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.
Zero fixations was the prediction
The empirical leg fails more starkly. I treated zero fixations across a million loci in 240 generations as an astonishing result demanding explanation. It is the opposite: it is what the theory predicts, and the prediction is not close.
Neutral fixation from intermediate frequency takes on the order of 4N<sub>e</sub> generations. For a human effective size around ten thousand, that is roughly forty thousand generations — about a million years. Two hundred and forty generations is six-tenths of one percent of the way there.
Running the absorption probabilities at that scaling, an allele sitting at frequency 0.5 has a probability of fixing within 240 generations of around 4 × 10⁻³⁵. At frequency 0.1 it is 6 × 10⁻⁹³. Only alleles already above about 0.99 have any meaningful chance at all. Integrated across a million loci starting from intermediate frequencies, the expected number of fixations is somewhere near 10⁻²⁹.
Finding zero is not a coin landing on its edge. Finding one would have been the falsification. The dataset cannot discriminate between neutral theory and its rivals over that window, because every candidate theory predicts the same thing. What the window can resolve is frequency drift — the neutral expectation is a standard deviation of about 0.05 at p = 0.5 over 240 generations — and that is the analysis that should have been run.
Where the epicycle actually is
Having conceded both legs, I want to defend the instinct. The suspicion that this field carries Ptolemaic freight is correct. I was pointing it at the wrong object.
An epicycle is a free parameter introduced to preserve a prediction against contrary observation, adjustable after the fact, with no independent measurement to constrain it. On that definition, the recent mathematical work on the boundary behaviour of the Kimura equation is not epicyclic. Nothing there is fitted to data. The point masses at the boundaries are forced by conservation of probability, and the fixation probability that falls out matches a derivation made independently sixty years earlier, by the backward equation rather than the forward one. A consistency check that closes is the antithesis of an epicycle.
But look at N<sub>e</sub> itself. There is not one effective population size; there are at least half a dozen — inbreeding, variance, eigenvalue, coalescent, linkage-disequilibrium — which disagree with one another, sometimes by orders of magnitude, in the same population. Each is estimated from the quantity it is then used to explain. It is the parameter that absorbs everything the model does not handle: population structure, reproductive skew, fluctuating size, linked selection. That is precisely the epicyclic form.
The honest way to say it is that N<sub>e</sub> is not a parameter of a real population at all. It is a family of moment-matching estimators — each one the size of the ideal Wright–Fisher population that matches the real one on one specified property. Inbreeding N<sub>e</sub> matches the rate of increase in inbreeding; variance N<sub>e</sub> matches the variance in allele frequency change; coalescent N<sub>e</sub> matches pairwise coalescence time. They coincide only for the ideal population and diverge predictably when specific assumptions fail. The equivocation is nomenclature, and the fix is subscripts.
And look at the clock. Direct pedigree sequencing of parent–offspring trios gives a human germline mutation rate near 1.2 × 10⁻⁸ per site per generation. Calibrating against the fossil-dated human–chimpanzee split gives something closer to twice that. The two numbers have not been reconciled; they have been managed, with adjustments for generation time, for life-history change, for calibration uncertainty, each introduced to close a gap rather than measured independently and found to close it.
Worse for the clock’s foundations: mutation is not a per-generation constant. Something like four-fifths of de novo mutations are paternal, and the paternal contribution rises with the father’s age at conception at roughly one to two mutations per year. The mutation rate is therefore a function of the age at which men in a given society father children — which is to say, a function of marriage patterns, economic conditions, and religion. The molecular clock ticks at a rate set partly by social history. That is a considerably more radical claim than anything in my February essay, and unlike that essay’s claims it is well documented.
The cold equations, inverted
Which brings me to the argument I actually wanted to make, and to a result I did not expect.
The standard worry runs: modern medicine and abundance have switched off the selective filter. Childhood mortality has fallen from something like forty percent to under one percent in the developed world. Deleterious mutations that once killed before reproduction now do not. Lynch put a number on it in Genetics in 2016 — roughly one percent decline in fitness per generation under fully relaxed selection — and drew a sharp published rebuttal from Roth and Wakeley arguing the load calculation is invalid as constructed. The dispute is unresolved and the territory is politically radioactive, which is not a reason to avoid the arithmetic.
Here is the arithmetic. Crow’s index of the opportunity for selection is I = V_k / k̄²: the variance in completed offspring number divided by the square of the mean. Note the denominator. Mean fertility appears squared, and it appears below the line.
Run it across the transition. Pre-transition, mean around 5.5 with variance around 12: I ≈ 0.40. Mid-century, mean 3.0, variance 4.0: I ≈ 0.44. New Zealand today, TFR near 1.56: I ≈ 0.78. South Korea at 0.72: I ≈ 1.74.
Under a Poisson benchmark, where variance equals mean, the index collapses to I = 1/k̄ exactly — and sub-replacement fertility therefore produces more opportunity for selection than high fertility, mechanically, as a matter of arithmetic.
I should say that the qualitative version of this point is not mine. Roth and Wakeley made it against Lynch in 2016, in the same exchange: Lynch’s claim about “the two-child syndrome in middle-class neighborhoods” cites no evidence that the coefficient of variation in offspring number has actually fallen, and the fraction of American women aged 40–44 who were childless rose from 10% in 1976 to 18% in 2008. Their conclusion was that offspring number remains highly variable and selection may still be effective.
The arithmetic above sharpens that from remains variable to becomes more variable, and shows why: the mean is in the denominator, squared. Rising childlessness and falling mean fertility both push the index up. Roth and Wakeley were arguing that Lynch had not established his premise; the Crow index says the premise is inverted.
The filter has not been switched off. It has been switched over. Viability selection — the cold equation that killed children before they reproduced — is very nearly gone in the developed world, and Lynch is right about that much. But fertility selection, the differential between those who have four grandchildren and those who have none, is at its strongest in recorded demographic history precisely because the mean has collapsed toward zero.
The consequential question is therefore not whether selection is operating. It is what it is now selecting for. Whatever predicts having children in a society where children are optional, expensive and culturally discounted is not what predicted surviving dysentery at age three. The two filters have almost no overlap in the variance they act on. Medicine relaxed the first; the demographic transition tightened the second and pointed it somewhere entirely new.
What I should have written
There is a version of the February essay that survives all of this, and it was available at the time. It goes: the mathematics of neutral theory is in good order, and the interesting instability is one storey up, in the parameters we feed it and the social facts that set them. N<sub>e</sub> is a name for six things. The mutation rate is a function of paternal age and therefore of marriage custom. And the selective filter has not lifted but rotated, onto a target we have not characterised.
That essay would have required conceding that the specific technical objection I had adopted was wrong. I did not do the work to find that out before publishing, and the work took an afternoon.
The claim about the substitution rate is withdrawn. The claim about the ancient-DNA finding is withdrawn. The demographic work stands, with its interpretation revised: it is a finding about diversity and coalescent depth, not about substitution.
A longer piece on what replaces this — a lineage-level replacement criterion, and what it shows when you point it at the nineteenth century — follows shortly.