Dennis McCarthy noted an interesting statistical fact about the geneology of Charlemagne.
Every person of European descent is a direct descendant of Charlemagne. How can this possibly be true?
Well, remember you have 2 parents, 4 grandparents, 8 grandparents, etc. Go back 48 generations (~1200 years), and that would equate to 248 ancestors for that generation in the time of Charlemagne, which is roughly 281 trillion people.
The actual population of Europe in 800 AD was roughly 30 million. So what happened? After roughly 10 to 15 generations, your family tree experiences “pedigree collapse.” That is, it stops being a tree and turns into a densely interconnected lattice that turns back on itself thousands of times—with the same ancestors turning up multiple times in your family tree.
Which, of course, is true, but considerably less significant in the genetic sense than one might think.
Because the even more remarkable thing about population genetics is that despite every European being a descendant of Charlemagne, very, very few of them inherited any genes from him. Every European is genealogically descended from Charlemagne many thousands of times over due to pedigree collapse. That’s correct. But genealogical ancestry ≠ genetic ancestry. Recombination limits how many ancestors actually contribute DNA to you.
Which means approximately 99.987% of Europeans inherited zero gene pairs from Charlemagne.
And this got me thinking about my previous debate with Mr. McCarthy about Probability Zero, Kimura, and neutral theory, and led me to another critical insight: because Kimura’s equation was based on the fixation of individual mutations, it almost certainly didn’t account for the way in which gene pairs travel in segments, and that this aspect of mutational transmission was not accounted for in the generational overlap constraint independently identified by me in 2025, and prior to that, by Balloux and Lehmann in 2012.
Which, of course, necessitates a new constraint and a new paper: The Transmission Channel Capacity Constraint: A Cross-Taxa Survey of Meiotic Bandwidth in Sexual Populations. Here is the abstract:
The molecular clock treats each nucleotide site as an independent unit whose substitution trajectory is uncorrelated with neighboring sites. This independence assumption requires that meiotic recombination separates linked alleles faster than mutation creates new linkage associations—a condition we formalize as the transmission channel capacity constraint: μ ≤ r, where μ is the per-site per-generation mutation rate and r is the per-site per-generation recombination rate. We survey the μ/r ratio across six model organisms spanning mammals, birds, insects, nematodes, and plants. The results reveal a sharp taxonomic divide. Mammals (human, mouse) operate at or above channel saturation (μ/r ≈ 1.0–1.5), while non-mammalian taxa (Drosophila, zebra finch, C. elegans, Arabidopsis) maintain 70–90% spare capacity (μ/r ≈ 0.1–0.3). The independent-site assumption underlying neutral theory was developed and validated in Drosophila, where it approximately holds. It was then imported wholesale into mammalian population genetics, where the channel is oversubscribed and the assumption systematically fails. The constraint is not a one-time packaging artifact but a steady-state throughput condition: every generation, mutation creates new linkage associations at rate μ per site while recombination dissolves them at rate r per site. When μ > r, the pipeline is perpetually overloaded regardless of how many generations elapse. The channel capacity C = Lr is a physical constant of an organism’s meiotic machinery—independent of population size, drift, or selection. For species where μ > r, the genome does not transmit independent sites; it transmits linked blocks, and the number of blocks per generation is set by the crossover count, not the mutation count.
There are, of course, tremendous implications that result from the stacking of these independent constraints. But we’ll save that for tomorrow.