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 
. 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, for the human lineage and to use it as a diagnostic comparator for empirically inferred effective population size .
The motivation is that is commonly interpreted—explicitly or implicitly—as reflecting a long‑term or historical population size. If that interpretation is valid, then should be meaningfully related to an explicit time‑average of census population size
. Computing
from known census estimates removes ambiguity about what “long‑term” means and allows a direct comparison.
Importantly, is not proposed as a replacement for in population‑genetic equations. It is used strictly as a bookkeeping quantity to test whether
corresponds to any reasonable long‑term average of census population size or not.
Definition and derivation of
Let denote the census population size at time , measured backward from the present, with
at present and
in the past.
For any starting time , define the time‑averaged census population size from
to the present as:
Because 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 between adjacent points.
- Integrate each segment exactly.
- Divide by the total elapsed time .
This produces an explicit, reproducible value of for each starting time
.
Census anchors used
- Census population sizes 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 (
) are used in the construction of
.
Present is taken as 2026 CE.
Results: from
to present
All values are people. is the census size at the start time. is the time‑average from
to 2026 CE.
| Start time | Years before present | (low / mid / high) | (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
- is orders of magnitude larger than empirical human
(typically
) for all plausible averaging windows. - This remains true even when averaging over millions of years and even under conservative census assumptions.
- Therefore, cannot be interpreted as:
- an average census size,
- a long‑term census proxy,
- or a time‑integrated representation of .
The comparison holds regardless of where the averaging window begins, reinforcing the conclusion that
is not a demographic population size but a fitted parameter summarizing drift under complex, non‑stationary dynamics.
Kimura’s cancellation requires N = N_e. CN has shown that N ≠ N_e 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 N_e—always, in every species, in every era. Reproductive variance, population structure, and fluctuating population size all push N_e below N. There’s no compensating mechanism that pushes N_e 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.