The Ancestors Who Left Nothing

Essay · biology · September 2026

Thirty generations back you have a billion slots in your family tree and about two thousand pieces of DNA to fill them with. Everyone of European descent is descended from Charlemagne; almost no one carries anything of his. Ancestry runs on two clocks, one exponential and one linear, and the gap between them is where genealogy and genetics part company — and where the newest object in population genetics, a graph that holds the whole history of a species’ DNA, comes in.

Reading time ≈ 25 min Donnelly 1983 → Chang 1999 → Wohns 2022 A standalone essay · see also The Complete Parts List and The Frozen Accident

Two clocks

You have two parents, four grandparents, eight great-grandparents. The sequence is the most familiar one in arithmetic, and it is also the fastest, and the two facts do not sit comfortably together. Ten generations back, around the time of the Thirty Years’ War, you have 1,024 slots in your family tree. Twenty back, a little over a million. Thirty back, in the twelfth century, you have 1,073,741,824 slots, and the whole world at that time held perhaps 350 million people. Forty generations back, in the reign of Charlemagne, the tree has a trillion slots and the planet has a few hundred million people to put in them.[1]

The arithmetic is not wrong; it is counting slots, not people. Long before the tree outgrows the planet, the same individuals begin to appear in it more than once, through different lines, and the number of distinct people in a generation of your ancestry stops doubling and flattens out. Genealogists call this pedigree collapse, a term coined by Robert Gunderson, who spent a career on the ancestry of American families and noticed that past a certain depth every line he followed ran into every other.[2] The consequence, once you follow it through, is startling and has been known in outline for a century: go back far enough and every person who left descendants at all is an ancestor of everyone alive today. Charlemagne, who died in 814 and had at least eighteen children, is the standard example. If you have European ancestry, he is in your tree. So, with the same certainty, is nearly every peasant in his empire whose line did not die out.[3]

That is one clock, and it runs exponentially until the world stops it. The other clock is the one your DNA runs on, and it is linear. Your genome does not double its sources every generation, because it is not a tree of copies; it is a finite object, cut into a slowly growing number of pieces, and each piece comes from exactly one person in each generation. By the time the pedigree has a billion slots, the genome has about two thousand pieces to distribute among them. The rest of the slots are, genetically, empty. The person in them is your ancestor by every legal and historical definition and has left nothing in you at all.

This essay is about the gap between those two clocks. It works out both of them from first principles, by simulation, and then asks what the gap means: for Charlemagne, for the cousins that a DNA test finds and the ones it cannot, for the phrase “common ancestor”, which turns out to mean three quite different things, and for the object that population genetics now uses to hold all of this at once.

The first clock

A billion slots

The proper treatment of pedigree collapse is surprisingly recent. In 1999 Joseph Chang, a statistician at Yale, wrote down the simplest possible model — a population of fixed size n, each person in each generation choosing two parents at random from the generation before — and proved two things about it.[4] The first is that the most recent person who is an ancestor of everyone alive lived only about log₂ n generations ago. For a million people that is twenty generations; for the whole planet, about thirty-three. The second is that a little further back, at about 1.77 log₂ n generations, the population splits cleanly in two: about 80 per cent of the people alive then are ancestors of every single person alive now, and the remaining 20 per cent are ancestors of no one. There is no middle. Chang called the second date the identical-ancestors point, because past it everyone alive today has exactly the same set of ancestors, and only the number of times each appears in the tree differs from person to person.

The 80 per cent is not a fitted number. It is the probability that a random person’s line of descent does not die out, which in this model is the survival probability of a branching process with an average of two children per person, and it satisfies q = e^{−2(1 − q)} for the extinction probability q. That gives q = 0.2032, and so 0.7968 for the survivors. The same year, and independently, three physicists — Derrida, Manrubia and Zanette — found the same fraction by treating the pedigree as a problem in statistical mechanics, which is a fair indication that the result belongs to the arithmetic rather than to anything about humans.[5]

Figure 3 is Chang’s model run rather than proved, for a population of ten thousand. The share of a past generation who are ancestors of one chosen person climbs from nothing to 80 per cent within about fifteen generations and stays there, exactly on the line. The share who are ancestors of everyone rises a few generations later and meets it. Between them, briefly, is the only period in which it is meaningful to ask whose ancestor a given person is; before it, nobody is anybody’s; after it, everyone who counts is everybody’s. Chang’s constants are limits for large populations; at ten thousand the first universal ancestor arrives at generation 14 against a predicted 13.3, and the identical-ancestors point at about 27 against 23.5, a few generations late, as one expects of a result proved in the limit of large populations.

Fig. 3 — Chang’s theorem, simulated. A population of 10,000 per generation, random mating, two parents each. Going back in time, the share of a generation who are ancestors of one chosen present-day person (black) climbs to about 80 per cent and stops: the other 20 per cent are the lines that died out. The first person who is an ancestor of everyone alive today appears at generation 14 (log₂ of 10,000 is 13.3); by generation 27 every individual is an ancestor of all or of none. Mean of 20 runs; the dashed line is the exact limit 1 − q, where q = e^−2(1−q).

The model is a caricature: real people do not choose parents uniformly from the whole planet, and the whole force of geography, class and language is to make them choose from close by. In 2004 Douglas Rohde, Steve Olson and Chang put the geography back — continents, migration rates between them estimated from history, port towns, the peopling of the Americas and of the Pacific — and ran the model for the real world.[6] The dates moved, but not by much. In their models the most recent common ancestor of everyone now alive lived a few thousand years ago, most likely between two and five thousand, and the identical-ancestors point some thousands of years before that, plausibly in the range of five to fifteen thousand years. The reason is the one Chang’s logarithm already contains: it takes very little mixing to connect two populations, because a single migrant, a few generations on, is ancestral to a large fraction of the place they moved to. The dates are dominated by the last isolated groups to be reached, not by the size of the world.

It follows that “descended from Charlemagne” is not a distinction. Charlemagne lived about forty generations ago, well past the identical-ancestors point for Europe, and the honest statement is that every person of European descent is descended from every eighth-century European who has living descendants, and Charlemagne happens to be the one whose name we know.[3] Something stronger is true. There are a trillion slots in your tree at that depth and perhaps twenty or thirty million distinct people to fill them, so each of those people occupies, on average, tens of thousands of positions in your ancestry. Charlemagne is not your ancestor once. He is your ancestor along tens of thousands of separate lines, and so is the peasant.

The second clock

Seventy cuts a generation

Now the other clock. You carry two copies of each of your twenty-two autosomes, one from each parent, and each copy is not a copy of one of your parent’s chromosomes but a splice of the two they carried. When an egg or a sperm is made, the two copies of each chromosome pair up, exchange material at one or more points, and a single spliced version goes into the cell. The exchange points are called crossovers, and there are not many of them: roughly twenty-seven per genome in a man’s meiosis and forty-four in a woman’s, a difference nobody fully explains, for an average of about thirty-five.[7] Geneticists measure chromosome length in the unit that counts these: one Morgan is the length over which one crossover is expected per meiosis, and the human autosomes total about thirty-five Morgans, sex-averaged, with chromosome 1 at nearly three and chromosome 21 at a little over half a Morgan.

Follow one of your copies of chromosome 1 backwards. It came from your father entire, as a single piece with a single owner. It is a splice of his two copies, so at the grandparents’ generation it is already in pieces, about three of them on a chromosome this long, each owned by one of his parents. Each grandparent’s piece was itself a splice, so at the great-grandparents’ generation there are more pieces, and so on. Every generation adds about as many new cuts as there are Morgans, so across both copies of the whole genome your ancestry gets about seventy new cuts per generation, and the number of pieces at generation g is about 44 + 70(g − 1). That is a straight line. It is competing with 2^g.

Figure 2 shows one simulated chromosome 1, both copies, painted by owner for eight generations. The colours are assigned so that shades of teal are the father’s side and shades of magenta the mother’s; a piece keeps its position on the bar and changes colour only when a crossover in some ancestor’s meiosis hands it to a different grandparent. By the fourth row, sixteen ancestors are sharing nineteen pieces; by the eighth, 256 ancestors are sharing forty-nine, and it is already plain that most of them hold nothing on this chromosome.

Fig. 2 — One chromosome, eight generations, one simulated pedigree. Each row is your two copies of chromosome 1, the one from your father above and the one from your mother below, coloured by the ancestor each stretch descends from at that generation: teal shades on the father’s side, magenta on the mother’s. At the right, the number of pieces against the number of ancestors in that generation. The pieces grow by about six a row, because the two copies take about six crossovers a generation between them on a chromosome this long; the ancestors double. By the eighth row most of the 256 ancestors own nothing here.

Add the other twenty-one chromosomes and count, over many simulated pedigrees, how many distinct ancestors in each generation own at least one piece of your genome anywhere. That is the second clock, and Figure 1 draws it against the first. The two agree perfectly for five generations: all thirty-two great-great-great-grandparents contribute. At six, one in a hundred slots is empty; at eight, one in six; at ten, more than half. At 453 of your 1,024 tenth-generation ancestors — a typical figure; the spread between simulated pedigrees is about 15 either way — have left a piece of DNA in you, and the other 571 have left none. By fifteen generations, 938 of 32,768 slots are filled. By twenty, 1,307 of a million. By thirty, 2,017 of a billion. By forty, the Charlemagne depth, 2,724 of a trillion.

Fig. 1 — Two clocks. The dashed line counts the slots in a pedigree, 2^g at generation g. The solid lines count what the genome can fill them with: the number of pieces the autosomes have been cut into by g generations of recombination, and the number of distinct ancestors who contribute at least one piece. Mean of 300 simulated pedigrees. By ten generations most slots are already empty; by forty, all but about 2,724 of a trillion are.

The calculation is not new; only the framing is. Peter Donnelly worked out in 1983 the probability that two relatives share any stretch of genome by descent, which is the same problem seen from the side, and Carsten Wiuf and Jotun Hein in 1997 counted the ancestors of a single DNA sequence and found that the count grows linearly in time and then saturates.[8] Graham Coop, at Davis, put the numbers in front of a general audience in 2013 in a series of posts that are still the clearest treatment, and the figures here reproduce his by an independent route.[9] Simon Gravel and Mike Steel later gave the missing ancestors a name, ghost ancestors, and showed that in a finite population the number of genealogical ancestors who are also genetic ones does not merely fall as a fraction but is bounded above by the number of pieces, whatever the population does.[10]

The pieces are not shared evenly, either. Among the tenth-generation ancestors who do contribute, the largest single share in a typical pedigree is about 1.4 per cent of your genome and the median about 0.16 per cent, which on a genome of some seven thousand centimorgans across both copies is a stretch of ten centimorgans or so: one piece, on one chromosome, from one person, and nothing else. Being a genetic ancestor at that depth mostly means having left one fragment. The slot in the pedigree says great-great-great-great-great-great-great-great-grandparent. The DNA says a paragraph.

Charlemagne, arithmetically

In the tree, not in the blood

Put the two clocks together and the emperor’s case can be settled. At forty generations your genome is in about 2,802 pieces, owned by about 2,724 people. Your genealogical ancestors in that generation are, past the identical-ancestors point, essentially the whole surviving population of early medieval Europe, of the order of twenty to thirty million. If the pieces were handed out uniformly among them, the chance that a named individual holds one is about 2,724 divided by twenty-five million, a little over one in ten thousand. Charlemagne’s tens of thousands of positions in your tree are already counted in that figure; his share of your slots is what the average ancestor gets, and it buys him one chance in ten thousand of a fragment. Kings outbreed peasants, and the tail of his descent is longer than the average person’s, so the honest range is one in ten thousand to perhaps one in a few hundred. Either way, nearly everyone of European descent is descended from Charlemagne and nearly no one carries anything of him. Both sentences are exact, and they are about different things.

It is worth being clear about what has been computed, because the pedigree simulation makes one simplification that the population one does not: it treats every slot as a different person. In a real population the pieces can reach the same individual by different routes, and two of your fragments might coalesce in one ancestor. At forty generations, with a couple of thousand lineages in a population of millions, that happens a handful of times and changes nothing. In a small or isolated population it happens constantly, and the number of distinct genetic ancestors falls below the number of pieces. The line in Figure 1 is therefore an upper bound on people and an exact count of pieces.

The same arithmetic explains a confusion that appears whenever the phrase “common ancestor” is used. Two parts of the genome do not recombine at all: the mitochondrial DNA, which passes from mother to child, and most of the Y chromosome, which passes from father to son. Each traces a single line through the tree, one ancestor per generation out of 2^g, and the point where all living people’s lines meet is what is called Mitochondrial Eve or Y-chromosomal Adam. Those two lived on the order of two hundred thousand years ago.[11] Chang’s common ancestor lived a few thousand years ago. Both statements are true, because they answer different questions: the first asks where one particular piece of DNA converges, the second asks where any line at all does, and with a trillion lines to choose from the second is answered enormously sooner. Eve is a genetic ancestor of everyone through one piece. Chang’s ancestor is a genealogical ancestor of everyone and, in all probability, a genetic ancestor of nobody.

What is measurable

Cousins and segments

All of this became testable, rather than a theorem about a model, when it became cheap to read the genome at a million or so positions. Two people who share a recent ancestor carry, from that ancestor, a stretch of chromosome that is identical in both of them, letter for letter, because it has not had time to be broken up or mutated. Such stretches are said to be identical by descent, and their length gives the date: the expected length of a segment from an ancestor g generations back is about 100/(2g) centimorgans, so first cousins share pieces of twenty-five centimorgans or so and eighth cousins, when they share anything, pieces of five or six.[9] The consumer testing companies work entirely from this. Their “DNA relatives” are the people in their database with whom you share at least one segment above a threshold of a few centimorgans, and the estimated relationship is read from the total shared length.

The second clock sets the limit of what they can see. Second cousins always share DNA; third cousins nearly always; by fourth cousins the chance of sharing any detectable segment is about even, by fifth it is somewhere between one in ten and one in five, and by seventh cousins most pairs, who are related by every definition a genealogist uses, share nothing at all.[9] A documented seventh cousin who shares no DNA with you is not a mistake in the records. It is the expected case.

Turned around, the segments are an instrument for reading the first clock in real populations. In 2013 Peter Ralph and Coop took the genomes of about two thousand Europeans and, for every pair, counted the segments they shared and dated them from their lengths.[12] Two people from the same country typically share segments from dozens of genetic common ancestors in the last thousand years. Two from opposite ends of the continent — Ireland and Turkey, say — still share a handful of genetic ancestors from the last fifteen hundred years, and upwards of a hundred from the thousand years before that. Those are genetic ancestors, the ones who left a piece; each stands for an enormous number of genealogical ones who did not. It was the first direct measurement of what Chang’s model had predicted: that the continent is one pedigree, connected within a millennium, and that the connection is written, thinly but legibly, in the DNA.

The object underneath

The graph that holds it all

Figure 2 is a picture of something that has a name. Follow every piece of every chromosome in a population back through the generations, recording each time a piece is cut by a crossover and each time two pieces meet in a common ancestor, and the record is a graph: the ancestral recombination graph, introduced by Richard Hudson in 1983 as an extension of Kingman’s coalescent and given its name and its formal shape by Griffiths and Marjoram in 1997.[13] At any single position on the genome the graph is a tree, the genealogy of that position. Move along the chromosome and the tree changes, a little at each point where some ancestor had a crossover, so the whole object is a sequence of correlated trees, one per stretch of DNA, glued together by the recombinations that separate them. The first simulation in this essay built one pedigree’s slice of it. A population’s graph is the same thing for everyone at once.

Two properties of the graph are the point of this essay. First, it contains the genetic ancestors and only them. A person who left no piece of DNA in anyone living is not a node in the graph, by construction; the ghost ancestors of Gravel and Steel are exactly the people the graph cannot see. The genealogical tree, with its trillion slots, is not recoverable from DNA even in principle, and the graph is the largest part of it that is. Second, the graph is finite in a way the pedigree is not. The pedigree at generation g has 2^g slots; the graph at generation g has as many lineages as there are pieces, a few thousand at most, and going further back the lineages coalesce and the number shrinks toward one. The whole ancestry of a species’ genome is a bounded object. It is enormous, but it is not exponential, and it can, in principle, be written down.

Since about 2019 it is being written down. Jerome Kelleher and colleagues at Oxford showed that the graph for hundreds of thousands of genomes could be inferred and stored compactly, as a “tree sequence”, by exploiting the fact that adjacent trees differ by only a few edges.[14] In 2022 Anthony Wilder Wohns and the same group built a single graph for some 3,600 modern genomes and eight ancient ones, from 215 populations, and dated its nodes; the oldest ancestral pieces in it reach back beyond a million years, and the geography of the nodes, plotted through time, retraces the movement out of Africa without being told about it.[15] In 2023 a method from Pier Palamara’s group inferred the graph for the 337,000 genomes of the UK Biobank.[16] The graph is now the standard object in which the field expects to store, compare and analyse genomes, and the open questions about it are the ones one would expect of a new mathematical object: how to infer it well, how to know when an inference is wrong, and how to say, of two such graphs, how far apart they are.

An earlier essay here argued that biology’s scarcest resource is a theory that rules things out in advance, and that the parts of the subject where the mathematics is real are the parts where a result can be demonstrated rather than announced.[17] This is one of those parts. Nothing in this essay depends on a mechanism nobody understands. The two clocks are arithmetic; the 80 per cent is a fixed point of an exponential; the linear growth of pieces is a Poisson process; the graph is a well-defined object with theorems about it. The biology supplies the two numbers that turn the arithmetic into a statement about people, thirty-five Morgans and two parents, and everything else follows.

Coda

What the two clocks say

Three things follow from putting the clocks side by side, and none of them is a comfort to the family historian. The first is that ancestry, past a dozen generations, is a property of populations, not of persons. The tree does not narrow to the named and the notable; it widens until it includes everyone who left descendants, and it includes each of them so many times that the count is meaningless. To be descended from someone in the ninth century is to be alive.

The second is that the genome remembers a tiny, biased sample of that tree. Of the ancestors ten generations back, half are absent from you altogether. Of the ancestors forty generations back, all but a few thousand of a trillion slots are empty, and the few thousand are chosen by the fall of crossovers, not by rank, name or deed. The DNA test that finds a fourth cousin is reading that sample, and the seventh cousin it cannot find is a true relative who fell outside it.

The third is that the sample is the part that can be known. The pedigree with its trillion slots is a fiction, in the precise sense that no possible evidence could reconstruct it. The graph of the pieces is not; it is finite, it is being inferred at the scale of whole biobanks, and its nodes are the ancestors who left something. They are the only ancestors, in the end, of whom there is anything to say. The others are in the tree. They are not in the blood. Both are true, and it took a statistician, a few physicists and about forty Morgans to see why.

Open threads

Where this could go

Distance between two graphs. An ancestral recombination graph inferred by two methods from the same data will differ, and there is no agreed way to say by how much. The distance metrics for single trees do not extend cleanly to sequences of correlated trees. That is a self-contained mathematical problem with no laboratory in it, and it is the one this blog is most likely to return to.

The real pedigree, where it exists. Iceland and Québec have genealogies going back centuries alongside genotyped populations, so the ghost ancestors can be counted rather than simulated: which documented ancestors of a living person left DNA, which did not, and whether the fall-off matches the Poisson arithmetic here or departs from it.

Sex-specific recombination. The female map is 1.6 times the male, so a line of mothers cuts your ancestry into pieces faster than a line of fathers. That should bias which ancestors survive genetically in a way the simulation here averages over and could instead resolve.

Chang for the world, again. The 2004 model of human history was built before ancient DNA existed. The identical-ancestors point for the planet could now be estimated from the inferred graph rather than from a migration model, which is a different kind of evidence, and it would be worth knowing whether the dates agree.

On method and tools

This piece was written collaboratively with Claude Fable 5.1 (Anthropic): human specification, editorial direction and critical review; machine synthesis, drafting, computation and figure generation.

The figures are computed, not traced. The script scripts/ancestors.py does two things. It traces one person’s two haploid autosomal genomes back forty generations through a pedigree in which every slot is a distinct individual, drawing for each ancestor one meiosis per chromosome as a Poisson process of crossovers (no interference, no obligate crossover) on a sex-specific map: sex-averaged chromosome lengths rounded from the deCODE map, total 35.4 Morgans, scaled by 1.25 for women and 0.75 for men. Figures 1 and 2 and the counts in the text come from 300 such pedigrees. It also runs Chang’s model directly, a population of 10,000 per generation with two distinct parents chosen uniformly, tracking for every past individual the set of present-day descendants; Figure 3 is the mean of 20 runs. The numerical output is in docs/ancestors-results.json. The population sizes for medieval Europe and the twelfth-century world are the McEvedy and Jones estimates, and are used only as orders of magnitude.

Authored by: Luis Matos Ferreira — Physicist, Developer, Writer

Related essays on this blog

1. The Complete Parts List — what biology can enumerate and what it can predict; the essay this one answers.
2. The Frozen Accident — how much of life is found and how much is the tape happening to run that way.
3. Codes That Were Never Tried — the space of genetic codes, recomputed from scratch.
4. Where Number Comes From — the approximate number sense, and the exactness laid over it.

Sources

1. World and regional population estimates from McEvedy & Jones, Atlas of World Population History, Penguin, 1978: roughly 350 million worldwide in 1150 and 30 million in Europe around 800. A generation is taken as 28 to 30 years throughout.
2. Gunderson’s term and his work are described in Shoumatoff, The Mountain of Names: A History of the Human Family, Simon & Schuster, 1985.
3. Olson, “The Royal We”, The Atlantic, May 2002; Rutherford, A Brief History of Everyone Who Ever Lived, Weidenfeld & Nicolson, 2016, ch. 3, for the popular statement of the Charlemagne argument.
4. Chang, “Recent common ancestors of all present-day individuals”, Advances in Applied Probability 31, 1002 (1999), with discussion. The two results are Theorems 1 and 2; the 80 per cent is in the discussion of Theorem 2.
5. Derrida, Manrubia & Zanette, “Statistical Properties of Genealogical Trees”, Physical Review Letters 82, 1987 (1999); and “On the genealogy of a population of biparental individuals”, Journal of Theoretical Biology 203, 303 (2000).
6. Rohde, Olson & Chang, “Modelling the recent common ancestry of all living humans”, Nature 431, 562 (2004). The date ranges quoted are across their model variants; the paper gives point estimates for its preferred model and stresses their dependence on migration assumptions.
7. Kong et al., “A high-resolution recombination map of the human genome”, Nature Genetics 31, 241 (2002), the deCODE map, with the female map about 1.6 times the male; Bhérer, Campbell & Auton, “Refined genetic maps reveal sexual dimorphism in human meiotic recombination at multiple scales”, Nature Communications 8, 14994 (2017), for the modern sex-specific maps and the crossover counts per meiosis.
8. Donnelly, “The probability that related individuals share some section of genome identical by descent”, Theoretical Population Biology 23, 34 (1983); Wiuf & Hein, “On the number of ancestors to a DNA sequence”, Genetics 147, 1459 (1997).
9. Coop, “How many genetic ancestors do I have?” and “How much of your genome do you inherit from a particular ancestor?”, gcbias (blog), November 2013; the segment-length rule of thumb and the cousin-sharing probabilities are from the same posts and from the 23andMe relationship tables they discuss.
10. Gravel & Steel, “The existence and abundance of ghost ancestors in biparental populations”, Theoretical Population Biology 101, 47 (2015).
11. Cann, Stoneking & Wilson, “Mitochondrial DNA and human evolution”, Nature 325, 31 (1987); Poznik et al., “Sequencing Y chromosomes resolves discrepancy in time to common ancestor of males versus females”, Science 341, 562 (2013). Current estimates for both put the coalescence in the range of one to three hundred thousand years, depending on the mutation rate assumed.
12. Ralph & Coop, “The Geography of Recent Genetic Ancestry across Europe”, PLoS Biology 11, e1001555 (2013).
13. Hudson, “Properties of a neutral allele model with intragenic recombination”, Theoretical Population Biology 23, 183 (1983); Griffiths & Marjoram, “An ancestral recombination graph”, in Donnelly & Tavaré (eds), Progress in Population Genetics and Human Evolution, IMA Volumes in Mathematics and its Applications 87, Springer, 1997, p. 257; Kingman, “The coalescent”, Stochastic Processes and their Applications 13, 235 (1982).
14. Kelleher, Wong, Wohns, Fadil, Albers & McVean, “Inferring whole-genome histories in large population datasets”, Nature Genetics 51, 1330 (2019); Kelleher, Etheridge & McVean, “Efficient coalescent simulation and genealogical analysis for large sample sizes”, PLoS Computational Biology 12, e1004842 (2016), for the tree-sequence encoding.
15. Wohns et al., “A unified genealogy of modern and ancient genomes”, Science 375, eabi8264 (2022).
16. Zhang, Biddanda, Gunnarsson, Cooper & Palamara, “Biobank-scale inference of ancestral recombination graphs enables genealogical analysis of complex traits”, Nature Genetics 55, 768 (2023).
17. “The Complete Parts List”, on this blog, September 2026.