Nothing says “rare glacial erratic from the last Ice Age” quite like a bulk bag of Mendip Grey Gabion Stone, available by the tonne, with free delivery on larger orders.
Five hundred years after the main phases were finished, the people who raised it had vanished. No clear genetic trace left behind. A thin residue of similar ancestry turns up later, maybe a few per cent, up to about 8%, but it could just as easily have arrived already mixed from the continent.
This was a population wipeout more complete than any colonial catastrophe we can document from history. We don’t know what caused it: disease, violence, or being outbred. No historical parallel or evidence yet explains it. The only close match is the unexplained and complete disappearance of the Dorset people of the Arctic.
The stones stood. The bloodline did not.
More detail and the data:
This is a technical discussion draft relating to the Population Replacement Evidence post.
Revised 3 August 2026. This version replaces the draft posted on
2 August. The substantive change is in §9 and §10: the earlier version
described the four England_BB_highEEF outliers as surviving
independent testing, which read as a claim that they are demonstrable
British survivals. That does not follow from the argument of §7, and has
been corrected. A new §2 states the estimand explicitly, so everything
after it is renumbered by one. Full list of changes at the foot of the
post.
Two studies appear to disagree about whether ancestry from Neolithic Britain persisted after the Bell Beaker migrations of the mid-third millennium BC. Booth et al. (2021) reported a small but rising Neolithic-related component through the Chalcolithic and Early Bronze Age, and read it as evidence that the resident population left descendants. Olalde et al. (2026), using a Lower Rhine–Meuse source population unavailable in 2021, found the main English Beaker group indistinguishable from that source and bounded any surviving British Neolithic ancestry between zero and eight per cent.
Re-analysis of the supplementary data of both papers shows that they do not disagree about how much ancestry survived. Placing Booth’s individual-level estimates inside Olalde’s group definitions returns 2.52 per cent for the Beaker group and 8.55 per cent for the later Chalcolithic–Early Bronze Age group, against Olalde’s zero and 7.3–7.9 per cent. The pattern of estimates censored at the zero boundary in Booth’s own table independently distinguishes the two groups, recovering the 2026 result from data available five years earlier. Both papers find the same rise over time, and weighted regression on calibrated date shows it to be a continuous gradient rather than a step between burial categories: fitting date and a group indicator jointly, the date term survives and the group term does not.
What the 2026 data cannot establish is where the surviving ancestry came from. Seven candidate source populations — from Yorkshire to Andalusia — return estimates spanning only 7.3 to 9.0 per cent and fit the data comparably well, with the English Neolithic source ranking third of seven. The reason is substantive rather than technical: English Neolithic and continental Middle/Late Neolithic populations are genuinely alike in hunter-gatherer content, the only axis on which this design separates them. The published bound of nought to eight per cent is therefore a statement of non-identifiability, not a confidence interval, and no amount of additional sampling in the candidate source regions will narrow it.
The quantity is well measured; the address is unknown. One extension of an analysis already performed in the 2026 paper — identity-by-descent between the English Chalcolithic–Early Bronze Age individuals and the English Neolithic ones, a comparison absent from the published table by construction — could settle it. A power calculation from that table’s own detection rates indicates the test would be underpowered at the coverage used and decisive at full group coverage.
The genetic transformation of Britain between roughly 2450 and 2000 BC is one of the most dramatic events in European prehistory. People associated with the Bell Beaker complex arrived from the continent, and within a few centuries the ancestry of the population buried in Britain had changed by something like ninety per cent. What is disputed is not that this happened but what it means: whether a resident population was displaced, absorbed, or had already dwindled to the point where "replacement" is the wrong word.
The residual matters more than its size suggests. If a few per cent of Neolithic British ancestry survived into the Bronze Age, some Neolithic families had descendants and the transition involved intermarriage rather than pure displacement. If that residual rose over time, as Booth and colleagues argued, then descendants of the Neolithic population were not merely surviving but were increasingly represented — which is very hard to reconcile with rapid violent replacement. And if the residual is actually zero, the record is silent on all of this, because you cannot infer anything from a signal you cannot see.
Two obstacles make the question hard, and both are statistical rather than archaeological.
The first is that these ancestry proportions are estimated by a method — qpAdm — that answers a narrow question. It asks whether a proposed mixture of source populations is rejected by the data, given a set of reference populations held out for comparison. A model that is not rejected is not thereby confirmed. A proportion estimated at zero may mean the ancestry is absent or may mean the method cannot see it. Distinguishing these requires knowing how small a contribution the design could have detected, which nobody has calculated for this case.
The second obstacle is that ancestry proportions are estimated for a named source. When you ask how much English Neolithic ancestry a population carries, you get an answer conditioned on English Neolithic being the right source. If a different but genetically similar population were the true contributor, the estimate would come back much the same. Booth and colleagues flagged exactly this, writing that it was difficult to distinguish ancestry from Neolithic Britain from ancestry from more western parts of continental Europe — northern France in particular — where archaeogenetic data were then almost absent. They judged that this was unlikely to account for much.
That judgement was the weakest joint in their argument. The 2026 paper supplies part of the missing data. What follows tests both.
Most of the confusion in this area comes from treating two different quantities as one. It is worth separating them before any numbers are quoted.
Olalde et al. (2026) model England_CA_EBA (n = 39, median 1865 cal BC) as requiring a second Middle/Late Neolithic source at 7.3–7.9 per cent alongside RhineMeuse_LNB_BB; the one-source model fails at P = 2.8 × 10⁻¹⁰. The paper states the interpretive range explicitly: nought per cent British if the component is wholly continental, eight per cent if wholly British. That range is not uncertainty about the size of the component. It is uncertainty about its provenance.
The quantity at issue is therefore not the magnitude of the residue — which is well measured, to within about two percentage points — but a mixing parameter:
β = the proportion of the 7.3–7.9 per cent second-source component that derives from populations resident in Britain before c. 2450 BC, as against populations resident in north-west continental Europe.
β = 1 means the entire component is indigenous survival: Neolithic Britons contributed to the later gene pool at group level. β = 0 means the incomers arrived carrying farmer ancestry already, and no Neolithic Briton did. Everything published to date is consistent with any value in [0, 1], and the substantive disagreement between the two papers — to the extent there is one — is entirely a disagreement about β.
Three distinct processes can produce a 7–8 per cent second source with β = 0, and they are worth naming because they have different implications and different remedies:
Sections 3 to 6 establish that the residue is real and that its magnitude is agreed. Sections 7 and 8 establish that β is not identified by the published design. Section 10 returns to the three mechanisms above; §11 sets out the one test that would separate them.
Three published supplementary datasets are used. All figures below can be regenerated from them by the accompanying script.
Booth et al. 2021, Supplementary Table 1 gives, for each of 98 individuals from Chalcolithic to Late Bronze Age Britain, a point estimate of ancestry related to the Neolithic populations of Britain, a standard error, and a calibrated radiocarbon median. These estimates originate in the qpAdm models of Olalde et al. 2018.
Olalde et al. 2026, Supplementary Tables 1, 3, 7, 9, 10, 12 and 14 give group assignments by genetic identifier, qpAdm models for the English groups with the Lower Rhine–Meuse source in place, distal three-way models decomposing ancestry into Balkan Neolithic, western hunter-gatherer and Corded Ware components, and identity-by-descent connections.
Olalde et al. 2018, Supplementary Table 1 gives autosomal SNP counts.
Merging on individual identifiers places Booth’s per-individual estimates inside Olalde’s 2026 group boundaries. Coverage is 26 of the 28 individuals in England_BB, 28 of the 39 in England_CA_EBA, and 3 of the 4 in England_BB_highEEF — I14200 is not in Booth’s table. The remaining 44 Booth individuals are Scottish, Welsh, northern English or Middle-to-Late Bronze Age and fall outside the English groups; they are used only where the temporal analysis is stated as covering all dated individuals.
Group summaries are inverse-variance weighted means. Heterogeneity is assessed by Cochran’s Q, I², and the random-effects between-individual standard deviation τ. Temporal models are weighted least squares of ancestry proportion on calibrated median date, with the residual scale factor φ = χ²/df used to correct slope standard errors for over- or under-dispersion.
| Group | Booth per-individual, pooled | Olalde 2026, group-level qpAdm |
|---|---|---|
| England_BB (26 of 28) | 2.52 % ± 0.84 | 0 % — cladal with RhineMeuse_LNB_BB, P = 0.61 |
| England_CA_EBA (28 of 39) | 8.55 % ± 0.85 | 7.3–7.9 % (setup 1); 7.5–9.0 % (setups 2–3) |
| England_BB_highEEF (3 of 4) | 26.98 % ± 2.44 | 27–39 % in three-way models |
The estimates are not in conflict. They are the same numbers, reached through different model structures — one without a Rhine–Meuse proxy and one with it — and the concordance holds across an order-of-magnitude range. Whatever else the 2026 paper achieved, it did not overturn Booth’s measurements.
Two qualifications, both of which tighten the agreement rather than loosening it. Booth’s per-individual standard errors derive from a common set of f₄ statistics and are correlated; treating them as independent makes the pooled standard errors above optimistic, plausibly by a factor of order √2. And 19 of the 98 estimates sit exactly on the zero boundary, which biases the pooled means upward.
qpAdm can return negative admixture proportions. Olalde 2018 reported them censored at zero. This is normally a nuisance; here it carries information.
If a group’s true British Neolithic ancestry were exactly zero, half its individual estimates would fall below zero by chance and be reported as zero. In England_BB, 13 of 26 estimates sit on the boundary — a fraction of 0.500 against an expectation of 0.500 (binomial p = 1.00). In England_CA_EBA, 4 of 28; under a true value of zero the expectation is 14 (p = 1.8 × 10⁻⁴).
Simulating the censoring directly — drawing each individual from a normal centred on an assumed common true value with that individual’s own reported standard error, censoring at zero, re-pooling, 200,000 times:
| Assumed true value | England_BB pooled, median (95 %) | Expected zeros | P(pooled ≥ 2.52 %) |
|---|---|---|---|
| 0 % | 1.66 % (0.81–2.72) | 13.0 | 0.053 |
| 1 % | 2.21 % (1.21–3.39) | 10.6 | 0.294 |
| 2 % | 2.85 % (1.72–4.15) | 8.4 | 0.706 |
| 3 % | 3.58 % (2.33–4.98) | 6.4 | 0.949 |
Inverting the simulation gives a consistency interval of 0 to 3.25 per cent for England_BB’s true common value. Truncation accounts for most but not all of the observed 2.52 per cent: under a true value of exactly zero the expected pooled estimate is 1.66 per cent, and the observation sits at the 95th percentile. Zero survives, but only just.
For England_CA_EBA the same simulation excludes every true value below about 7 per cent (P < 10⁻³ throughout); at 7.15 per cent the observed pooled estimate sits at p = 0.058. The later group’s residue is real and is not an artefact of censoring.
So Booth’s own data, read through nothing but their boundary counts, say that the Beaker group’s Neolithic ancestry is indistinguishable from zero and the Chalcolithic–Early Bronze Age group’s is not. That is the 2026 result, obtainable five years earlier from data already in hand.
England_BB has a median calibrated date of 2146 cal BC; England_CA_EBA, 1865 cal BC (Mann–Whitney p = 1.8 × 10⁻⁵). Olalde 2026 assigns the earlier group no additional Middle/Late Neolithic ancestry and the later group 7.3–7.9 per cent. That is a rise, in the same direction and of similar magnitude to Booth’s reported increase from about 6 to about 12 per cent, obtained from independent model specifications with the correct proxy in place.
Booth’s trend also survives being tested more stringently than they tested it. A rank-sum comparison across a split chosen partly by inspection is weak evidence; weighted regression on the continuous date variable is not.
| Sample | n | Slope (pp/century) | Z (naive) | Z (corrected) | φ |
|---|---|---|---|---|---|
| All dated individuals | 87 | +0.62 | 5.84 | 3.35 (p = 0.0008) | 3.03 |
| Excluding the highEEF outliers | 84 | +0.82 | 7.52 | 5.20 (p < 10⁻⁶) | 2.09 |
| England_BB + England_CA_EBA only | 48 | +1.83 | 5.69 | 4.15 (p = 3 × 10⁻⁵) | 1.88 |
| …excluding I2462 as well | 47 | +1.73 | 5.39 | 6.85 (p < 10⁻⁵) | 0.62 |
Spearman’s ρ is 0.509 (p = 4.9 × 10⁻⁷) across all dated individuals and 0.592 (p = 9.5 × 10⁻⁶) on the restricted set. Restricting to the 48 individuals inside the two English groups removes any confounding from Scottish, Welsh or Middle Bronze Age individuals, and strengthens the slope roughly threefold — the full-sample figure is depressed by the later Bronze Age tail, where the trajectory plateaus.
The rise is a gradient, not a step. Within England_BB alone the slope is +1.26 pp/century (corrected Z = 1.77); within England_CA_EBA alone, +1.38 (corrected Z = 1.92). Fitting date and a group indicator together, the date term survives (Z = 5.02 excluding I2462) and the group step does not (+1.5 pp, Z = 1.24). Olalde’s two groups are not two states of the population but two windows onto one continuous trajectory. This is the strongest available vindication of Booth’s actual thesis, which was gradualism: a group-level model with two categories cannot represent a gradient, and the gradient is in the data.
Olalde 2026 tested nine candidate second sources for England_CA_EBA across three SNP-filtering setups. The two Lower Rhine–Meuse sources fail decisively. The remaining seven:
| Second source | n | P (setup 1) | P (setup 2) | P (setup 3) | Proportion range |
|---|---|---|---|---|---|
| TRB_N (Funnel Beaker) | 6 | 7.4 × 10⁻³ | 0.514 | 0.488 | 7.3–8.2 % |
| S_France_LN | 18 | 1.8 × 10⁻³ | 0.220 | 0.351 | 7.3–8.6 % |
| England_N | 27 | 2.2 × 10⁻³ | 0.205 | 0.334 | 7.6–8.8 % |
| Germany_Baalberge_MN | 3 | 1.4 × 10⁻³ | 0.115 | 0.244 | 7.8–8.9 % |
| NE_France_LN | 14 | 1.2 × 10⁻³ | 0.083 | 0.199 | 7.4–8.6 % |
| GlobularAmphora_LN | 10 | 6.3 × 10⁻⁴ | 0.051 | 0.099 | 7.9–9.0 % |
| N_Spain_LNCA | 45 | 3.1 × 10⁻⁴ | 0.037 | 0.100 | 7.6–8.7 % |
Across all 21 fits the point estimate ranges from 7.3 to 9.0 per cent — a spread of 1.7 percentage points — while the candidate source ranges geographically from Yorkshire to Andalusia. The quantity is estimated to within about two points; the source is not estimated at all. England_N ranks third of seven where models pass. There is no statistical basis for preferring the British source and none for excluding it.
This is why Booth’s caveat matters more, not less, after 2026. They worried that western continental Neolithic populations could not be told apart from British ones. NE_France_LN returns 7.4–8.6 per cent with fits comparable to England_N’s. Their caveat has been tested and confirmed as unresolvable with this design, not refuted.
The 2026 bound of 0 to 8 per cent is a statement of that non-identifiability. It is not a confidence interval and should not be collapsed into one. Sampling uncertainty on the quantity is roughly ±2.4 points; the 0-to-8 range is source-attribution ambiguity, a different kind of ignorance that does not shrink with more British individuals.
England_BB’s cladality with RhineMeuse_LNB_BB at P = 0.61 licenses a bound, not an absence. The bound can be calculated (Appendix A).
With six outgroups the one-way model carries 5 degrees of freedom, and P = 0.61 corresponds to an observed χ² of 3.61. Inverting the non-central χ² gives a one-sided 95 per cent upper bound on the non-centrality of λ ≤ 6.88. The mapping λ ≈ κ(α/SE)² is calibrated against the two groups that demonstrably require a second source, giving κ = 0.96 across six checks. The standard error for a two-source England_BB model, obtained by cross-target calibration against Supplementary Table 10, is 0.0130–0.0140 — about 8 per cent larger than England_CA_EBA’s, since England_BB’s higher per-individual coverage does not compensate for eleven fewer individuals. Hence:
England_BB can conceal up to 3.5 per cent local Neolithic ancestry (95 per cent upper bound; 3.46–3.75 per cent across the three SNP setups).
The power curve matters as much as the bound. The design reaches 50 per cent power only at 3.5 per cent, 80 per cent power at 4.7–5.1 per cent, and 95 per cent power at 5.9–6.4 per cent. A genuine residue of four per cent would have been missed more often than not.
Booth’s pooled England_BB estimate of 2.52 per cent, and the 0–3.25 per cent consistency interval from the censoring simulation, sit entirely inside that window. Two independent routes converge on the same ceiling. The two papers do not disagree here either; the 2026 design simply cannot see a residue this small.
Cochran’s Q on Booth’s estimates within each Olalde group:
| Group | Q | df | p | I² | τ |
|---|---|---|---|---|---|
| England_BB | 20.9 | 25 | 0.70 | 0 % | 0.00 pp |
| England_CA_EBA | 81.0 | 27 | 2.6 × 10⁻⁷ | 66.7 % | 6.36 pp |
England_BB, once Olalde’s four outliers are removed, is statistically homogeneous. Olalde’s decision to excise England_BB_highEEF is therefore vindicated on Booth’s own numbers: those four individuals carried the entirety of the Beaker-period signal, and the magnitude of their Neolithic-related component — 27 to 39 per cent, against 27.0 % ± 2.4 pooled from Booth’s estimates — survives independent testing with a proper Rhine–Meuse proxy in place.
It does not follow that these are demonstrable British survivals, and the paper should not be read as claiming so. The non-identifiability of §7 applies to the outliers with undiminished force: their component is better measured than the group-level residue, being an order of magnitude larger, but it is no better addressed. Nothing in the design distinguishes a British Neolithic contribution from a continental one at 27 per cent any more than at 8 per cent. Appendix D.2 makes the point directly — England_BB_highEEF’s WHG/(WHG + Balkan_N) ratio is 0.239, inside the 0.217–0.246 continental Middle/Late Neolithic cluster and 0.03 from England_N’s 0.207, a separation smaller than the individual-level scatter in England_N itself (0.180–0.230). These four individuals are people with substantially more farmer ancestry than their contemporaries. Whether that ancestry was acquired in Britain or carried across the Channel is exactly the question this paper argues cannot presently be answered.
England_CA_EBA is not homogeneous, and its heterogeneity has a single cause. Removing one individual — I2462, Sk 220053 from East Kent Access Road, 35.3 ± 3.8 per cent, z = +7.04, dated 2131–1890 cal BC — leaves Q = 28.9 on 26 df (p = 0.32), τ = 1.51 pp, and a group mean of 7.15 per cent in close agreement with Olalde’s 7.3–7.9. Booth identified her as the latest individual carrying substantial British Neolithic ancestry. Olalde 2026 separated four outliers from the Beaker group and retained this statistically comparable fifth inside a 39-individual pool where she contributes about 0.7 points to the group mean. That is not an error — a group-level analysis is entitled to pool — but it means the count of individually demonstrable Neolithic-descended people in the record is five, not four.
The rise, however, is not carried by the outliers. Decomposing the 4.72-point gap between the two groups’ unweighted means:
| Individuals removed from England_CA_EBA | Group mean | Gap retained |
|---|---|---|
| none | 7.40 % | 4.72 pp (100 %) |
| I2462 | 6.36 % | 3.68 pp (78 %) |
| top 2 | 6.06 % | 3.38 pp (72 %) |
| top 3 | 5.76 % | 3.08 pp (65 %) |
| top 5 | 5.13 % | 2.45 pp (52 %) |
The Hodges–Lehmann shift between the groups is 3.90 points, and a quarter of England_CA_EBA individuals exceed England_BB’s 95th percentile. This is a broad distributional shift, not a handful of surviving families visible against an unchanged background — a distinction that matters, because a broad shift is hard to explain by lineage survival and easy to explain by continuing gene flow from wherever the source turns out to be.
Two further features of England_CA_EBA, both invisible at group level. The within-group residual correlates with date even after I2462 is removed (r = +0.50, p = 0.009), so the gradient of §6 runs inside the group as well as between the groups. And there is a hint of regional structure: Amesbury Down (n = 5) averages 2.10 per cent against East Kent Access (n = 3) at 10.43 per cent. At these sample sizes that is a hypothesis rather than a finding, but it is precisely the kind of pattern a pooled model is guaranteed to erase.
Both papers are correct. They are describing different phases of one sequence, and the appearance of conflict comes from a difference in what each was able to name.
The Beaker phase (England_BB, median 2146 cal BC). Local Neolithic ancestry is indistinguishable from zero, at a detection ceiling of about 3.5 per cent. Booth’s boundary structure and Olalde’s cladality test say the same thing. Four individuals carry a large and robustly measured Neolithic-related component, and a fifth (I2462) sits inside the later group; whether any of the five descends from Neolithic Britain is subject to the same non-identifiability as the group-level residue.
The Chalcolithic–Early Bronze Age phase (England_CA_EBA, median 1865 cal BC). A second Middle/Late Neolithic component of 7 to 9 per cent is required, robustly, in both analyses. The increase from the earlier phase is real, is reproduced by Olalde’s own models, is a continuous gradient rather than a step, and is a broad shift rather than an outlier effect.
The source of that component is unidentified and, with this design, unidentifiable. It lies somewhere between 0 and 8 per cent British. Booth’s inference that it represents a resurgence of specifically British Neolithic ancestry is unsupported; so is the inference that it represents continental ancestry arriving pre-mixed. The honest statement is that the ancestry rose and we do not know whose it was.
The consequence for the wider argument is uncomfortable for both sides. A rising Neolithic-related component after about 2100 BC no longer functions as evidence against rapid displacement, because a continental origin for that component is fully consistent with the data and would say nothing about British survival. It also does not function as evidence for displacement. The quantity is well measured and interpretively inert until the source is pinned down.
In the terms of §2, both papers measure the same residue and neither identifies β. Booth’s reading assumes β close to 1; the natural reading of Olalde’s cladality result assumes β close to 0. Neither assumption is licensed. Of the three mechanisms that would produce the observed component with β = 0, none can currently be excluded: founder sampling is unconstrained because the thirteen sequenced RhineMeuse_LNB_BB individuals are a thin sample of a large source region; continental top-up is positively suggested by the broad distributional shift of §9, which is easier to explain by continuing gene flow than by the survival of a few families; and proxy error, while made less likely by the cladality of England_BB with the same source, is not eliminated by it. It is worth noting that the second of these — continental top-up — is the mechanism that the temporal gradient of §6 most naturally suggests, and it is the one under which the entire interpretive weight placed on the rising residue collapses.
Four interventions suggest themselves. Three can be tested against the supplementary data, and the results (Appendix D) reorder them substantially.
1. Identity-by-descent between England_CA_EBA and England_N. The only proposal that can work, it is a one-line extension of an analysis already performed, and it would be decisive. Supplementary Table 14 reports 5,846 IBD pairs. Checked against the table’s own population labels, England-by-England pairs number zero — not because none exist, but because the table’s scope is pairs featuring a Lower Rhine–Meuse individual, so the comparison was never attempted. Shared segments carry information about specific recent common ancestors that no f₄ ratio supplies, and this is the one axis on which a British source and a continental one must differ.
Appendix E works out what the test would yield. In outline: the method has ample resolution at the required time depth (nineteen detected pairs in the table span date gaps above 1,600 years), the near-contemporary full-ancestry detection rate is 17.0 per cent of possible pairs, and the expected yield for a 7.6 per cent ancestry component runs from 3.4 to 13.6 detections at full group coverage depending on how severely one assumes segment detection decays across 1,600 years. Against a continental-source null that predicts no England_N-specific excess, that is a decisive test. At the coverage actually used in Supplementary Table 14 — 19 English Neolithic and 11 English Chalcolithic–EBA individuals — the expected yield is 0.7 to 2.7 and the test is underpowered.
Three design points. Relatedness among the English Neolithic individuals is substantial: at least eight of the nineteen belong to one annotated pedigree, so the effective number of independent pairs is well below the nominal count and the sample should be pruned. The coverage threshold admitting only 12 of 39 England_CA_EBA individuals will need relaxing. And the arm against England_N should be run simultaneously against date-matched continental candidates so that time decay, the largest nuisance parameter, cancels in the ratio — Ireland_MN (n = 8, median 5410 BP) and MN_Wartberg (n = 26, median 5140 BP) are the best-matched to England_N’s median of about 5500 BP and are already in the table.
2. Individual-level re-analysis of England_CA_EBA with the Rhine–Meuse proxy. The group-level result conceals one demonstrable survival, a within-group temporal gradient, and possible regional structure. Section 8 does what can be done from Booth’s estimates; doing it properly requires running the 2026 models per individual.
3. A rotating qpAdm competition with a right set chosen to separate the candidate sources. Less promising than it looks, and Appendix D explains why. The 2026 paper contains an orthogonal discriminator it does not apply here: the ratio WHG/(WHG + Balkan_N) from the distal three-way models. That statistic separates the Rhine–Meuse sources (0.397–0.492) from everything else very cleanly, and the paper uses it to good effect. But England_N’s own value, computed from the 14 England_N individuals in Supplementary Table 7, is 0.207 with an individual range of 0.180 to 0.230 — sitting inside the continental Middle/Late Neolithic cluster at 0.217 to 0.246. English Neolithic and continental Middle/Late Neolithic populations are genuinely alike in hunter-gatherer content, to within about the individual scatter. The seven candidates behave alike in the qpAdm models because they are alike on the axis this design resolves. A better right set helps only if it resolves something other than hunter-gatherer level, and it is not obvious what that would be. The non-identifiability is substantive, not a defect of outgroup choice.
It is worth noting separately that England_CA_EBA does not appear in Supplementary Table 9 at all. The one orthogonal statistic in the paper was never computed for the group that carries the residue.
4. More northern French Middle/Late Neolithic genomes. This will not help. Across the seven candidate sources, group size ranges from 3 to 45 — a fifteen-fold span — and the standard error on the second-source proportion is 0.011 to 0.013 throughout. There is no relationship between how many individuals a source group contains and how well it fits: N_Spain_LNCA at n = 45 ranks sixth or seventh, Germany_Baalberge_MN at n = 3 ranks fourth, TRB_N at n = 6 ranks first (Spearman ρ between source n and log P = −0.32, p = 0.48 in setup 2; −0.18, p = 0.70 in setup 3). Sampling density in the source is not the binding constraint. More French genomes are worth having for other reasons — a northern French population genuinely distinct from those sampled would change the picture — but they will not tighten this estimate.
The disagreement between these two papers was never about arithmetic. Placed side by side under a common set of group definitions, they return the same numbers to within their standard errors: near-zero surviving Neolithic ancestry among the Beaker-associated dead, seven to nine per cent among their Chalcolithic and Early Bronze Age successors, and a continuous rise between the two that neither paper’s model structure was designed to represent but which both papers’ data contain.
What separates them is a question about naming rather than counting. Booth and colleagues measured a quantity and called it British; Olalde and colleagues measured the same quantity and declined to call it anything, because with a proper Rhine–Meuse proxy in the model, seven Neolithic populations spread across western Europe fit it equally well. The 2026 bound of nought to eight per cent is not a measurement with wide error bars. It is a statement that the question has two answers and the data cannot choose.
That is a genuinely awkward place for the field to be, and it is worth being clear about how awkward. The rising Neolithic-related component after about 2100 BC has done a lot of interpretive work in recent discussion of the Beaker transition — as evidence that displacement was incomplete, that intermarriage was common, that the population of Neolithic Britain left descendants. None of that is refuted here. But none of it is supported either, because a component whose source is unknown cannot tell us whether anyone survived. The number is solid and the inference is suspended.
The most useful thing to take from the exercise is that the non-identifiability turns out to be real rather than fixable by more of the same. Neolithic Britain and Neolithic northern France, Germany and Iberia are alike in the one respect that this method resolves. Sequencing more of them will not separate them.
What might is a technique that looks at shared segments of chromosome rather than aggregate ancestry proportions — and here the news is better than the rest of this paper. The relevant data already exist, in a table in the 2026 paper’s own supplement. The comparison it needs has never been run, because that table was built to answer a different question and by construction contains no English-by-English pairs at all. But the same table contains everything required to work out what the comparison would yield: how well identity-by-descent survives the sixteen centuries between Neolithic and Early Bronze Age Britain, and how often a seven-per-cent ancestry component leaves a detectable segment. The answer is that at the coverage used in the published table the test would be underpowered, and at full group coverage it would be decisive — somewhere between three and fourteen detections where a continental source predicts essentially none.
That is an unusually cheap way to settle a question this old. No new excavation, no new sequencing, no new samples: one comparison, on data already published, between two sets of people who lived in the same country sixteen hundred years apart.
Booth’s per-individual standard errors derive from a common set of f₄ statistics and are correlated. Inverse-variance pooling assumes independence and therefore understates the pooled standard errors, plausibly by around √2. All pooled intervals should be read as wider than quoted.
Booth’s estimates come from Olalde 2018’s models, which lacked a Rhine–Meuse source. They are not independent measurements of the same parameter as Olalde 2026’s; the agreement in §4 is consistency under a change of specification, not replication.
Cochran’s Q has low power at these sample sizes. I² = 0 for England_BB is a failure to detect heterogeneity, not a demonstration of its absence.
Residual z-scores fail normality tests in both groups (Shapiro–Wilk W = 0.70 and 0.76), driven by the zero floor in England_BB and by I2462 in England_CA_EBA. Normal-theory intervals throughout are indicative; the permutation and simulation results are more reliable.
11 of the 39 England_CA_EBA individuals and 2 of the 28 England_BB individuals are newly reported in 2026 and absent from Booth’s table.
The detection floor in §8 rests on the empirically calibrated mapping λ ≈ κ(α/SE)². The six calibration checks scatter between κ = 0.83 and 1.21. A direct recomputation would require the 1240k genotype data.
The censoring simulation in §5 assumes a single common true value per group and independent errors. The first assumption is falsified for England_CA_EBA (§9), so that interval applies to the group’s central tendency rather than to individuals.
qpAdm P-values across Supplementary Tables 9–12 come from a large number of tested models. None quoted here has been corrected for multiple testing, and none should be read as a probability that a model is true.
The regional contrast in §9 rests on 5 and 3 individuals respectively and should not be relied on.
No new data are generated here. All results derive from three published supplementary datasets:
The script reproduce.py takes these three workbooks as
input and prints every number reported here, in the order it appears,
including the appendix tables. Dependencies: Python 3,
numpy, pandas, scipy,
openpyxl. The random seed for the censoring simulation is
fixed at 20260802; simulation figures are stable to the second decimal
place across seeds.
Script: https://claude.ai/public/artifacts/97845468-1666-45b9-877c-c9ac70cf7fc0
GitHub Repo: https://github.com/TimDaw37/beaker-neolithic-residue
Degrees of freedom. With n_right outgroups and n_left populations on the left, qpAdm’s rank test has df = 1 × (n_right − n_left + 1). Supplementary Tables 9–12 use six outgroups (OldAfrica, IronGates_HG, Turkey_Neo, WSHG, CHG_Iran_N, Russia_Afanasievo), so the one-way model has 5 df and the two-way model 4 df.
Table A1 — cross-target standard-error calibration. Supplementary Table 10 runs one model family against several targets, allowing standard errors to be compared with model structure held fixed. Values are medians across the ten source pairings.
| Target | n | Setup 1 | Setup 2 | Setup 3 |
|---|---|---|---|---|
| England_BB | 28 | 0.0150 | 0.0160 | 0.0160 |
| England_CA_EBA | 39 | 0.0140 | 0.0145 | 0.0150 |
| England_BB_highEEF | 4 | 0.0220 | 0.0230 | 0.0230 |
| RhineMeuse_LNB_BB | 13 | 0.0155 | 0.0160 | 0.0170 |
Ratio SE(England_BB)/SE(England_CA_EBA) = 1.071, 1.103, 1.067; mean 1.081. Applied to the Supplementary Table 12 standard errors for England_CA_EBA (0.012, 0.012, 0.013), this gives 0.0130–0.0140 for a two-source England_BB model.
Table A2 — calibrating λ ≈ κ(α/SE)². For the two groups that demonstrably require a second source, the one-way model’s failure should have non-centrality proportional to the squared Wald statistic of the omitted component.
| Target | one-way P | implied χ²(5) | α | SE | (α/SE)² | κ |
|---|---|---|---|---|---|---|
| England_CA_EBA (s1) | 2.79 × 10⁻¹⁰ | 53.4 | 0.076 | 0.012 | 45.1 | 1.21 |
| England_CA_EBA (s2) | 1.49 × 10⁻⁹ | 49.8 | 0.088 | 0.012 | 58.8 | 0.83 |
| England_CA_EBA (s3) | 3.82 × 10⁻⁷ | 38.0 | 0.079 | 0.013 | 41.9 | 0.89 |
| England_BB_highEEF (s1) | 1.64 × 10⁻²¹ | 107.1 | 0.243 | 0.024 | 107.5 | 1.00 |
| England_BB_highEEF (s2) | 1.01 × 10⁻²² | 112.9 | 0.260 | 0.025 | 113.2 | 1.00 |
| England_BB_highEEF (s3) | 8.23 × 10⁻²⁰ | 99.1 | 0.241 | 0.024 | 105.8 | 0.93 |
Median κ = 0.96. The fit is near-exact for the high-signal outlier group and scatters by roughly ±20 per cent for England_CA_EBA.
Table A3 — the bound and the power curve. England_BB’s observed one-way statistic is χ² = 3.61 on 5 df (P = 0.607); the critical value at α = 0.05 is 11.07.
| Quantity | λ | Implied England_N ancestry |
|---|---|---|
| One-sided 95 % upper bound on λ given χ² = 3.61 | ≤ 6.88 | ≤ 3.46–3.75 % |
| λ giving 50 % power | 6.99 | 3.49–3.78 % |
| λ giving 80 % power | 12.83 | 4.73–5.12 % |
| λ giving 95 % power | 19.78 | 5.87–6.36 % |
Autosomal SNP coverage (Olalde 2018 Supplementary Table 1): England_BB, 26 of 28 individuals with counts, median 663,686, range 14,794–913,255. England_CA_EBA, 28 of 39, median 491,782, range 17,178–891,333. England_BB_highEEF, 3 of 4, median 700,532, range 136,956–729,987.
200,000 draws per row. Each individual is drawn from N(true, SE_i²) using its own reported standard error, censored at zero, and re-pooled by inverse variance.
| True value | England_BB pooled, median (95 %) | E[zeros] | P(≥ 2.52 %) | England_CA_EBA pooled, median | E[zeros] | P(≥ 8.55 %) |
|---|---|---|---|---|---|---|
| 0 % | 1.66 (0.81–2.72) | 13.0 | 0.053 | 1.71 | 14.0 | < 0.001 |
| 1 % | 2.21 (1.21–3.39) | 10.6 | 0.294 | 2.26 | 11.6 | < 0.001 |
| 2 % | 2.85 (1.72–4.15) | 8.4 | 0.706 | 2.90 | 9.3 | < 0.001 |
| 3 % | 3.58 (2.33–4.98) | 6.4 | 0.949 | 3.62 | 7.3 | < 0.001 |
| 5 % | 5.25 (3.80–6.77) | 3.4 | ≈ 1 | 5.29 | 4.2 | < 0.001 |
| 7.15 % | 7.24 (5.67–8.84) | 1.5 | ≈ 1 | 7.27 | 2.1 | 0.058 |
Observed: England_BB, 13 zeros of 26, pooled 2.52 %. England_CA_EBA, 4 zeros of 28, pooled 8.55 %.
Inverted consistency interval for England_BB’s true common value: 0.00 % to 3.25 %.
Group chronology (Booth calibrated medians, cal BC):
| Group | n dated | Median | IQR |
|---|---|---|---|
| England_BB_highEEF | 3 | 2289 | 2340–2190 |
| England_BB | 21 | 2146 | 2261–2091 |
| England_CA_EBA | 27 | 1865 | 2064–1814 |
England_BB earlier than England_CA_EBA: Mann–Whitney p = 1.8 × 10⁻⁵.
Weighted regression on date. Slope in percentage points per century; corrected Z uses the residual scale factor φ.
| Sample | n | Slope | Z naive | Z corrected | p | φ |
|---|---|---|---|---|---|---|
| All dated | 87 | +0.62 | 5.84 | 3.35 | 8 × 10⁻⁴ | 3.03 |
| Excluding highEEF | 84 | +0.82 | 7.52 | 5.20 | < 10⁻⁶ | 2.09 |
| England_BB + CA_EBA | 48 | +1.83 | 5.69 | 4.15 | 3 × 10⁻⁵ | 1.88 |
| …excluding I2462 | 47 | +1.73 | 5.39 | 6.85 | < 10⁻⁵ | 0.62 |
| England_BB alone | 21 | +1.26 | 1.44 | 1.77 | 0.077 | 0.66 |
| England_CA_EBA alone | 27 | +1.38 | 3.19 | 1.92 | 0.055 | 2.75 |
Note that φ < 1 in two rows: the reported standard errors there slightly overstate the scatter, so the corrected Z is conservative.
Gradient versus step (date and a group indicator fitted jointly):
| Model | Date slope | Group step | φ |
|---|---|---|---|
| With I2462 | +1.35 pp/century (Z = 2.60) | +3.40 pp (Z = 1.63) | 1.81 |
| Without I2462 | +1.52 pp/century (Z = 5.02) | +1.52 pp (Z = 1.24) | 0.61 |
Distribution-free checks. Spearman ρ = 0.509 (p = 4.9 × 10⁻⁷) on all dated individuals; 0.592 (p = 9.5 × 10⁻⁶) on the restricted set. Split at 2000 cal BC: earlier n = 39, mean 5.9 %; later n = 48, mean 11.5 %; Mann–Whitney one-tailed p = 3.9 × 10⁻⁶. Restricted Mann–Whitney, England_CA_EBA > England_BB: p = 0.0011. Permutation test on the restricted slope excluding I2462, 20,000 draws: p < 5 × 10⁻⁵.
England_CA_EBA, two-way models with RhineMeuse_LNB_BB (Supplementary Table 12), against source group sizes from Supplementary Tables 1 and 3:
| Source | n | P s1 | P s2 | P s3 | α s1 | α s2 | α s3 | SE s1 |
|---|---|---|---|---|---|---|---|---|
| MLN_Belgium | 18 | 1.3 × 10⁻⁹ | 1.4 × 10⁻⁸ | 2.2 × 10⁻⁶ | 3.2 % | 3.6 % | 3.7 % | 0.013 |
| MN_Wartberg | 40 | 6.5 × 10⁻⁷ | 1.7 × 10⁻⁵ | 5.7 × 10⁻⁴ | 5.9 % | 6.8 % | 6.4 % | 0.013 |
| Germany_Baalberge_MN | 3 | 1.4 × 10⁻³ | 0.115 | 0.244 | 7.8 % | 8.9 % | 7.9 % | 0.012 |
| TRB_N | 6 | 7.4 × 10⁻³ | 0.514 | 0.488 | 7.3 % | 8.2 % | 7.5 % | 0.011 |
| GlobularAmphora_LN | 10 | 6.3 × 10⁻⁴ | 0.051 | 0.099 | 7.9 % | 9.0 % | 8.3 % | 0.013 |
| England_N | 27 | 2.2 × 10⁻³ | 0.205 | 0.334 | 7.6 % | 8.8 % | 7.9 % | 0.012 |
| S_France_LN | 18 | 1.8 × 10⁻³ | 0.220 | 0.351 | 7.3 % | 8.6 % | 7.7 % | 0.012 |
| NE_France_LN | 14 | 1.2 × 10⁻³ | 0.083 | 0.199 | 7.4 % | 8.6 % | 7.7 % | 0.012 |
| N_Spain_LNCA | 45 | 3.1 × 10⁻⁴ | 0.037 | 0.100 | 7.6 % | 8.7 % | 7.9 % | 0.012 |
One-way model with RhineMeuse_LNB_BB alone: England_CA_EBA P = 2.79 × 10⁻¹⁰, 1.49 × 10⁻⁹, 3.82 × 10⁻⁷. England_BB P = 0.607, 0.838, 0.881, 0.188 (setup 4).
Correlations between log₁₀ source n and log₁₀ P across the seven non-Rhine-Meuse sources: Pearson r = −0.348 (p = 0.445) setup 2, −0.321 (p = 0.482) setup 3; Spearman ρ = −0.321 (p = 0.482) and −0.179 (p = 0.702).
Group-level distal models (Balkan_N + WHG + Germany_CordedWare), Supplementary Table 9, setup 1:
| Group | Balkan_N | WHG | CordedWare | WHG/(WHG+Balkan_N) |
|---|---|---|---|---|
| RhineMeuse_LNA_Vlaardingen/CW | 0.424 | 0.410 | 0.165 | 0.492 |
| RhineMeuse_LNB_BB | 0.105 | 0.069 | 0.826 | 0.397 |
| England_BB | 0.116 | 0.068 | 0.816 | 0.370 |
| RhineMeuse_EBA | 0.146 | 0.066 | 0.788 | 0.311 |
| France_BB_Steppe | 0.288 | 0.094 | 0.618 | 0.246 |
| England_BB_highEEF | 0.267 | 0.084 | 0.649 | 0.239 |
| Czechia_BB | 0.237 | 0.072 | 0.691 | 0.233 |
| France_BB_NoSteppe | 0.755 | 0.220 | 0.026 | 0.226 |
| SEGermany_BB | 0.252 | 0.070 | 0.677 | 0.217 |
England_CA_EBA does not appear in this table.
England_N’s own composition, from the 14 England_N individuals present in Supplementary Table 7: mean Balkan_N 0.785, mean WHG 0.204, ratio 0.207, individual median 0.207, range 0.180–0.230. That places English Neolithic inside the 0.217–0.246 continental cluster and about 0.19 away from the Rhine–Meuse sources.
Limitation. Differencing England_BB against England_BB_highEEF by mass balance gives the added component a ratio of 0.089–0.101 across setups, below England_N’s 0.207 and below every candidate. The three-way distal model is evidently not cleanly additive across groups with different Corded Ware fractions (0.816 versus 0.649), so that figure should not be treated as an estimate of the added component’s true hunter-gatherer content. The comparison that stands is the direct one between England_N and the continental groups.
Supplementary Table 14 contains 5,846 pairs with segments above 12 cM. Pairs involving an English individual, by group:
| Pair type | Pairs |
|---|---|
| England_BB × RhineMeuse_LNB_BB | 26 |
| England_CA_EBA × RhineMeuse_LNB_BB | 20 |
| England_BB × RhineMeuse_EBA | 7 |
| England_N × RhineMeuse_MN_Tiel | 7 |
| England_BB_highEEF × RhineMeuse_LNB_BB | 5 |
| England_CA_EBA × RhineMeuse_EBA | 4 |
| England_CA_EBA × RhineMeuse_MN_Wartberg | 2 |
| England_BB × RhineMeuse_MLN_Belgium | 2 |
| England_CA_EBA × RhineMeuse_LNA_Vlaardingen/CW | 1 |
| England_BB × RhineMeuse_LNA_Vlaardingen/CW | 1 |
| England_N × RhineMeuse_MN_Swifterbant | 1 |
| England_N × RhineMeuse_MN_Wartberg | 1 |
| England_BB × RhineMeuse_MN_Wartberg | 1 |
| England × England | 0 |
Individuals available, counted by mapping the table’s identifiers onto Olalde’s Supplementary Table 3 group assignments: England_BB 16 of 28, England_CA_EBA 12 of 39, England_N 6 of 27, England_BB_highEEF 1 of 4. Counted instead by the table’s own population-label columns — which are more inclusive, absorbing England_N_Megalithic and the kinship-annotated variants — the England_N and England_CA_EBA figures rise to 19 of 27 and 11 of 39. Both counts are reported throughout, because the two mappings do not agree and the difference matters for the power calculation in Appendix E; on either count, roughly two-thirds of the relevant individuals are absent. The nine England_N × Rhine–Meuse Neolithic pairs have longest segments of 12.7–20.0 cM.
Largest absolute residuals against the common-effect estimate of 8.55 per cent:
| ID | Site | Estimate | z |
|---|---|---|---|
| I2462 | East Kent Access | 35.3 ± 3.8 % | +7.04 |
| I6777 | Wilsford G.54 | 0.0 ± 4.0 % | −2.14 |
| I2597 | Amesbury Down | 0.9 ± 4.0 % | −1.91 |
| I5373 | Carsington Pasture Cave | 0.0 ± 4.5 % | −1.90 |
| Mean | Q / df | p | I² | τ | |
|---|---|---|---|---|---|
| With I2462 | 8.55 % | 81.0 / 27 | 2.6 × 10⁻⁷ | 66.7 % | 6.36 pp |
| Without I2462 | 7.15 % | 28.9 / 26 | 0.32 | 10.0 % | 1.51 pp |
Residual against calibrated date, I2462 removed: r = +0.501, p = 0.009. By genetic sex, I2462 removed: male n = 16, mean 6.58 %; female n = 10, mean 6.35 % (Mann–Whitney p = 0.83). Sites with more than one dated individual, I2462 removed: Amesbury Down n = 5, mean 2.10 %; Baston and Langtoft n = 2, mean 7.60 %; East Kent Access n = 3, mean 10.43 %.
Checked against Supplementary Table 14’s own population-label columns rather than by external mapping: of 5,846 reported pairs, 1,157 carry population labels for both members, and no pair has English individuals on both sides. Every one of the 78 pairs involving an English individual has a Lower Rhine–Meuse partner, which is the table’s stated scope.
Individuals available under the table’s own labels: 19 England_N (including England_N_Megalithic and kinship-annotated variants), 11 England_C_EBA. Full group sizes from Supplementary Table 3 are 27 and 39.
Date gaps among the 1,157 pairs with dates for both members:
| Gap (years) | Pairs | Median longest segment |
|---|---|---|
| 0–200 | 483 | 16.1 cM |
| 200–400 | 224 | 14.3 cM |
| 400–800 | 274 | 14.1 cM |
| 800–1200 | 124 | 13.8 cM |
| 1200–1600 | 33 | 13.6 cM |
| 1600–1961 | 19 | 13.1 cM |
Spearman ρ between gap and longest segment = −0.341 (p = 5.9 × 10⁻³³). The required comparison spans roughly 1,600 years — England_N has a median date near 5500 BP, England_CA_EBA near 3800 BP. Detection at that depth occurs in the existing data: nineteen pairs exceed a 1,600-year gap, and the longest recorded is 1,961 years. Segment lengths compress toward the 12 cM reporting threshold but do not vanish. The clearest single instance is KD070.SG (England_Northumberland_EBA, 4306 BP) sharing 16.0 cM with an MN_Wartberg individual at 5164 BP across 858 years.
| Comparison | Individuals | Possible pairs | Detected | Rate | Median longest |
|---|---|---|---|---|---|
| England_BellBeaker × LNB_Bell_Beaker | 16 × 11 | 176 | 30 | 17.0 % | 15.6 cM |
| England_C_EBA × LNB_Bell_Beaker | 11 × 11 | 121 | 18 | 14.9 % | 14.2 cM |
| England_N × MN_Tiel | 19 × 2 | 38 | 17 | 44.7 % | 14.5 cM |
The first row is the appropriate benchmark: near-contemporary individuals with essentially all ancestry from the partner population. The third row’s high rate reflects only two MN_Tiel individuals, at least one of whom shares with many England_N individuals, and should not be used for scaling.
Modelling detections as Poisson with rate proportional to the ancestry fraction f = 0.076 times the benchmark rate of 0.170 (30 detections in 176 possible pairs), times a decay factor for the 1,600-year separation:
| Decay assumption | Coverage | Pairs | Expected detections | P(≥ 1) | P(≥ 3) |
|---|---|---|---|---|---|
| None | current (19 × 11) | 209 | 2.7 | 0.93 | 0.51 |
| None | full groups (27 × 39) | 1,053 | 13.6 | 1.00 | 1.00 |
| 50 % | current | 209 | 1.4 | 0.74 | 0.16 |
| 50 % | full groups | 1,053 | 6.8 | 1.00 | 0.97 |
| 75 % | current | 209 | 0.7 | 0.49 | 0.03 |
| 75 % | full groups | 1,053 | 3.4 | 0.97 | 0.66 |
Under the continental-source null the England_N-specific excess is zero, so any detection above the shared-deep-ancestry background is evidence for a British source. At full group coverage the test discriminates decisively across the whole range of decay assumptions; at the coverage used in the published table it does not.
Prune for relatedness. At least three of the 19 England_N individuals carry explicit kinship annotations, and those annotations reference a further five identifiers that also appear in the table — I21393, I21389, I30334, I30304, I21395, I30332, I30302. A pedigree therefore spans something like eight of the nineteen. The Poisson calculation above treats pairs as independent and consequently overstates power; the effective independent sample is materially smaller and should be reduced to one representative per pedigree before testing.
Run a matched-baseline arm. Time decay is the largest nuisance parameter and cannot be estimated well from 19 pairs. Running the England_CA_EBA arm simultaneously against date-matched continental candidates makes decay common to both arms and cancels it in the ratio. Candidates already present in the table, with median dates:
| Population | n | Median date (BP) |
|---|---|---|
| MN_Wartberg | 26 | 5140 |
| France_MontAime | 5 | 5162 |
| Ireland_MN | 8 | 5410 |
| MN_Hazendonk | 2 | 5470 |
| MN_Tiel | 2 | 5650 |
| Czechia_C_Baalberge | 7 | 5850 |
| France_N | 18 | 6550 |
Ireland_MN and MN_Wartberg are the best-matched to England_N’s median of about 5500 BP. France_N at 6550 BP is a poor baseline and Denmark_SouthScandinavia_LN at 4124 BP is too late.
Relax the coverage threshold. Only 12 of the 39 England_CA_EBA individuals and 6 of the 27 England_N individuals appear in Supplementary Table 14 under the external group mapping; the table’s own labels raise this to 11 and 19. Either way, roughly two-thirds of the relevant individuals are excluded by whatever coverage criterion was applied. The power calculation above shows this is the difference between a decisive test and an inconclusive one.
Caveats on the calculation. The linear scaling of detection rate in the ancestry fraction is an approximation that ignores the distribution of segment lengths contributed by a minority ancestry component, which will be shifted downward relative to a full-ancestry comparison and so will lose disproportionately at a fixed 12 cM threshold. The benchmark rate itself rests on 30 detections. And the Poisson independence assumption is violated by relatedness in both arms. Taken together these push the true power below the tabulated figures, which is why the full-group coverage matters rather than being a refinement.
All figures in this paper are produced by the accompanying script
reproduce.py (see Data and code availability),
which takes the three published supplementary workbooks as input and
prints every number in the order it appears here. The random seed for
the censoring simulation is fixed at 20260802; simulation figures are
stable to the second decimal place across seeds.
Required input files:
Booth et al. 2021 Supplementary Table 1 (Cambridge Core, supplementary material to doi:10.1017/S0959774321000019)
Olalde et al. 2018 Supplementary Tables (Nature 555, doi:10.1038/nature25738)
Olalde et al. 2026 Supplementary Tables (Nature, doi:10.1038/s41586-026-10111-8)
Dependencies: Python 3, numpy, pandas, scipy, openpyxl.
Booth, T.J., Brück, J., Brace, S. & Barnes, I. 2021. Tales from the supplementary information: ancestry change in Chalcolithic–Early Bronze Age Britain was gradual with varied kinship organization. Cambridge Archaeological Journal 31(3): 379–400. doi:10.1017/S0959774321000019.
Harney, É., Patterson, N., Reich, D. & Wakeley, J. 2021. Assessing the performance of qpAdm: a statistical tool for studying population admixture. Genetics 217(4): iyaa045. doi:10.1093/genetics/iyaa045.
Maier, R., Flegontov, P., Flegontova, O., Işıldak, U., Changmai, P. & Reich, D. 2023. On the limits of fitting complex models of population history to f-statistics. eLife 12: e85492. doi:10.7554/eLife.85492.
Olalde, I., Brace, S., Allentoft, M.E. et al. 2018. The Beaker phenomenon and the genomic transformation of northwest Europe. Nature 555: 190–196. doi:10.1038/nature25738.
Olalde, I., Altena, E., Bourgeois, Q. et al. 2026. Lasting Lower Rhine–Meuse forager ancestry shaped Bell Beaker expansion. Nature 652: 938–946. doi:10.1038/s41586-026-10111-8.
Methodological background on qpAdm behaviour, model rejection and rotating source analysis is not cited above because none of the calculations here depend on it, but readers evaluating the degrees-of-freedom convention in Appendix A and the interpretation of non-rejection in §8 should consult Harney et al. 2021 and Maier et al. 2023.
New §2, “The estimand, stated precisely.” The paper previously moved between two quantities — the size of the residue and its provenance — without naming the second. It is now defined as β, the proportion of the second-source component deriving from populations resident in Britain before c. 2450 BC. Sections 2 to 11 of the earlier draft are renumbered 3 to 12.
§9, the outliers (was §8). The earlier text read: “those four individuals were the entirety of the Beaker-period signal, and they survive independent testing with a proper Rhine–Meuse proxy.” That sentence conflated measurement with attribution. The magnitude of the outliers’ Neolithic-related component is confirmed; its source is subject to exactly the same non-identifiability as the group-level residue, and arguably more visibly so, since Appendix D.2 puts England_BB_highEEF’s hunter-gatherer ratio at 0.239, inside the continental cluster and within the individual scatter of England_N. The paper cannot deny at eight per cent what it asserts at twenty-seven. Corrected.
§10, the reconciliation (was §9). The description of the Beaker phase carried the same error and has been amended. A closing passage now sets out the three mechanisms that would produce the observed residue with β = 0 — founder sampling, continental top-up and proxy error — and notes that the temporal gradient of §6 points most naturally at the second, which is the reading under which the interpretive weight placed on the rising residue collapses entirely.
Appendix F is now Appendix E. The earlier draft ran A, B, C, D, F.
Appendix D.3, IBD coverage. The draft gave two different sets of counts for the same quantity — 6 of 27 and 12 of 39 in D.3, against 19 and 11 in F.1 — without explaining that these come from two different mappings of the table’s identifiers. Both are now given at first mention, with the discrepancy explained; on either count roughly two-thirds of the relevant individuals are absent.
Appendix A, Table A2. The column headed “df + (α/SE)²” did not describe the quantity actually tabulated. κ is computed as (χ² − df)/(α/SE)²; the column now shows (α/SE)² and the header is correct. The values are unchanged and the derived bound of 3.46–3.75 per cent is unaffected.
Bullet formatting in the Limitations and Sources lists, which had collapsed on the earlier post.
Nothing in the numerical results has changed. Sections 3 to 8, and Appendices B and C, are as posted.
