diff --git a/paper/pnas/body.tex b/paper/pnas/body.tex
index 41449b7..aef65a4 100644
--- a/paper/pnas/body.tex
+++ b/paper/pnas/body.tex
@@ -12,7 +12,7 @@ AI development increasingly resembles a population process. Models are specialis
 
 Machine learning has become a population-scale phenomenon. Public repositories host millions of models (Hugging Face alone grew past three million by 2026), and these are not independent creations: the overwhelming majority are fine-tunes, distillations, or merges of a small number of foundation models, forming large family trees whose lineage structure, inherited traits, and mutation dynamics are already being mapped with explicitly phylogenetic methods (1--3). This population also reproduces. Weight-space \emph{model merging}, the direct combination of trained parents into a new model, is mainstream community practice with standard tooling and thousands of hybrid checkpoints, including leaderboard-topping ones (4--7), and the engineering literature describes it in evolutionary vocabulary: ``crossover,'' ``mutation,'' ``mate choice,'' populations of merging models that climb benchmarks (5, 8--10).
 
-The generations are coupled through data as well as through weights. Successive models increasingly learn from model output rather than from fresh human experience: frontier alignment pipelines are now predominantly synthetic (over 98\% in documented cases; 11, 12), self-generated instruction data seeds whole lineages of descendants (13), a large and growing share of the public web is machine-generated or machine-translated text (14, 15), and the stock of human text is projected to be exhausted by frontier training within this decade (16). Meanwhile persistent multi-agent systems and emerging agent economies put many interacting models into sustained contact (17--20). A population whose members inherit from one another, recombine, and retransmit under these conditions is an evolving population in the technical sense. The claim of this paper is that the vocabulary should be given its mathematics: multigenerational model populations are systems whose inheritance, diversity, and compatibility must be managed, not merely collections of models to optimise, and the branch of biology that studies exactly this problem, the population genetics of the evolution of sex, transfers as a quantitative framework.
+The generations are coupled through data as well as through weights. Successive models increasingly learn from model output rather than from fresh human experience: frontier alignment pipelines are now predominantly synthetic (over 98\% in documented cases; 11, 12), self-generated instruction data seeds whole lineages of descendants (13), a large and growing share of the public web is machine-generated or machine-translated text (14, 15), and the stock of human text is projected to be exhausted by frontier training within this decade (16). Meanwhile persistent multi-agent systems and emerging agent economies put many interacting models into sustained contact (17--20). A population whose members inherit from one another, recombine, and retransmit under these conditions is an evolving population in the technical sense, and that observation motivates this work. Here we transfer the quantitative framework of the branch of biology built for exactly this situation, the population genetics of the evolution of sex, and use it to treat multigenerational model populations as systems whose inheritance, diversity, and compatibility can be measured, predicted, and managed.
 
 The diagnosis comes first. Training each generation of a model on the previous generation's output degrades it (\emph{model collapse}): rare capabilities vanish first, and the lineage drifts toward its own most common behaviour (21). That degradation is, mathematically, \emph{genetic drift}, the loss of rare variants that any finite population suffers when each generation is a finite sample of the last. The identification has been made repeatedly and independently: for sequential inference chains before deep learning (22), for language-model text ecosystems (23), as a closed-form first-extinction law placing collapse onset at the Wright--Fisher first-extinction time (24), and in quantitative-genetic form for self-consuming diffusion models (25). A diagnosis reached so often, from such different starting points, marks population genetics as the natural mathematics of the setting. It is also only the entry point. Population genetics is not, at heart, a theory of decay; it is a theory of the mechanisms that maintain and build populations despite decay (immigration, recombination, selection, population structure) and of where those mechanisms reach their limits. This paper develops that fuller structure for model populations: the arc from drift through its remedies to its limit, reproductive isolation, carried as one framework from closed forms to trained networks to language models.
 
diff --git a/paper/pnas/main.md b/paper/pnas/main.md
index e09cf2f..b7ae8fc 100644
--- a/paper/pnas/main.md
+++ b/paper/pnas/main.md
@@ -61,11 +61,10 @@ machine-generated or machine-translated text (14, 15), and the stock of human te
 exhausted by frontier training within this decade (16). Meanwhile persistent multi-agent systems and
 emerging agent economies put many interacting models into sustained contact (17–20). A population
 whose members inherit from one another, recombine, and retransmit under these conditions is an
-evolving population in the technical sense. The claim of this paper is that the vocabulary should be
-given its mathematics: multigenerational model populations are systems whose inheritance,
-diversity, and compatibility must be managed, not merely collections of models to optimise, and the
-branch of biology that studies exactly this problem, the population genetics of the evolution of sex,
-transfers as a quantitative framework.
+evolving population in the technical sense, and that observation motivates this work. Here we
+transfer the quantitative framework of the branch of biology built for exactly this situation, the
+population genetics of the evolution of sex, and use it to treat multigenerational model populations
+as systems whose inheritance, diversity, and compatibility can be measured, predicted, and managed.
 
 The diagnosis comes first. Training each generation of a model on the previous generation's output
 degrades it (*model collapse*): rare capabilities vanish first, and the lineage drifts toward its own
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