament texts applied to specific circumstances, that changed conditions required changed applications, that moderate interest on commercial loans was permissible where the borrower was not destitute. Calvin’s position was hedged with qualifications, but the qualifications were soon forgotten while the permission was remembered. The Protestant nations became laboratories for liberalized finance.

England, after Henry VIII’s break with Rome, began relaxing usury restrictions almost immediately. The Act of 1545 legalized interest up to 10 percent, technically as a pragmatic measure, but effectively turned out to be the abandonment of the principle. The rate ceiling was adjusted over the following centuries, always in the direction of liberalization, until the Usury Laws Repeal Act of 1854 abolished restrictions entirely. What had been sin became policy; what had been crime became commerce.

The Financial Revolution

The full consequences of usury’s legitimization emerged with the development of central banking and the instruments of modern finance.

The Bank of Amsterdam, founded in 1609, pioneered the model: a central institution that accepted deposits, transferred payments, and provided a stable currency for commercial transactions. It was a modest innovation compared to what followed. The Bank of England, established in 1694, added something new: the bank was created to lend money to the government, and the loan was funded by the creation of money that had not previously existed. The national debt was born—a permanent obligation of the state to its creditors, serviced by taxation, rolled over in perpetuity.

The implications were revolutionary. A government that can borrow against future revenues can spend beyond its current means. It can fund wars, projects, and patronage that would be impossible if limited to present taxation. And if the lenders can create the money they lend—as fractional reserve banking permits—then the constraint of actual savings is removed. Money becomes an abstraction, created by ledger entries, backed by promises, untethered from the production of real goods.

The eighteenth and nineteenth centuries elaborated these instruments. Central banks multiplied across Europe. Fractional reserve lending became standard practice: banks lent out more than they held in deposits, creating money through the act of lending. National debts grew, funded by bonds that became the foundation of financial markets. The gold standard imposed some discipline—currency was nominally redeemable in precious metal—but the discipline was progressively relaxed and finally abandoned in the twentieth century. Fiat currency, backed by nothing but government decree, became the norm. Money was now purely abstract: a number in an account, a promise from an institution, a claim on future production that might or might not be honored.

The twentieth century completed the transformation. The Federal Reserve, established in 1913, gave the United States a central bank with the power to expand and contract the money supply at will. The abandonment of the gold standard—partially in 1933, completely in 1971—removed the last constraint on money creation. Deficit spending became not merely possible but routine. Governments discovered that they could fund present consumption by borrowing from the future, that they could create money to purchase political support, that the costs would be dispersed through inflation while the benefits would be concentrated among the recipients of spending.

The Consequences

The usury revolution transformed the material conditions of intellectual life. Ideas require resources; resources could now be generated without limit by those who controlled the mechanisms of credit creation. The long game—patient investment over generations to capture institutions and shape minds—became possible in a way it had never been before.

Consider what is required to shift the intellectual orientation of a civilization. Scholars must be funded; chairs must be endowed; journals must be subsidized; books must be published; students must be supported. The process takes decades at minimum, generations in full. It requires patient capital, deployed consistently, according to a long-term strategy. Under the old dispensation—when wealth accumulated slowly through production and trade, when lending at interest was restricted, when money could not be created by fiat—such a project was difficult to sustain. Patrons died; fortunes dispersed; priorities shifted.

The usury revolution removed these constraints. Those who controlled credit creation had access to functionally unlimited resources. They could fund the salons, the academies, the journals, the chairs. They could sustain the funding across generations, with compound interest working in their favor. They could outspend any opponent operating on honest money and real savings. The tradition’s patrons—the old aristocracy, the Church—were increasingly constrained by the new financial order. The Enlightenment’s patrons had discovered infinite leverage.

This is not to reduce the intellectual contest to mere economics. The ideas mattered; the arguments mattered. But ideas need vectors, arguments need platforms, and truth needs defenders who can sustain the fight. The tradition brought dialectic to a financial war. It was outspent before it was outargued.

The consequences extend beyond the propagation of ideas. Usury transforms the structure of society. Wealth flows from debtors to creditors, from the productive to the financial, from the young to the old. Communities that once owned their land and tools become tenants and employees. Independence gives way to dependence; proprietorship gives way to wage labor; stability gives way to the anxiety of those who owe more than they own.

The Enlightenment promised liberation; the usury that funded it delivered a new form of bondage. The serf owed labor to his lord; the modern debtor owes money to institutions he has never seen, created through mechanisms he does not understand, compounding at rates that ensure the debt can never be fully repaid. The chains are invisible, but they are chains nonetheless.

The Inversion Complete

The trajectory is now complete. What was prohibited has become mandatory. Modern economies do not merely permit usury; they require it. The entire financial system rests on debt: consumer debt, corporate debt, government debt. Money itself is debt—a liability of the central bank, created through lending, destroyed through repayment. An economy that repaid its debts would be an economy without money. The system requires perpetual expansion of debt to function; deleveraging is not an option but a crisis.

What was vice has become virtue. Borrowing is “investment.” Saving is “hoarding.” The debtor is a contributor to economic growth; the saver is an obstacle to prosperity. The moral vocabulary has been inverted along with the practice. Prudence, the ancient virtue of providing for the future, is now deemed to be an economic drag. Profligacy, once considered the ancient vice of consuming beyond one’s means, has become the primary engine of economic growth through consumer and government spending.

The Enlightenment’s intellectual victory was underwritten by this financial revolution. The ideas could not have propagated without the resources; the resources could not have been generated without the legitimization of usury; the legitimization of usury required the abandonment of the tradition’s moral and economic framework. The battles were connected. The tradition lost on multiple fronts simultaneously, and the losses reinforced one another.

Understanding this history is essential for any project of renewal. The tradition was not merely out-argued; it was out-spent. Any attempt to recover what was lost must reckon with the material conditions of intellectual life. Ideas need institutions; institutions need funding; funding, in the modern world, is controlled by those who control credit. The tradition cannot simply reassert its truths and expect them to prevail. It must build alternative structures, cultivate alternative resources, play the long game with the same patience and persistence that its opponents displayed.

The usury revolution was not incidental to the Enlightenment’s triumph. It was foundational. And the financial, social, and moral consequences of its acceptance remain with us, shaping the conditions under which any attempt at civilizational renewal must operate.

DISCUSS ON SG

The Outing of Nick Fuentes

In possibly the least surprising confirmation of what everyone has known for at least two years, Nick Fuentes is confirmed to be gay by one of his longtime associates.

I’ve already had an extremely successful career and do the podcast as a hobby. I’ve been nothing but respectful to Nick and got him on Big Podcast when nobody would touch him. I even ignored him being as homosexual (you all know This is true) but apparently he wants war with me.

I don’t know if Nick is also a Fed or not, but I’ve always known that he’s a fake. This is always the case when someone rapidly comes up out of nowhere. So the idea that a manufactured Clown World puppet would be fake and gay is not exactly surprising.

DISCUSS ON SG

Mailvox: the N/Ne Divergence

It’s easy to get distracted by the floundering of the critics, but those who have read and understood Probability Zero and The Frozen Gene are already beginning to make profitable use of them. For example, CN wanted to verify my falsification of Kimura’s fixation equation, so he did a study on whether N really was confirmed to be reliably different than Ne. His results are a conclusive affirmation of my assertion that the Kimura fixation equation is guaranteed to produce erroneous results and has been producing erroneous results for the last 58 years.

I’ll admit it’s rather amusing to contrast the mathematical ineptitude of the critics with readers who actually know their way around a calculator.

The purpose of this analysis is to derive a time‑averaged census population size, Navg for the human lineage and to use it as a diagnostic comparator for empirically inferred effective population size Ne.

The motivation is that Ne is commonly interpreted—explicitly or implicitly—as reflecting a long‑term or historical population size. If that interpretation is valid, then Ne should be meaningfully related to an explicit time‑average of census population size Nt. Computing Navg from known census estimates removes ambiguity about what “long‑term” means and allows a direct comparison.

Importantly, Navg is not proposed as a replacement for Ne in population‑genetic equations. It is used strictly as a bookkeeping quantity to test whether Ne corresponds to any reasonable long‑term average of census population size or not.

Definition and derivation of Navg

Let Nt denote the census population size at time t, measured backward from the present, with t=0 at present and T>0 in the past.

For any starting time ti, define the time‑averaged census population size from ti to the present as:

Because Nt is only known at discrete historical points, the integral is evaluated using a piecewise linear approximation:

Select a set of times at which census population estimates are available.
Linearly interpolate Nt between adjacent points.
Integrate each segment exactly.
Divide by the total elapsed time ti

This produces an explicit, reproducible value of Navg for each starting time ti.

Census anchors used

Census population sizes Nt are taken from published historical and prehistoric estimates.
Where a range is reported, low / mid / high scenarios are retained.
For periods of hominin coexistence (e.g. Neanderthals), census counts are summed to represent the total human lineage.
No effective sizes Ne are used in the construction of Nt.

Present is taken as 2026 CE.

Results: Navg from ti to present

All values are people.
Nti is the census size at the start time.
Navg is the time‑average from ti to 2026 CE.

Start time ti	Years before present	Nti (low / mid / high)	Navg(ti – present) (low / mid / high)
2,000,000 BP (H. erectus)	2,000,000	500,000 / 600,000 / 700,000	2.48 M / 2.86 M / 3.24 M
50,000 BCE (sapiens + Neanderthals)	52,026	2.01 M / 2.04 M / 2.07 M	48.5 M / 60.6 M / 72.7 M
10,000 BCE (early Holocene)	12,026	5.0 M / 5.0 M / 5.0 M	198 M / 250 M / 303 M
1 CE	2,025	170 M / 250 M / 330 M	745 M / 858 M / 970 M
1800 CE	226	813 M / 969 M / 1.125 B	2.76 B / 2.83 B / 2.90 B
1900 CE	126	1.55 B / 1.66 B / 1.76 B	4.02 B / 4.04 B / 4.06 B
1950 CE	76	2.50 B / 2.50 B / 2.50 B	5.33 B (all cases)
2000 CE	26	6.17 B / 6.17 B / 6.17 B	7.24 B (all cases)

Interpretation for comparison with Ne

Navg is orders of magnitude larger than empirical human Ne, typically ~10(4) for all plausible averaging windows.
This remains true even when averaging over millions of years and even under conservative census assumptions.
Therefore, Ne cannot be interpreted as:
- an average census size,
- a long‑term census proxy,
- or a time‑integrated representation of Nt

The comparison Navg > Ne holds regardless of where the averaging window begins, reinforcing the conclusion that Ne is not a demographic population size but a fitted parameter summarizing drift under complex, non‑stationary dynamics.

Kimura’s cancellation requires N = Ne. CN has shown that N ≠ Ne at every point in human history, under every averaging window, by orders of magnitude. The cancellation has never been valid. It was never a simplifying assumption that happened to be approximately true, it was always wrong, and it was always substantially wrong.

The elegance of k = μ was its selling point. Population size drops out! The substitution rate is universal! The molecular clock ticks independent of demography! It was too beautiful not to be true—except it isn’t true, because it depends on a variable identity that has never held for any sexually reproducing organism with census populations larger than its effective population. Which is all of them.

And the error doesn’t oscillate or self-correct over time. N is always larger than Ne—always, in every species, in every era. Reproductive variance, population structure, and fluctuating population size all push Ne below N. There’s no compensating mechanism that pushes Ne above N. The error is systematic and unidirectional.

Which means every molecular clock calibration built on k = μ is wrong. Every divergence time estimated from neutral substitution rates carries this error. Every paper that uses Kimura’s framework to predict expected divergence between species has been using a formula that was derived from an assumption that the author’s own model parameters demonstrate to be false.

DISCUSS ON SG

A Tale of the Council of Elrond

The morning light fell upon Rivendell like a benediction, gold and pale through the leaves of the ancient trees, and the sound of waterfalls threaded through the air like music half-remembered. The council had been called in the great terrace overlooking the valley, and representatives of every Free People sat arranged in a wide crescent of carved chairs. Elves of Rivendell and the Woodland Realm, Dwarves from Erebor, Men of Gondor and the wild North, and a Hobbit who looked as if he very much wished he were anywhere else.

Frodo Baggins sat in a chair that was slightly too tall for him and tried not to let his feet swing. Beside him, Gandalf the Grey leaned on his staff and surveyed the assembly with an expression Frodo had learned, over many months, to associate with a man who has already made up his mind but intends to let everyone else talk themselves into exhaustion first.

Elrond Half-elven stood and opened the proceedings with a history of the Ring. He spoke at considerable length. He spoke of Sauron’s forging of the One in the fires of Orodruin, of the Last Alliance and the fall of Gil-galad, of Isildur’s bane and the creature Gollum and the extraordinary improbability of the Ring passing to a Hobbit of the Shire. He spoke with the unhurried gravity of someone who has lived six thousand years and sees no reason to abridge.

Boromir, son of Denethor, shifted in his seat. He had ridden many weeks from Minas Tirith and was not accustomed to being a member of an audience.

“Let us use the Ring against Sauron,” he said, at the first breath Elrond drew. “Give it to the armies of Gondor and let us —”

“No,” said Elrond.

“But —”

“No.”

Gandalf lifted one hand. “Boromir. The Ring answers to Sauron alone. Any who wield it will be consumed by it. It cannot be used. It can only be destroyed.”

“And it can only be destroyed in the place where it was made,” said Elrond. “In the fires of Mount Doom, in the land of Mordor.”

A silence followed this pronouncement — or rather, a silence attempted to follow it, but was immediately interrupted by several people speaking at once. Gimli the Dwarf suggested that they simply smash the thing with an axe, but when this was attempted, the axe shattered spectacularly and Gimli sat down again looking more than a little chagrined. Legolas mentioned that the Elves would never be safe while the Ring endured. Boromir brought up Gondor’s need again, and once more, everyone ignored him.

Through all of this, Frodo felt the Ring against his chest, hanging on its chain, and a strange certainty had been growing in him since before the council began. It was the kind of certainty that arrives not as a comfort but as a weight, pressing down on the shoulders with quiet and terrible patience. He knew, with a clarity that surprised him, what he was going to say. He had known it, perhaps, since Weathertop, or since the Ford, or since the day Bilbo had given him the Ring and gone away.

He stood up.

“I will take the Ring to Mordor.”

The words fell into the assembly like a stone into a pond. Frodo felt every eye turn to him — the tall, ageless eyes of the Elves, the shrewd eyes of the Dwarves, the complicated eyes of Aragorn, the frankly skeptical eyes of Boromir. He drew a breath. His voice, when it came again, was small but steady.

“I will take the Ring to Mordor. Though I do not know the way.”

He stood there in the silence that followed, three feet six inches of determination, and waited for someone to say something. The moment stretched. Gandalf was looking at him with an expression that was not quite pity and not quite admiration and was, if Frodo was reading it correctly, largely preoccupied with something else entirely.

“That is a very noble offer, Frodo,” said Gandalf.

“Thank you,” said Frodo.

“Very noble. Very brave. And completely unnecessary.”

Frodo blinked. “I’m sorry?”

Gandalf rose from his seat and addressed the council with the air of a man who has been waiting for exactly the right moment and is rather pleased with himself for having found it.

“My friends,” he said. “As many of you know, I was recently imprisoned atop the tower of Orthanc by Saruman the White, who has turned to darkness and now serves the Enemy. I was rescued from that imprisonment by Gwaihir the Windlord, the chieftan of the Eagles of the Misty Mountains.”

“We are aware,” said Elrond, with the faintest trace of impatience.

“Gwaihir bore me through the sky at tremendous speed,” Gandalf continued, as if Elrond had not spoken. “From Orthanc to the fields of Rohan in a matter of hours. A journey that would take a company on foot many weeks, if not months, and which would require passage through some of the most dangerous territory in Middle-earth.”

He paused and looked around the council with bright, expectant eyes.

“The distance from here to Mordor is approximately four hundred leagues,” he said. “On foot, through the wilderness, over mountains and through marshes, past enemy fortifications and patrolled borders, the journey would take months. It would be fraught with danger at every step. The Ring-bearer would need a company of protectors. Even then, the odds of success would be vanishingly small.”

Aragorn was watching Gandalf with an expression of dawning comprehension. Frodo was watching him with an expression of dawning alarm.

“Gwaihir,” said Gandalf, “can fly four hundred leagues in less than a day.”

The silence that followed this statement was qualitatively different from the silences that had preceded it. It was the silence of an idea so obvious that everyone present was rapidly calculating whether they could claim to have thought of it first.

“The eagles,” said Elrond.

“The eagles,” said Gandalf.

“Gandalf,” said Frodo, and there was a faint note of desperation in his voice that he was not entirely proud of. “I said that I would take the Ring. I have offered to bear it.”

“And it was a magnificent offer,” said Gandalf warmly. “Truly. The courage of Hobbits never ceases to amaze me. But consider, Frodo — you would walk for months through trackless wilderness, facing Ringwraiths and Orcs and untold hardship, when instead we might simply have the Ring flown directly to Mount Doom in the span of an afternoon.”

“But surely,” said Boromir, who had been growing increasingly restless, “the Enemy would see an eagle approaching. His Eye watches from the tower of Barad-dûr. The Nazgûl ride fell beasts through the air. An eagle would be spotted and intercepted.”

Gandalf smiled. “Gwaihir flies higher than any fell beast can reach. The eagles are creatures of the high airs, the uttermost peaks. The Nazgûl patrol the lower skies on their winged mounts, but they cannot match the altitude or speed of one of the Great Eagles. Gwaihir could fly above the very clouds, invisible from below, and descend upon Orodruin before Sauron could muster his response.”

“But the entrance,” said Gimli, who was a practical sort. “The Sammath Naur — the Crack of Doom — it is within the mountain. Can an eagle enter it?”

Every head turned to Elrond. The lord of Rivendell was quiet for a long moment. His eyes had gone distant, as they did when he was consulting the vast and impeccably organized archive of his memory.

“I have been to Orodruin,” he said at last. “I stood at the threshold of the Sammath Naur with Isildur after the fall of Sauron. I recall the entrance well.” He paused. “It is wide. Very wide. It was carved — or rather, torn open — by volcanic force. The passage into the mountain is high-vaulted and broad. An eagle, even one of the Great Eagles, with a wingspan of some thirty fathoms —” He paused again, and there was something almost reluctant in his voice, as if he would have preferred the logistics to be more complicated. “An eagle could enter it. With room to spare.”

“There you are,” said Gandalf.

Frodo sat down slowly. He was experiencing an emotion he could not quite name — something between relief and an obscure sense of redundancy, as if he had spent weeks steeling himself to lift a great boulder only to watch someone roll it aside with a lever.

“I should like to ride the eagle,” said Aragorn. “I can bear the Ring.”

This declaration produced another brief silence, though of a different character. Aragorn, heir of Isildur, Chieftain of the Dúnedain, sat straight-backed in his chair with the composed dignity of a man who has spent decades wandering the wild places of the world in deliberate preparation for a moment of destiny and does not intend to be left out of it on a technicality.

“Someone must ensure that the Ring is cast into the fire,” he said. “The eagle cannot do it alone. It has no hands. I will ride Gwaihir into Mordor, bearing the Ring, and throw it into the Crack of Doom myself.”

“A brave proposal and one well worthy of your line,” said Gandalf. “But consider: you are the heir of Isildur. Isildur himself could not resist the Ring’s call. The Ring would know you. It would whisper to you of the throne of Gondor, of the reunited kingdoms, of your right to rule Middle Earth. The temptation, for you above all others, would be —”

“I can resist it,” said Aragorn firmly.

“With all respect, my son,” said Elrond, and the phrase carried the particular weight it always does when spoken by someone who has watched civilizations rise and fall, “that is what Isildur thought too.”

Aragorn’s jaw tightened, but he did not argue further and nodded in silent acquiescence. He knew the history as well as anyone.

“This raises the essential question,” said Gandalf. “Who — or what — should bear the Ring on this flight? The great advantage of the eagle is not merely its speed. It is resistance. Gwaihir is not a creature of ambition. He desires no kingdom, no power, no dominion over others. He is a bird. An exceedingly large and noble bird, to be sure, but a bird nonetheless. The Ring’s power lies in its appeal to the will — to the desire for mastery. What does an eagle desire? Updrafts. Thermals. The occasional mountain goat. The Ring would have very little purchase on such a mind.”

“You are suggesting,” said Elrond, “that we tie the Ring to an eagle’s leg and let it fly unaccompanied into the heart of Mordor?”

“I am suggesting,” said Gandalf, “that we place the Ring in a pouch secured to Gwaihir’s talons, and that Gwaihir fly at maximum altitude directly to Orodruin, enter the Sammath Naur, and release the pouch into the fire. The entire operation need take no more than six hours.”

“And if the Ring tempts the eagle to turn aside?” asked Legolas.

“To what end?” said Gandalf. “What would the Ring promise an eagle? Dominion over the skies? Gwaihir already has that. A hoard of gold? Eagles have no use for gold. An army of servants? Eagles are solitary creatures who find the company of most other beings tedious. The Ring’s entire mechanism of corruption depends on exploiting desire, and the desires of an eagle are so thoroughly alien to the desires of the Ring’s maker that the two are, for all practical purposes, incompatible.”

“The wind does not desire a crown,” murmured Elrond, and something in his ancient voice suggested that he was quite taken with the elegance of this.

“But the Quest,” said Frodo. He was aware that his voice sounded rather small. “The journey. The sacrifice. Bilbo always said that adventures were the making of a Hobbit —”

“Bilbo,” said Gandalf gently, “also said that adventures made you late for dinner. I think, Frodo, that in this case, being home in time for dinner is rather the point.”

Sam Gamgee, who had been lurking behind a pillar in open defiance of the council’s protocols, leaned forward and whispered, “He’s got you there, Mr. Frodo.”

Frodo looked around the council one last time. He saw the faces of the great and the wise, the warriors and the kings, and on every one of them he saw the same expression: the faintly embarrassed recognition that the answer had been, all along, absurdly simple.

“Then it is decided,” said Elrond, rising. “Gwaihir the Windlord shall bear the One Ring to Orodruin. Gandalf shall speak with him and make the arrangements. The Ring shall be secured to his person by means yet to be determined — I suggest we consult with the leatherworkers of my household — and he shall depart at first light tomorrow.”

“And the rest of us?” said Boromir, who looked as if he had been cheated of something but was not entirely sure what.

“The rest of us,” said Elrond, “shall wait.”

“I hate waiting,” said Gimli.

“You may pass the time in my halls,” said Elrond. “The kitchens are beyond compare. The library is extensive. The gardens are in late bloom.”

“I was willing to carry the Ring,” said Frodo quietly, to no one in particular.

Gandalf placed a hand on his shoulder. “And that willingness, Frodo, is precisely why you were the right Hobbit to offer. The courage to give one’s life is no less real for being, in the end, unnecessary. You would have carried the Ring all the way to Mordor on foot, through fire and darkness, and that is a thing worth honoring.”

“But you’re not going to let me.”

“No. Most certainly not.”

Frodo looked up at the sky, where high above the valley of Rivendell, a distant shape circled on broad wings in the morning light. It was Gwaihir, called by some means that Gandalf had no doubt arranged in advance, already descending toward the terrace with the unhurried confidence of a creature who has never in his long life had reason to fear anything below him.

“Right,” said Frodo. “Well. I suppose I’ll have another cup of tea, then.”

And the council, having solved in a single morning the problem that would have otherwise consumed the better part of a year and the lives of a considerable number of good people, adjourned for an early lunch.

In the Tower of Barad-dûr

DISCUSS ON SG

Veriphysics: The Treatise 004

V. The Economic Failures

The Enlightenment extended its confidence to the economic realm. Just as reason could discern the laws of nature and the principles of just government, so too could it uncover the mechanisms by which wealth is created and distributed. The result was classical economics, with its promise of prosperity through rational organization of production and exchange.

Adam Smith’s Wealth of Nations, published in 1776, became the founding text of this enterprise. At its heart lay the law of supply and demand: the elegant mechanism by which prices adjust to balance what producers offer and what consumers desire. Let the market operate freely, Smith argued, and an invisible hand would guide individual self-interest toward collective prosperity. The baker bakes not from benevolence but from self-love, and yet we all have bread.

The law of supply and demand became the bedrock of economic reasoning. It appeared in every textbook, was taught in every university, and informed the policy of every nation that aspired to modernity. For two centuries, it seemed as solid as Newton’s laws.

It was an illusion. In 1953, the economist William Gorman demonstrated mathematically that individual demand curves cannot be aggregated into a coherent market demand curve under the conditions that actually obtain in real economies. The proof is technical, but its implications are devastating: the supply and demand curves that generations of economists drew on their chalkboards, the intersecting lines that determined equilibrium prices, do not correspond to anything that exists in actual markets. The law of supply and demand, as commonly understood, is not a law at all. It is a pedagogical simplification that fails precisely when applied to the phenomena it was meant to explain.

This was not a minor qualification or a boundary case. It was a falsification of the foundational model. Yet the economics profession continued teaching supply and demand as though Gorman had never written. Decades later, Steve Keen brought Gorman’s work to wider attention and documented the discipline’s remarkable capacity to ignore what it could not answer. The emperor had been shown to be naked in 1953, even though in 2025, the textbooks still describe his magnificent robes that supposedly improve things for everyone.

David Ricardo’s theory of comparative advantage suffered a similar fate. Published in 1817, the theory purported to demonstrate that free trade benefits all parties, even when one nation is more efficient at producing everything. Each nation should specialize in what it produces relatively best, trade for the rest, and all will prosper. This elegant argument became the intellectual foundation of free trade policy for two centuries.

The argument contains a fatal assumption: that the factors of production, and especially labor, do not move between nations. Ricardo’s proof works only if English cloth-workers cannot become Portuguese wine-makers, and vice versa. In the early nineteenth century, this assumption was approximately true. In the twenty-first century, it is obviously false. Cheap transportation and communication have made labor mobility a defining feature of the global economy. The assumption upon which the entire edifice rests no longer obtains, and with it falls the conclusion.

Ian Fletcher systematically demolished the theoretical foundations of comparative advantage. The assumptions required for the theory to hold—not only labor immobility but perfect competition, no economies of scale, no externalities, no strategic behavior—describe no economy that has ever existed. More recently, Steve Keen has identified the amphiboly that rendered the proof invalid from the start. Comparative advantage is not a law of nature; it is a fictional fantasy describing a hypothetical world, and our world is not that world.

The empirical verdict has been equally damning. After three decades of trade agreements, including NAFTA, the EEA, and the WTO, the prosperity that was promised by free trade has proven highly selective. The nations that preached free trade most fervently have watched their manufacturing bases erode, their working classes immiserated, their trade deficits balloon. The nations that practiced strategic protectionism have prospered at the expense of those who didn’t. The correlation between free trade ideology and the flourishing of a nation runs precisely opposite to what the theory predicts.

No one who has watched the hollowing-out of the American industrial heartland, the stagnation of Western wages, the rise of the Chinese export machine, can believe that free trade has delivered on its promises. The economists who assured the public that the gains would be shared, that the dislocations would be temporary, that retraining would absorb the displaced, these economists were not necessarily all lying. But they were reasoning from simplified models that did not describe reality and mistook the coherent elegance of their mathematics for the truth.

And finally, for three hundred years, we have been assured by the economists that debt did not matter. They even omitted it from their most complicated equations and declared that it did not matter if Peter owed Paul or Paul owed Peter, that debt was just a variable on both sides of the equation that cancelled itself out. Now the entire Western world awash in debts it cannot pay and institutional investors now own 20 million private homes, 15 percent of the total housing stock in the United States.

This, too, is a consequence of the Enlightenment’s successful war on the laws that once prevented people from falling into debt servitude. Now debts are increasingly noncancelable even by bankruptcy, the total U.S. debt is $106 trillion, and each and every native-born U.S. citizen’s share of that debt is $365,500. Instead of making everyone wealthy as was promised, the economics of the Enlightenment have turned a once-free people into a collection of debt slaves.

DISCUSS ON SG

Rod Walker	Nick Cole
Martin van Creveld	Jonathan Moeller
John C. Wright	Rolf Nelson
Mike Cernovich	David the Good
David VanDyke	B.V. Larson
Jerry Pournelle	Cheah Kai Wai
Peter Grant	Sarah Salviander
Ivan Throne	Steve Keen
Robert Beale	LawDog
C.R. Hallpike	Jon Del Arroz