introduction rewritten: setting, diagnosis, question — nothing else

The Introduction is halved (1,360 -> 654 words, four paragraphs): the
model-population setting; the data-coupled generations + the thesis
sentence; the drift diagnosis placed in the literature; and the
motivating question (the four operator decisions with no principled
guidance + the continual-learning framing), closing on the value
anticipation without disclosing results. Evicted and rehomed: the
interpretation/explanation/prediction ladder (deleted — its content
lives in the calibrated Results and ledger); the answers-list (deleted —
results belong in Results); the correspondence walk-through (Muller's
ratchet moved to the minimal-model section with its scope clause;
immigration/Fisher-Muller/BDM citations anchored where the concepts are
developed in Results; the Livnat precursor and predictor-delta moved to
the Discussion ledger); the tiers-of-evidence and negative-results-
prominence sentences (deleted). The continual-learning operator mapping
moved into the Discussion block, retitled "Continual learning at the
population scale", deduplicated against its five offers. All 66
references wholesale-renumbered to the new first-appearance order and
the list reordered (invariant verified: in-text order = 1..66 = list).
Main text 4.7k words; 19 pp.

Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01BkRLcc18rwT2Lysu6PbG7v

paper/pnas/body.tex

@@ -16,21 +16,13 @@ The generations are coupled through data as well as through weights. Successive
 
 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.
 
-In machine learning's own terms, the problem this frame addresses is the field's oldest, \emph{continual learning}, reappearing one level up. Within a single network, sequential learning overwrites prior knowledge (catastrophic forgetting; 26, 27), and the discipline's remedies are, one by one, the population operators of this paper in single-model form: \emph{rehearsal and replay} of past data is grounding's within-lineage counterpart, and the field's empirically settled replay fractions, on the order of 1\% for instruction tuning (28) and 5\% to 25\% by distribution-shift strength in continual pretraining (29), sit where the minimal model's operational grounding threshold lies, a correspondence for which the framework supplies the missing theory (equilibrium diversity, and a per-capability survival law). \emph{Pseudo-rehearsal}, the replay of the network's own generated samples, proposed as a cure in 1995 (30) and revived as generative replay (31), is this paper's ungrounded null: immigration from a drifting source, benign for one hop and compounding into collapse over generations; verifier-filtering (32, 33) converts it back into grounding. \emph{Parameter isolation} (34, and frozen-base adapters, which forget far less; 35) is the engineered decorrelation our specialists use; \emph{complementary-learning-systems consolidation} (36--38) is our periodic adapter-into-base merge; the recent turn to \emph{merging as a continual-learning mechanism} (39--42) applies recombination within one lineage over time, where we apply it across lineages; and the observation that rare examples and long-tail knowledge are forgotten first (43--45) is tail-allele extinction observed one model at a time. One distinction is kept explicit throughout: catastrophic forgetting is largely deterministic interference from shifted training, whereas collapse is stochastic sampling drift; the two phenomena share their victims, the rare, and their remedies, but not their mechanism. To our knowledge, no prior work carries population-genetic formalism into continual learning itself; that bridge (replay as immigration with a survival law, merging as recombination with a compatibility criterion, consolidation as the slow store of a two-speed memory) is where this framework may matter most.
+The question this paper addresses is what to do with that diagnosis. An operator of a model population faces recurring decisions for which there is no principled guidance: how much verified real data does retraining need before a lineage decays; will combining two particular models compose their abilities or damage them; can incompatibility be detected before paying for a failed merge; and when should specialists be kept separate rather than consolidated? In practice these are settled by convention and by trial-and-error search. They are also, recognisably, machine learning's oldest problem at a new scale: \emph{continual learning}, the struggle to acquire new abilities without losing old ones (26, 27), transposed from a single network to a population whose members inherit from one another. Population genetics, we will argue, prices these decisions. Table 1 summarises the correspondences on which the argument runs; the sections that follow develop them from closed-form theory to experiments in trained networks and language models.
-
-We are explicit about what kind of contribution each claim is, distinguishing \emph{interpretation} (an existing result understood in population-genetic terms), \emph{explanation} (the transferred mechanism accounts for observations existing accounts leave open), and \emph{prediction} (the framework forecasts an unmeasured outcome). The paper is strongest on the first; makes concrete progress on the second (separating merge failures that are coordinate artefacts from those that are functional); and reports a first, bounded step on the third: a controlled predictive test in which pre-merge functional-disagreement measures, chosen by the framework, predicted merge damage on a constructed task grid while the tested weight-geometry baselines showed no detectable association.
-
-Stated as a problem: an operator of a model population today has no principled answer to four recurring decisions. How much verified real data does retraining need before a lineage decays? Will combining two particular models compose their abilities or damage them? Can incompatibility be detected before paying for a failed merge? And when should specialists be kept separate rather than consolidated? Current practice answers these with folklore constants and trial-and-error searches. The framework prices each decision, and several of its answers are not the intuitive ones. Averaging, the default combining operator, cancels the benefit of multiple parents to first order precisely in the regime where that benefit matters most, the rare-capability tail. Specialisation and divergence, widely treated as the threat to mergeability, produced no incompatibility in any regime we tested; conflicting conventions always did. Weight distance, the field's default compatibility signal, carried no detectable predictive signal in our controlled test, while a cheap behavioural measure did. And where the framework's numbers can be checked against settled practice, they land on it: the replay fractions that continual learning converged on empirically sit at the minimal model's threshold.
-
-The correspondences we develop, summarised in Table 1: single-teacher retraining is \emph{asexual reproduction}, and the irreversible arm of its decay shares the defining consequence of \emph{Muller's ratchet} (46): once every copy of a rare capability is gone from all parents and sources, no recombination can rebuild it, which is why remedies must act before fixation-by-loss (a consequence- level correspondence: the minimal model lacks the ratchet's recurrent deleterious-mutation mechanism, so irreversible loss alone does not identify that specific mechanism). Injecting verified real data is \emph{immigration} from a non-drifting source (32, 47, 48). Model merging is \emph{recombination}, and its central payoff, a merged model exceeding every parent, is the \emph{Fisher--Muller effect} (49, 50). Merging entangled skills courts \emph{outbreeding depression}; screening many candidate merges is engineered recombination with unusually flexible parent choice and pre-deployment screening (we use the shorthand \emph{directed sex}); restricting who merges with whom is \emph{population structure}. Merging's hard limit, models too diverged in function to combine, is \emph{reproductive isolation}, for which the Bateson--Dobzhansky--Muller theory of incompatibilities (51, 52) supplies the structure. The nearest precursor to this programme reads sex as an algorithm for mixability in the theory of computation (53), pre-dating model merging; the model-merging literature itself has strong empirical operators (4, 54, 55) and emerging merge-success predictors (56, 57), to which our delta is mechanism: \emph{when and why} failure is coordinate versus functional, and what moves the boundary.
-
-We support the framework at three tiers of evidence, in ascending realism and descending exactness: a \emph{minimal analytic model} validated against closed forms to a fraction of a percent; \emph{small trained networks} (MLPs, recurrent networks, an MNIST image generator) where the operators are measured in real weights; and \emph{language models} (LoRA-specialised Qwen models, 0.5B locally and 7B on a compute cluster) where the claims are tested as signs under seed replication. Negative results are reported with the same prominence as confirmations; they include the failure of an internal pre-registered prediction, a null on emergent speciation that bounds the analogy, and the sensitivity analyses on the predictive test.
 
 \section*{The minimal model, and where its exactness ends}
 
 Knowledge is modelled as a distribution \texttt{p\_t} over \texttt{K} discrete items (capabilities, facts, modes of behaviour), with a fixed true distribution \texttt{p*} whose rare tail carries the knowledge most at risk. One generation is: *draw \texttt{n} samples from the parent's distribution, optionally mix in \texttt{m} verified real samples (``grounding'', \texttt{g = m/(n+m)}), and refit the child\emph{. In this minimal inheritance model the resampling step }is* the Wright--Fisher process: the same equations, which we exploit as an engineering gate: our simulator reproduces the classical closed forms (heterozygosity decay \texttt{E[H\_t] = H\_0(1 − 1/n)\textasciicircum{}t}; the exact immigration--drift equilibrium; the closed-form multi-teacher union) to within 0.5\%, and these are standing tests in the codebase, not one-off checks.
 
-The boundary of the exactness matters, and we measured it rather than assumed it. Real training adds approximation, optimisation noise, and inductive bias, and when trained networks are fit against the exact drift null they deviate in \emph{opposite, architecture-specific} directions: a smoothing recurrent network resists collapse (keeping spurious variants alive), while a sharpening image generator accelerates it. A one-parameter \emph{learning kernel} (a smoothing knob and a sharpening knob on the refit) reproduces both. Throughout, a real learner is therefore treated as Wright--Fisher \emph{plus a signed, measurable estimator bias}, and the drift signs (rare-first loss; the grounding response) survived that bias in every architecture we tested, including a convolutional VAE retrained on its own generated digits, where the dry lineage collapses to a single blurred digit class while 10\% grounding holds all thirty modes (Fig. 1).
+The boundary of the exactness matters, and we measured it rather than assumed it. Real training adds approximation, optimisation noise, and inductive bias, and when trained networks are fit against the exact drift null they deviate in \emph{opposite, architecture-specific} directions: a smoothing recurrent network resists collapse (keeping spurious variants alive), while a sharpening image generator accelerates it. A one-parameter \emph{learning kernel} (a smoothing knob and a sharpening knob on the refit) reproduces both. Throughout, a real learner is therefore treated as Wright--Fisher \emph{plus a signed, measurable estimator bias}, and the drift signs (rare-first loss; the grounding response) survived that bias in every architecture we tested, including a convolutional VAE retrained on its own generated digits, where the dry lineage collapses to a single blurred digit class while 10\% grounding holds all thirty modes (Fig. 1). One consequence of drift deserves its genetic name. Retraining on a single parent is \emph{asexual reproduction}, and sustained loss under it carries the defining consequence of \emph{Muller's ratchet} (28): once every copy of a rare capability is gone from all parents and sources, no recombination can rebuild it, so remedies must act before fixation-by-loss (a consequence-level correspondence; the minimal model lacks the ratchet's recurrent-mutation driver).
 
 \textbf{Table 1.} The dictionary. Each correspondence is stated with the level of support it currently has (exact = closed form in the minimal model; empirical = measured in trained systems; hypothesis = stated with a falsifier, untested or unconfirmed). The full claim-by-claim ledger with assumptions and known limits is SI Appendix, Table S1.
 
@@ -53,7 +45,7 @@ Selection on a fitness function & Verifier-anchored selection (``reality that ca
 
 \subsection*{Grounding is immigration: cheap, with a floor}
 
-In the minimal model, grounding from a fixed real source is immigration into a drifting population, and the equilibrium diversity has a closed form our simulator matches exactly. That equilibrium is \emph{smooth} in the grounding fraction (there is no phase transition in aggregate diversity), so the practical number is an operational threshold, and we define it as such: under the tested population size and Zipf source distribution, \texttt{g \(\approx\) 0.05} retained most (\(\geq\)95\%) of equilibrium diversity indefinitely, with the required fraction depending on sample size, source distribution, and the chosen retention target (dependencies in SI). The engineering point survives the definition: verified real data is cheap insurance at fractions far below one. But the same analysis yields a floor the field's average-loss framing misses: under unstratified sampling from the source, a capability of rarity \texttt{p} appears in a real-data batch of size \texttt{m} with probability \texttt{1 − e\textasciicircum{}{−m\(\cdot\)p}}, so \texttt{m\(\cdot\)p \(\approx\) 1} marks roughly a 63\% chance of one example per batch: a soft observation floor, with higher confidence priced accordingly, and with distinct consequences for continuous retention, stationary occupancy, and reintroduction after loss (immigration can restore an absent item; SI separates these). Protecting the rarest knowledge under unstratified grounding is therefore priced per item at cost \texttt{\(\propto\) 1/p}; targeted or stratified sampling changes that cost, and recombination can recover rare capabilities \emph{that are still retained across complementary parents} (next section). In trained networks the \emph{sign} of the grounding response transfers everywhere we looked, with two deviations, both traced to the estimator bias above: sharp thresholds soften, and support-counting metrics decouple from truth (forward-KL is the operative collapse metric for a smoothing learner). On real images (Fig. 1B), dry self-training collapses a convolutional VAE to one mode while \textasciitilde{}10\% grounding holds all thirty (the trained model needs roughly twice the exact-operator fraction, the measured price of the estimator bias).
+In the minimal model, grounding from a fixed real source is \emph{immigration} into a drifting population (29--31), and the equilibrium diversity has a closed form our simulator matches exactly. That equilibrium is \emph{smooth} in the grounding fraction (there is no phase transition in aggregate diversity), so the practical number is an operational threshold, and we define it as such: under the tested population size and Zipf source distribution, \texttt{g \(\approx\) 0.05} retained most (\(\geq\)95\%) of equilibrium diversity indefinitely, with the required fraction depending on sample size, source distribution, and the chosen retention target (dependencies in SI). The engineering point survives the definition: verified real data is cheap insurance at fractions far below one. But the same analysis yields a floor the field's average-loss framing misses: under unstratified sampling from the source, a capability of rarity \texttt{p} appears in a real-data batch of size \texttt{m} with probability \texttt{1 − e\textasciicircum{}{−m\(\cdot\)p}}, so \texttt{m\(\cdot\)p \(\approx\) 1} marks roughly a 63\% chance of one example per batch: a soft observation floor, with higher confidence priced accordingly, and with distinct consequences for continuous retention, stationary occupancy, and reintroduction after loss (immigration can restore an absent item; SI separates these). Protecting the rarest knowledge under unstratified grounding is therefore priced per item at cost \texttt{\(\propto\) 1/p}; targeted or stratified sampling changes that cost, and recombination can recover rare capabilities \emph{that are still retained across complementary parents} (next section). In trained networks the \emph{sign} of the grounding response transfers everywhere we looked, with two deviations, both traced to the estimator bias above: sharp thresholds soften, and support-counting metrics decouple from truth (forward-KL is the operative collapse metric for a smoothing learner). On real images (Fig. 1B), dry self-training collapses a convolutional VAE to one mode while \textasciitilde{}10\% grounding holds all thirty (the trained model needs roughly twice the exact-operator fraction, the measured price of the estimator bias).
 
 \begin{figure*}[p]\centering  % fig1
 \includegraphics[width=\textwidth]{figs/fig1.pdf}
@@ -62,11 +54,11 @@ In the minimal model, grounding from a fixed real source is immigration into a d
 
 \subsection*{Recombination: a conservation law, its operators, and offspring that exceed every parent}
 
-The largest returns from the transfer concern merging. We begin with a result about the most common operator, stated with its assumptions. \textbf{Proposition (blending inheritance, rare-item regime).} Let K parents independently retain a rare item (mass \texttt{p} when retained), and let the child draw \texttt{n} samples either from one parent chosen at random or from the \emph{mean of the parents' output distributions}. Expected item mass is identical under the two schemes; and in the rare-item regime \texttt{n\(\cdot\)p/K \(\ll\) 1}, where per-item survival is first-order in sampled mass, expected \emph{survival} is also identical: the 1/K dilution of averaging cancels the K-parent union gain to first order, so in this regime adding parents through the output-mean does not increase expected tail retention. Two boundaries: outside that regime, survival is a convex function of mixed mass, so the variance reduction from averaging can \emph{reduce} extinction relative to a randomly chosen single parent; the cancellation is a first-order result about rare items, not a universal impossibility; and the contrasting union operator (keep each item's strongest source, then renormalise, which itself redistributes mass and presupposes a verifier or oracle to identify the strongest source) increases expected retention with K in all regimes in the minimal model. The practically important operators, \emph{weight averaging} (a nonlinear network's weight-mean does not compute its parents' output-mean) and \emph{routing among intact specialists} (58) (different storage and inference budgets from a single child), are its empirical cousins, and the measured bridge is a \emph{headroom rule}, stated qualitatively: in language models, union-preserving operators beat the weight-average where that average falls short of attainable performance, and add nothing where it does not (easy-versus-hard contrasts at two scales; a quantitative form of the relationship is untested). On easy tasks a capable base's average is already at ceiling and refinements add nothing; on hard tasks the average dilutes a fragile specialist below even the best single parent and routing wins by a wide margin (Fig. 6A--B).
+The largest returns from the transfer concern merging. We begin with a result about the most common operator, stated with its assumptions. \textbf{Proposition (blending inheritance, rare-item regime).} Let K parents independently retain a rare item (mass \texttt{p} when retained), and let the child draw \texttt{n} samples either from one parent chosen at random or from the \emph{mean of the parents' output distributions}. Expected item mass is identical under the two schemes; and in the rare-item regime \texttt{n\(\cdot\)p/K \(\ll\) 1}, where per-item survival is first-order in sampled mass, expected \emph{survival} is also identical: the 1/K dilution of averaging cancels the K-parent union gain to first order, so in this regime adding parents through the output-mean does not increase expected tail retention. Two boundaries: outside that regime, survival is a convex function of mixed mass, so the variance reduction from averaging can \emph{reduce} extinction relative to a randomly chosen single parent; the cancellation is a first-order result about rare items, not a universal impossibility; and the contrasting union operator (keep each item's strongest source, then renormalise, which itself redistributes mass and presupposes a verifier or oracle to identify the strongest source) increases expected retention with K in all regimes in the minimal model. The practically important operators, \emph{weight averaging} (a nonlinear network's weight-mean does not compute its parents' output-mean) and \emph{routing among intact specialists} (32) (different storage and inference budgets from a single child), are its empirical cousins, and the measured bridge is a \emph{headroom rule}, stated qualitatively: in language models, union-preserving operators beat the weight-average where that average falls short of attainable performance, and add nothing where it does not (easy-versus-hard contrasts at two scales; a quantitative form of the relationship is untested). On easy tasks a capable base's average is already at ceiling and refinements add nothing; on hard tasks the average dilutes a fragile specialist below even the best single parent and routing wins by a wide margin (Fig. 6A--B).
 
-The generative payoff is the Fisher--Muller effect: recombination assembles, in one offspring, complementary variants that arose in different lineages, producing a genotype fitter than any parent. In the multi-locus model, sexual merging of decorrelated specialists climbs to the global optimum, a genotype no parent held, while the best single parent and the blended average both plateau below (Fig. 2). In real language models the signature replicates under seed replication: merges of three LoRA (59) specialists beat every parent overall (decisively at 7B: 0.87 vs 0.77), and on the sharper worst-family metric the merged models are the only ones competent everywhere, in every seed (Fig. 6A).
+The generative payoff is the \emph{Fisher--Muller effect} (33, 34): recombination assembles, in one offspring, complementary variants that arose in different lineages, producing a genotype fitter than any parent. In the multi-locus model, sexual merging of decorrelated specialists climbs to the global optimum, a genotype no parent held, while the best single parent and the blended average both plateau below (Fig. 2). In real language models the signature replicates under seed replication: merges of three LoRA (35) specialists beat every parent overall (decisively at 7B: 0.87 vs 0.77), and on the sharper worst-family metric the merged models are the only ones competent everywhere, in every seed (Fig. 6A).
 
-Sex has risks and, for AI, an unfair advantage, both quantified on rugged (epistatic) NK landscapes (60) (Fig. 3). When skills are entangled, blind recombination produces offspring \emph{below} their parents (outbreeding depression), worsening with ruggedness, and the optimal recombination rate shrinks as entanglement grows. But an engineered population can do what biology cannot: recombine unbounded parents, choose complementary mates, and \emph{screen many candidate offspring against a verifier before keeping one}. This directed sex converts the outbreeding catastrophe into a reliable gain in the model (tracking or exceeding the best parent at every ruggedness) and replicates as a sign in language models: bred-and-screened merges beat the a-priori blend in every seed on headroom tasks, including one seed where the blend failed catastrophically and selection was immune (Fig. 6A). Finally, population \emph{structure} is itself a knob: sweeping the mate-pool breadth from monogamous (local) to promiscuous (panmictic) against ruggedness, wide mixing maximises the population mean while monotonically destroying diversity, and the best \emph{champion} shifts from wide breadth on smooth landscapes to intermediate breadth on rugged ones (Fig. 3C), the mating-system phenomenon known to structured-population search, mapped onto merging populations.
+Sex has risks and, for AI, an unfair advantage, both quantified on rugged (epistatic) NK landscapes (36) (Fig. 3). When skills are entangled, blind recombination produces offspring \emph{below} their parents (outbreeding depression), worsening with ruggedness, and the optimal recombination rate shrinks as entanglement grows. But an engineered population can do what biology cannot: recombine unbounded parents, choose complementary mates, and \emph{screen many candidate offspring against a verifier before keeping one}. This directed sex converts the outbreeding catastrophe into a reliable gain in the model (tracking or exceeding the best parent at every ruggedness) and replicates as a sign in language models: bred-and-screened merges beat the a-priori blend in every seed on headroom tasks, including one seed where the blend failed catastrophically and selection was immune (Fig. 6A). Finally, population \emph{structure} is itself a knob: sweeping the mate-pool breadth from monogamous (local) to promiscuous (panmictic) against ruggedness, wide mixing maximises the population mean while monotonically destroying diversity, and the best \emph{champion} shifts from wide breadth on smooth landscapes to intermediate breadth on rugged ones (Fig. 3C), the mating-system phenomenon known to structured-population search, mapped onto merging populations.
 
 \begin{figure*}[p]\centering  % fig2
 \includegraphics[width=\textwidth]{figs/fig2.pdf}
@@ -80,7 +72,7 @@ Sex has risks and, for AI, an unfair advantage, both quantified on rugged (epist
 
 \subsection*{The society: grounding, recombination, and diversity make complementary contributions}
 
-Composing the operators (Fig. 4) requires one definitional distinction first. In the inheritance model, grounding is \emph{grounded inheritance}: external samples added to the reproduction process (the data channel). In the society model, grounding is \emph{grounded evaluation}: selection weights true fitness against conformity to the population's own consensus, \texttt{g}\(\cdot\)true-fitness + (1−g)\(\cdot\)conformity, the analogue of scoring models by the crowd's approval (the fitness channel). These are related design ideas, since both couple the lineage to a non-drifting external signal, but they are different operators, and we name them separately. In the tested society (a finite agent population on a rugged NK landscape), a four-arm ablation separates the failure modes: the full system (grounded evaluation + directed recombination + diversity-preserving selection (61)) climbs to near the global optimum while keeping its specialists; removing grounded evaluation converges the population confidently on an unfit consensus (self-consumption); removing recombination strands it on local optima; removing diversity converges it prematurely to a worse answer. Each removal fails differently; the three implementations make complementary contributions \emph{under the tested conditions}; general joint necessity is not established (alternative mutation, restart, archive, or selection schemes could alter the picture). At language-model scale this composed loop remains unbuilt; it is the paper's largest stated gap.
+Composing the operators (Fig. 4) requires one definitional distinction first. In the inheritance model, grounding is \emph{grounded inheritance}: external samples added to the reproduction process (the data channel). In the society model, grounding is \emph{grounded evaluation}: selection weights true fitness against conformity to the population's own consensus, \texttt{g}\(\cdot\)true-fitness + (1−g)\(\cdot\)conformity, the analogue of scoring models by the crowd's approval (the fitness channel). These are related design ideas, since both couple the lineage to a non-drifting external signal, but they are different operators, and we name them separately. In the tested society (a finite agent population on a rugged NK landscape), a four-arm ablation separates the failure modes: the full system (grounded evaluation + directed recombination + diversity-preserving selection (37)) climbs to near the global optimum while keeping its specialists; removing grounded evaluation converges the population confidently on an unfit consensus (self-consumption); removing recombination strands it on local optima; removing diversity converges it prematurely to a worse answer. Each removal fails differently; the three implementations make complementary contributions \emph{under the tested conditions}; general joint necessity is not established (alternative mutation, restart, archive, or selection schemes could alter the picture). At language-model scale this composed loop remains unbuilt; it is the paper's largest stated gap.
 
 \begin{figure*}[p]\centering  % fig4
 \includegraphics[width=\textwidth]{figs/fig4.pdf}
@@ -89,11 +81,11 @@ Composing the operators (Fig. 4) requires one definitional distinction first. In
 
 \subsection*{The limit of sex: model speciation}
 
-Recombination presupposes compatible parents. In biology, lineages pushed far enough apart become separate species (reproductive isolation) through Bateson--Dobzhansky--Muller incompatibilities: changes harmless on their own background but deleterious in combination. A merged model is exactly the exposed hybrid. We built the analytic model (Fig. 5A): hybrid fitness tracks the parents while compatible, then peels off and crashes below the ancestor; the isolation cliff arrives earlier the denser the incompatibilities; and the incompatibility \emph{count} snowballs quadratically with divergence (52). We note that a super-linear count does not by itself entail a sharp performance cliff without the count-to-effect-size link, which the analytic model supplies under its assumptions and any neural test must establish separately.
+Recombination presupposes compatible parents. In biology, lineages pushed far enough apart become separate species (\emph{reproductive isolation}) through Bateson--Dobzhansky--Muller incompatibilities (38, 39): changes harmless on their own background but deleterious in combination. A merged model is exactly the exposed hybrid. We built the analytic model (Fig. 5A): hybrid fitness tracks the parents while compatible, then peels off and crashes below the ancestor; the isolation cliff arrives earlier the denser the incompatibilities; and the incompatibility \emph{count} snowballs quadratically with divergence (39). We note that a super-linear count does not by itself entail a sharp performance cliff without the count-to-effect-size link, which the analytic model supplies under its assumptions and any neural test must establish separately.
 
-In trained networks, the claim must survive a known alternative: merge barriers between independently trained networks are famously \emph{coordinate artefacts}, removable by re-aligning hidden units (62); richer symmetry groups remove more (63), with known failures beyond the shared-data regime (64). We therefore aligned under the composition of permutation matching and exact per-unit rescaling (the unit symmetry group of plain ReLU MLPs, as the search space) and decomposed the barrier (Fig. 5 C and D): two networks trained from different initialisations on the \emph{same} task have a barrier that this alignment removes essentially entirely (residual \(\approx\) 0.001, the aligned merge performing at parent level): coordinate, not functional; two networks trained on \emph{conflicting} label maps have a barrier the same alignment leaves largely unchanged (0.502 \(\rightarrow\) 0.497), with the merged model functionally dead. The tested alignment removes the same-task barrier but leaves the conflict-associated barrier intact, supporting a functional-conflict interpretation without proving optimal alignment: exact recovery of a permuted-and-rescaled copy validates a special case, so the removable share is a lower bound and the residual an upper bound. Sweeping conflict traces the cliff as hybrid fitness, 0.97 \(\rightarrow\) 0.03. The conflict floor itself is information-theoretic (no single model can satisfy contradictory conventions; SI Appendix, Proposition S2), with the framework's role being the \emph{structure around it}: which divergences generate conflict, and what moves the cliff.
+In trained networks, the claim must survive a known alternative: merge barriers between independently trained networks are famously \emph{coordinate artefacts}, removable by re-aligning hidden units (40); richer symmetry groups remove more (41), with known failures beyond the shared-data regime (42). We therefore aligned under the composition of permutation matching and exact per-unit rescaling (the unit symmetry group of plain ReLU MLPs, as the search space) and decomposed the barrier (Fig. 5 C and D): two networks trained from different initialisations on the \emph{same} task have a barrier that this alignment removes essentially entirely (residual \(\approx\) 0.001, the aligned merge performing at parent level): coordinate, not functional; two networks trained on \emph{conflicting} label maps have a barrier the same alignment leaves largely unchanged (0.502 \(\rightarrow\) 0.497), with the merged model functionally dead. The tested alignment removes the same-task barrier but leaves the conflict-associated barrier intact, supporting a functional-conflict interpretation without proving optimal alignment: exact recovery of a permuted-and-rescaled copy validates a special case, so the removable share is a lower bound and the residual an upper bound. Sweeping conflict traces the cliff as hybrid fitness, 0.97 \(\rightarrow\) 0.03. The conflict floor itself is information-theoretic (no single model can satisfy contradictory conventions; SI Appendix, Proposition S2), with the framework's role being the \emph{structure around it}: which divergences generate conflict, and what moves the cliff.
 
-The strongest constraint comes from the pre-registered \emph{emergent test}: true BDM incompatibilities are emergent (each lineage's changes harmless alone), so we let children diverge with \emph{no conflicting signal anywhere}, using complementary class specialists and divergent input conventions, to 6.4\(\times\) the base training. No isolation emerged (residual 0.000 throughout); instead the merge \emph{rescued} the two catastrophically-forgetting specialists (parents \(\approx\) 0.50, merge \(\approx\) 0.955, a sustained Fisher--Muller rescue). The same double result appears at the language-model tier (Fig. 5 E and F): conflicting conventions produce \emph{function-specific} hybrid breakdown (the merge scores below both parents on the conflicted function, while a budget-controlled design shows the disjoint skills merge unharmed), and over-training disjoint specialists 1\(\rightarrow\)12 epochs (cf. the merging literature's expert-duration effect; 65) produces no isolation at all --- the merge improves. Across every tier tested, isolation had to be provoked by functional conflict; specialisation alone did not speciate --- a bound on the analogy that sharpens the design rule: what breaks merging is conflicting conventions on shared circuitry, not divergence per se.
+The strongest constraint comes from the pre-registered \emph{emergent test}: true BDM incompatibilities are emergent (each lineage's changes harmless alone), so we let children diverge with \emph{no conflicting signal anywhere}, using complementary class specialists and divergent input conventions, to 6.4\(\times\) the base training. No isolation emerged (residual 0.000 throughout); instead the merge \emph{rescued} the two catastrophically-forgetting specialists (parents \(\approx\) 0.50, merge \(\approx\) 0.955, a sustained Fisher--Muller rescue). The same double result appears at the language-model tier (Fig. 5 E and F): conflicting conventions produce \emph{function-specific} hybrid breakdown (the merge scores below both parents on the conflicted function, while a budget-controlled design shows the disjoint skills merge unharmed), and over-training disjoint specialists 1\(\rightarrow\)12 epochs (cf. the merging literature's expert-duration effect; 43) produces no isolation at all --- the merge improves. Across every tier tested, isolation had to be provoked by functional conflict; specialisation alone did not speciate --- a bound on the analogy that sharpens the design rule: what breaks merging is conflicting conventions on shared circuitry, not divergence per se.
 
 \begin{figure*}[p]\centering  % fig5
 \includegraphics[width=\textwidth]{figs/fig5.pdf}
@@ -102,7 +94,7 @@ The strongest constraint comes from the pre-registered \emph{emergent test}: tru
 
 \subsection*{A controlled predictive test: functional conflict, measured pre-merge, predicts merge damage}
 
-The framework's prediction-level claim was put to a designed test (Fig. 6C). Thirty-nine parent pairs (13 conditions \(\times\) 3 seeds; rows are not independent --- parents share task-data seeds across conditions, so inference is condition-clustered, and because shared seeds also couple rows \emph{across} conditions we report per-seed and leave-one-seed-out sensitivity alongside) span three axes decorrelated by construction: \emph{conflict} (contradictory conventions on shared prompts, private budgets fixed), \emph{compatible overlap} (the same shared prompts under the same convention --- overlap and volume without conflict), and \emph{duration} (weight divergence with zero conflict). Before merging, six predictors are computed: \emph{confidence-weighted functional conflict} (bilateral confident disagreement on probes drawn blind to where conflict lives --- a proposed proxy for merge-relevant interactions, motivated by the observation that raw disagreement counts harmless complementation, one parent merely ignorant, as conflict), raw disagreement, gradient alignment at the shared base (56), LoRA-delta cosine and distance, and a cross-task performance baseline. The pre-registered outcome is the merge penalty against oracle parent potential (the hybrid-load analogue), also reported against best- and mean-parent references because the predictor ordering is sensitive to that choice.
+The framework's prediction-level claim was put to a designed test (Fig. 6C). Thirty-nine parent pairs (13 conditions \(\times\) 3 seeds; rows are not independent --- parents share task-data seeds across conditions, so inference is condition-clustered, and because shared seeds also couple rows \emph{across} conditions we report per-seed and leave-one-seed-out sensitivity alongside) span three axes decorrelated by construction: \emph{conflict} (contradictory conventions on shared prompts, private budgets fixed), \emph{compatible overlap} (the same shared prompts under the same convention --- overlap and volume without conflict), and \emph{duration} (weight divergence with zero conflict). Before merging, six predictors are computed: \emph{confidence-weighted functional conflict} (bilateral confident disagreement on probes drawn blind to where conflict lives --- a proposed proxy for merge-relevant interactions, motivated by the observation that raw disagreement counts harmless complementation, one parent merely ignorant, as conflict), raw disagreement, gradient alignment at the shared base (44), LoRA-delta cosine and distance, and a cross-task performance baseline. The pre-registered outcome is the merge penalty against oracle parent potential (the hybrid-load analogue), also reported against best- and mean-parent references because the predictor ordering is sensitive to that choice.
 
 The supported conclusion, stated conditionally: across this controlled grid, pre-merge functional disagreement predicted merge penalties (clustered bootstrap CIs excluding zero; held-out leave-one-condition-out \(\rho\) \(\approx\) 0.35--0.40), whereas LoRA-delta cosine and L2 showed no statistically detectable association; gradient alignment carried intermediate signal. Head-to-head predictor differences are not individually significant at this sample size; only these baselines were tested; and with three seeds, uncertainty about seed generalisation remains substantial --- though the seed sensitivity favours the functional measures (per-seed \(\rho\) stable at +0.37 to +0.53 in each seed alone, geometry \(\approx\) 0 in every seed, gradient alignment seed-unstable at −0.11 to −0.55). Two further results bound the claim: the initial two-axis grid's best predictor was delta-cosine (\(\rho\) = +0.60) --- an overlap artefact that the compatible-overlap control was added to expose, and did (collapse to +0.03); and the pre-registered internal prediction that confidence weighting would beat raw disagreement \emph{failed} (they are statistically indistinguishable as rank predictors), so the present evidence favours functional disagreement generally, not the DMI-specific refinement. The framework motivated the measurement and the controls; their success does not validate the specifically population-genetic mechanism. Whether the prediction improves a budget-matched operator choice, and whether it generalises to unfamiliar conflict structures and real task pairs, are the experiment's open front.
 
@@ -131,13 +123,13 @@ Predictive test & 13 conditions \(\times\) 3 seeds (0.5B) & Merge penalty vs ora
 
 \textbf{Design rules.} As engineering guidance, the results reduce to rules that an operator of a model population can apply, answering the four decisions posed in the Introduction. \emph{Ground every generation} in verified reality --- a few percent retained most diversity in our tested settings --- but price the rarest capabilities individually (observation probability \texttt{1 − e\textasciicircum{}{−m\(\cdot\)p}} per batch under unstratified sampling), consider targeted sampling for the deep tail, and use recombination to recover rare capabilities still retained across complementary parents. \emph{Merge, don't blend, when there is headroom}: keep specialists intact and route, or breed-and-screen candidate merges, whenever the naive average is far from ceiling; plain averaging is adequate only where a strong base has already composed the skills. \emph{Match the operator to entanglement}: merge freely when skills are additive; sparingly, with offspring selection, when they entangle; and expect the champion-optimal mating breadth to narrow as landscapes roughen. \emph{Preserve diversity as a first-class objective}, because selection can only preserve variety that exists, and in the tested society its removal produced a distinct failure mode. \emph{Before merging, measure functional conflict} --- cheap, pre-merge, and in our controlled setting predictive where the tested weight-distance baselines were not; and \emph{do not treat divergence or specialisation alone as evidence of incompatibility} --- in every regime we tested, what broke merging was conflicting conventions on shared circuitry, which is the thing to detect.
 
-\textbf{What this offers continual learning.} Read into the field where these results most directly land: (i) a first-principles account of the \emph{replay ratio}: the field's constants (\(\approx\)1\%, 5\%, 25\%; 28, 29) acquire an equilibrium theory and a sharper prediction, that the required fraction is set by the rarest capability one refuses to lose (the \texttt{1 − e\textasciicircum{}{−m\(\cdot\)p}} law) rather than by average loss, which is testable against published replay sweeps; (ii) a \emph{failure theory for generative replay}: self-generated rehearsal is safe for short horizons and compounds into collapse across generations unless verifier-filtered back into grounding (30--33); (iii) \emph{pre-merge interference prediction with a mechanism}: where the current state of the art fits regressions over candidate metrics (56), the functional-conflict measure arrives at a convergent signal from principle and comes with an operator prescription --- when conflict is high, do not average; route or breed-and-screen; (iv) a candidate \emph{decision rule for the consolidate-versus-stay-modular question} that currently splits the field's practice (keep adapters separate vs merge them; 54--58): union-preserving operators where headroom exists, fusion where the base composes, consolidation as the slow-store step; and (v) \emph{tail monitoring as the leading indicator}: continual-learning evaluation that averages over capabilities hides exactly the losses that drift theory says come first and, past a threshold, become irreversible. On that last point we note the standing objection that apparent forgetting can be skewed task-inference over latent capability rather than erasure (66); our irreversibility results concern oracle-measured behavioural distributions, and distinguishing latent from extinct capability at language-model scale is an open experiment whose outcome would be decisive for both readings.
+\textbf{Continual learning at the population scale.} Within a single network, the discipline's remedies for forgetting are this framework's operators writ small. Rehearsal and replay of stored data (26, 27) is grounded inheritance within one lineage, and the replay fractions the field settled on empirically, on the order of 1\% for instruction tuning (45) and 5\% to 25\% by distribution-shift strength in continual pretraining (46), sit where the minimal model's operational threshold lies. \emph{Pseudo-rehearsal}, the replay of a network's own generated samples, proposed as a cure in 1995 (47) and revived as generative replay (48), is precisely the ungrounded null studied here: immigration from a drifting source, benign for one hop, compounding over generations, with verifier-filtering (29, 49) converting it back into grounding. Parameter isolation (50), including frozen-base adapters, which forget far less (51), is engineered decorrelation; complementary-learning-systems consolidation (52--54) is the periodic adapter-into-base merge; the recent turn to merging as a continual-learning mechanism (55--58) applies recombination within one lineage over time, where this paper applies it across lineages; and the observation that rare examples and long-tail knowledge are forgotten first (59--61) is tail extinction seen one model at a time. The mechanisms differ (forgetting is largely deterministic interference, collapse is sampling drift) but the victims and the remedies coincide, and to our knowledge no prior work carries population-genetic formalism into continual learning. Read into that field, the results offer: (i) an equilibrium theory for the replay ratio, with the sharper prediction that the required fraction is set by the rarest capability one refuses to lose (the \texttt{1 − e\textasciicircum{}{−m\(\cdot\)p}} law) rather than by average loss, testable against published replay sweeps; (ii) a \emph{failure theory for generative replay}: self-generated rehearsal is safe for short horizons and compounds into collapse across generations unless verifier-filtered back into grounding (29, 47--49); (iii) \emph{pre-merge interference prediction with a mechanism}: where the current state of the art fits regressions over candidate metrics (44), the functional-conflict measure arrives at a convergent signal from principle and comes with an operator prescription --- when conflict is high, do not average; route or breed-and-screen; (iv) a candidate \emph{decision rule for the consolidate-versus-stay-modular question} that currently splits the field's practice (keep adapters separate vs merge them; 54--58): union-preserving operators where headroom exists, fusion where the base composes, consolidation as the slow-store step; and (v) \emph{tail monitoring as the leading indicator}: continual-learning evaluation that averages over capabilities hides exactly the losses that drift theory says come first and, past a threshold, become irreversible. On that last point we note the standing objection that apparent forgetting can be skewed task-inference over latent capability rather than erasure (62); our irreversibility results concern oracle-measured behavioural distributions, and distinguishing latent from extinct capability at language-model scale is an open experiment whose outcome would be decisive for both readings.
 
-\textbf{What is borrowed and what is ours.} The collapse-as-drift diagnosis is established prior work (21--25); so are the empirical facts that merges can beat parents, that decorrelated parents merge better, and that naive averaging loses to interference-aware or routed merges (4, 54, 55), that model populations can climb (5, 8--10), and that merge success admits ML-native predictors (56, 57). Ours is the framework-level synthesis --- inheritance, diversity, and compatibility as managed quantities --- together with: the conservation law for blending inheritance and its operator boundaries; the per-item grounding floor; the society ablation with its complementary failure modes; model speciation as a named, tested question, with the coordinate-versus-functional decomposition under permutation-and-rescaling alignment and the emergent null that bounds it; and the controlled predictive test with its controls. We claim the framework generated these measurements and experiments; we do not claim their outcomes validate a uniquely population-genetic mechanism, and one refinement it proposed was not supported.
+\textbf{What is borrowed and what is ours.} The collapse-as-drift diagnosis is established prior work (21--25); so are the empirical facts that merges can beat parents, that decorrelated parents merge better, and that naive averaging loses to interference-aware or routed merges (4, 63, 64), that model populations can climb (5, 8--10), and that merge success admits ML-native predictors (44, 65), correlational where this framework supplies mechanism; the reading of sex as an algorithm for mixability in the theory of computation (66) anticipated the transfer before model merging existed. Ours is the framework-level synthesis --- inheritance, diversity, and compatibility as managed quantities --- together with: the conservation law for blending inheritance and its operator boundaries; the per-item grounding floor; the society ablation with its complementary failure modes; model speciation as a named, tested question, with the coordinate-versus-functional decomposition under permutation-and-rescaling alignment and the emergent null that bounds it; and the controlled predictive test with its controls. We claim the framework generated these measurements and experiments; we do not claim their outcomes validate a uniquely population-genetic mechanism, and one refinement it proposed was not supported.
 
 \textbf{Limits and open problems.} The demonstrations are deliberately small: exact where small is a virtue, sign-level and seed-replicated at the language-model tier, on constructed task families with a trivially separable router and one model lineage (Qwen, 0.5B--7B). The composed society has not been built at language-model scale. The predictive test's next bars, in order of value: generalisation to \emph{unfamiliar} conflict structures and real task pairs; a demonstrably better \emph{budget-matched} merging decision; then scale replication. Beyond engineering, the framework's hardest open problem is the fitness function itself: selection optimises what is measured, and for knowledge systems the persuasive and the true compete --- grounding against a reality that can refuse is the only anchor we trust, and institutionalising that anchor (verification, replication, and challenge among models) is the society-level problem we pose but do not solve. What biology receives in return is a new model system: populations of learners where every genotype, environment, and mating decision is observable and manipulable --- where the evolution of sex can be studied with interventions (unbounded parents, offspring preview, directed mating) that no living system permits.
 
-\textbf{Outlook: the evolution of language models.} The Introduction's premise, that the model ecosystem is an evolving population, is also a forecast about where these results matter next. Language-model development is consolidating around exactly the operators studied here: synthetic-data flywheels (inheritance), merging and routing of specialist fine-tunes (recombination and population structure), verifier-gated data pipelines (grounded selection), and periodic consolidation of adapters into new bases. The framework's reading of that trajectory is concrete. If coming model generations remain what our tested regimes found, freely recombinable in the absence of conflicting conventions, then the ecosystem evolves as one interbreeding population, and the levers that matter are grounding budgets priced per rare capability and diversity preserved deliberately. If instead long-horizon specialisation at scale begins to produce emergent incompatibility, as the expert-training-duration observations hint (65) and our small-scale null does not rule out, then lineages will begin to speciate, and the ecosystem's future is a set of diverging species connected by routing rather than by merging. Which of the two it will be is measurable now, with the pre-merge conflict instruments this paper tested.
+\textbf{Outlook: the evolution of language models.} The Introduction's premise, that the model ecosystem is an evolving population, is also a forecast about where these results matter next. Language-model development is consolidating around exactly the operators studied here: synthetic-data flywheels (inheritance), merging and routing of specialist fine-tunes (recombination and population structure), verifier-gated data pipelines (grounded selection), and periodic consolidation of adapters into new bases. The framework's reading of that trajectory is concrete. If coming model generations remain what our tested regimes found, freely recombinable in the absence of conflicting conventions, then the ecosystem evolves as one interbreeding population, and the levers that matter are grounding budgets priced per rare capability and diversity preserved deliberately. If instead long-horizon specialisation at scale begins to produce emergent incompatibility, as the expert-training-duration observations hint (43) and our small-scale null does not rule out, then lineages will begin to speciate, and the ecosystem's future is a set of diverging species connected by routing rather than by merging. Which of the two it will be is measurable now, with the pre-merge conflict instruments this paper tested.
 
 \section*{Materials and Methods}
 
@@ -179,11 +171,27 @@ Predictive test & 13 conditions \(\times\) 3 seeds (0.5B) & Merge penalty vs ora
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+\item M. Gerstgrasser, et al., Is model collapse inevitable? Breaking the curse of recursion by accumulating real and synthetic data. \emph{Conf. Lang. Model.} (2024). https://doi.org/10.48550/arXiv.2404.01413.
+\item S. Wright, Evolution in Mendelian populations. \emph{Genetics} \textbf{16}, 97--159 (1931).
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+\item R. A. Fisher, \emph{The Genetical Theory of Natural Selection} (Clarendon Press, 1930).
+\item H. J. Muller, Some genetic aspects of sex. \emph{Am. Nat.} \textbf{66}, 118--138 (1932).
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+\item S. A. Kauffman, S. Levin, Towards a general theory of adaptive walks on rugged landscapes. \emph{J. Theor. Biol.} \textbf{128}, 11--45 (1987).
+\item J. Lehman, K. O. Stanley, Abandoning objectives: Evolution through the search for novelty alone. \emph{Evol. Comput.} \textbf{19}, 189--223 (2011).
+\item H. A. Orr, The population genetics of speciation: The evolution of hybrid incompatibilities. \emph{Genetics} \textbf{139}, 1805--1813 (1995).
+\item H. A. Orr, M. Turelli, The evolution of postzygotic isolation: Accumulating Dobzhansky--Muller incompatibilities. \emph{Evolution} \textbf{55}, 1085--1094 (2001).
+\item S. K. Ainsworth, J. Hayase, S. Srinivasa, Git Re-Basin: Merging models modulo permutation symmetries. \emph{Int. Conf. Learn. Represent.} (2023). https://doi.org/10.48550/arXiv.2209.04836.
+\item T. Li, Z. Shen, Scaling linear mode connectivity and merging to billion-parameter pretrained transformers. arXiv [Preprint] (2026). https://doi.org/10.48550/arXiv.2606.23607.
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+\item N. Kozodoi, Z. Afolabi, J. Butler, Are we merging the right models? Impact of expert training duration on model merging for LLMs. arXiv [Preprint] (2026). https://doi.org/10.48550/arXiv.2607.11997.
+\item L. Zhou, B. Zhao, R. Yu, E. Rodolà, Demystifying mergeability: Interpretable properties to predict model merging success. arXiv [Preprint] (2026). https://doi.org/10.48550/arXiv.2601.22285.
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-\item H. J. Muller, The relation of recombination to mutational advance. \emph{Mutat. Res.} \textbf{1}, 2--9 (1964).
+\item S. Kotha, J. M. Springer, A. Raghunathan, Understanding catastrophic forgetting in language models via implicit inference. \emph{Int. Conf. Learn. Represent.} (2024). https://doi.org/10.48550/arXiv.2309.10105.
-\item M. Gerstgrasser, et al., Is model collapse inevitable? Breaking the curse of recursion by accumulating real and synthetic data. \emph{Conf. Lang. Model.} (2024). https://doi.org/10.48550/arXiv.2404.01413.
-\item S. Wright, Evolution in Mendelian populations. \emph{Genetics} \textbf{16}, 97--159 (1931).
-\item R. A. Fisher, \emph{The Genetical Theory of Natural Selection} (Clarendon Press, 1930).
-\item H. J. Muller, Some genetic aspects of sex. \emph{Am. Nat.} \textbf{66}, 118--138 (1932).
-\item H. A. Orr, The population genetics of speciation: The evolution of hybrid incompatibilities. \emph{Genetics} \textbf{139}, 1805--1813 (1995).
-\item H. A. Orr, M. Turelli, The evolution of postzygotic isolation: Accumulating Dobzhansky--Muller incompatibilities. \emph{Evolution} \textbf{55}, 1085--1094 (2001).
-\item A. Livnat, C. Papadimitriou, Sex as an algorithm: The theory of evolution under the lens of computation. \emph{Commun. ACM} \textbf{59}, 84--93 (2016).
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-\item L. Zhou, B. Zhao, R. Yu, E. Rodolà, Demystifying mergeability: Interpretable properties to predict model merging success. arXiv [Preprint] (2026). https://doi.org/10.48550/arXiv.2601.22285.
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-\item J. Pari, S. Jelassi, P. Agrawal, Collective model intelligence requires compatible specialization. arXiv [Preprint] (2024). https://doi.org/10.48550/arXiv.2411.02207.
+\item A. Livnat, C. Papadimitriou, Sex as an algorithm: The theory of evolution under the lens of computation. \emph{Commun. ACM} \textbf{59}, 84--93 (2016).
-\item E. J. Hu, et al., LoRA: Low-rank adaptation of large language models. \emph{Int. Conf. Learn. Represent.} (2022). https://doi.org/10.48550/arXiv.2106.09685.
-\item S. A. Kauffman, S. Levin, Towards a general theory of adaptive walks on rugged landscapes. \emph{J. Theor. Biol.} \textbf{128}, 11--45 (1987).
-\item J. Lehman, K. O. Stanley, Abandoning objectives: Evolution through the search for novelty alone. \emph{Evol. Comput.} \textbf{19}, 189--223 (2011).
-\item S. K. Ainsworth, J. Hayase, S. Srinivasa, Git Re-Basin: Merging models modulo permutation symmetries. \emph{Int. Conf. Learn. Represent.} (2023). https://doi.org/10.48550/arXiv.2209.04836.
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-\item N. Kozodoi, Z. Afolabi, J. Butler, Are we merging the right models? Impact of expert training duration on model merging for LLMs. arXiv [Preprint] (2026). https://doi.org/10.48550/arXiv.2607.11997.
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 \end{enumerate}
 

paper/pnas/main.md

@@ -82,77 +82,17 @@ structure) and of where those mechanisms reach their limits. This paper develops
 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.
 
-In machine learning's own terms, the problem this frame addresses is the field's oldest,
+The question this paper addresses is what to do with that diagnosis. An operator of a model
-*continual learning*, reappearing one level up. Within a single network, sequential learning
+population faces recurring decisions for which there is no principled guidance: how much verified
-overwrites prior knowledge (catastrophic forgetting; 26, 27), and the discipline's remedies are, one
+real data does retraining need before a lineage decays; will combining two particular models compose
-by one, the population operators of this paper in single-model form: *rehearsal and replay* of past
+their abilities or damage them; can incompatibility be detected before paying for a failed merge; and
-data is grounding's within-lineage counterpart, and the field's empirically settled replay fractions,
+when should specialists be kept separate rather than consolidated? In practice these are settled by
-on the order of 1% for instruction tuning (28) and 5% to 25% by distribution-shift strength in
+convention and by trial-and-error search. They are also, recognisably, machine learning's oldest
-continual pretraining (29), sit where the minimal model's operational grounding threshold lies, a
+problem at a new scale: *continual learning*, the struggle to acquire new abilities without losing old
-correspondence for which the framework supplies the missing theory (equilibrium diversity, and a
+ones (26, 27), transposed from a single network to a population whose members inherit from one
-per-capability survival law). *Pseudo-rehearsal*, the replay of the network's own generated
+another. Population genetics, we will argue, prices these decisions. Table 1 summarises the
-samples, proposed as a cure in 1995 (30) and revived as generative replay (31), is this paper's
+correspondences on which the argument runs; the sections that follow develop them from closed-form
-ungrounded null: immigration from a drifting source, benign for one hop and compounding into
+theory to experiments in trained networks and language models.
-collapse over generations; verifier-filtering (32, 33) converts it back into grounding.
-*Parameter isolation* (34, and frozen-base adapters, which forget far less; 35) is the engineered
-decorrelation our specialists use; *complementary-learning-systems consolidation* (36–38) is our
-periodic adapter-into-base merge; the recent turn to *merging as a continual-learning mechanism*
-(39–42) applies recombination within one lineage over time, where we apply it across lineages; and
-the observation that rare examples and long-tail knowledge are forgotten first (43–45) is
-tail-allele extinction observed one model at a time. One distinction is kept explicit throughout:
-catastrophic forgetting is largely deterministic interference from shifted training, whereas collapse
-is stochastic sampling drift; the two phenomena share their victims, the rare, and their remedies, but not their mechanism. To our knowledge, no prior work carries population-genetic formalism into continual
-learning itself; that bridge (replay as immigration with a survival law, merging as recombination with a
-compatibility criterion, consolidation as the slow store of a two-speed memory) is where this
-framework may matter most.
-
-We are explicit about what kind of contribution each claim is, distinguishing *interpretation* (an existing result understood in population-genetic terms),
-*explanation* (the transferred mechanism accounts for observations existing accounts leave open),
-and *prediction* (the framework forecasts an unmeasured outcome). The paper is strongest on the
-first; makes concrete progress on the second (separating merge failures that are coordinate artefacts
-from those that are functional); and reports a first, bounded step on the third: a controlled
-predictive test in which pre-merge functional-disagreement measures, chosen by the framework,
-predicted merge damage on a constructed task grid while the tested weight-geometry baselines showed
-no detectable association.
-
-Stated as a problem: an operator of a model population today has no principled answer to four
-recurring decisions. How much verified real data does retraining need before a lineage decays?
-Will combining two particular models compose their abilities or damage them? Can incompatibility be
-detected before paying for a failed merge? And when should specialists be kept separate rather than
-consolidated? Current practice answers these with folklore constants and trial-and-error searches.
-The framework prices each decision, and several of its answers are not the intuitive ones. Averaging,
-the default combining operator, cancels the benefit of multiple parents to first order precisely in
-the regime where that benefit matters most, the rare-capability tail. Specialisation and divergence,
-widely treated as the threat to mergeability, produced no incompatibility in any regime we tested;
-conflicting conventions always did. Weight distance, the field's default compatibility signal, carried
-no detectable predictive signal in our controlled test, while a cheap behavioural measure did. And
-where the framework's numbers can be checked against settled practice, they land on it: the replay
-fractions that continual learning converged on empirically sit at the minimal model's threshold.
-
-The correspondences we develop, summarised in Table 1: single-teacher retraining is *asexual
-reproduction*, and the irreversible arm of its decay shares the defining consequence of *Muller's
-ratchet* (46): once every copy of a rare capability is gone from all parents and sources, no
-recombination can rebuild it, which is why remedies must act before fixation-by-loss (a consequence-
-level correspondence: the minimal model lacks the ratchet's recurrent deleterious-mutation mechanism,
-so irreversible loss alone does not identify that specific mechanism). Injecting verified real data is
-*immigration* from a non-drifting source (32, 47, 48). Model merging is *recombination*, and its
-central payoff, a merged model exceeding every parent, is the *Fisher–Muller effect* (49, 50).
-Merging entangled skills courts *outbreeding depression*; screening many candidate merges is
-engineered recombination with unusually flexible parent choice and pre-deployment screening (we use
-the shorthand *directed sex*); restricting who merges with whom is *population structure*. Merging's hard limit, models too diverged in function to combine, is *reproductive
-isolation*, for which the Bateson–Dobzhansky–Muller theory of incompatibilities (51, 52) supplies the
-structure. The nearest precursor to this programme reads sex as an algorithm for mixability in the
-theory of computation (53), pre-dating model merging; the model-merging literature itself has strong
-empirical operators (4, 54, 55) and emerging merge-success predictors (56, 57), to which our delta is
-mechanism: *when and why* failure is coordinate versus functional, and what moves the boundary.
-
-We support the framework at three tiers of evidence, in ascending realism and descending exactness: a
-*minimal analytic model* validated against closed forms to a fraction of a percent; *small trained
-networks* (MLPs, recurrent networks, an MNIST image generator) where the operators are measured in
-real weights; and *language models* (LoRA-specialised Qwen models, 0.5B locally and 7B on a compute
-cluster) where the claims are tested as signs under seed replication. Negative results are reported with the same prominence as confirmations; they include the failure of
-an internal pre-registered prediction, a null on emergent speciation that bounds the analogy, and the
-sensitivity analyses on the predictive test.
 
 ## The minimal model, and where its exactness ends
 
@@ -174,7 +114,11 @@ refit) reproduces both. Throughout, a real learner is therefore treated as Wrigh
 estimator bias*, and the drift signs (rare-first loss; the grounding response)
 survived that bias in every architecture we tested, including a convolutional VAE retrained on its own
 generated digits, where the dry lineage collapses to a single blurred digit class while 10% grounding
-holds all thirty modes (Fig. 1).
+holds all thirty modes (Fig. 1). One consequence of drift deserves its genetic name. Retraining on a
+single parent is *asexual reproduction*, and sustained loss under it carries the defining consequence
+of *Muller's ratchet* (28): once every copy of a rare capability is gone from all parents and sources,
+no recombination can rebuild it, so remedies must act before fixation-by-loss (a consequence-level
+correspondence; the minimal model lacks the ratchet's recurrent-mutation driver).
 
 **Table 1.** The dictionary. Each correspondence is stated with the level of support it currently has
 (exact = closed form in the minimal model; empirical = measured in trained systems; hypothesis =
@@ -197,8 +141,8 @@ known limits is SI Appendix, Table S1.
 
 ### Grounding is immigration: cheap, with a floor
 
-In the minimal model, grounding from a fixed real source is immigration into a drifting population,
+In the minimal model, grounding from a fixed real source is *immigration* into a drifting population
-and the equilibrium diversity has a closed form our simulator matches exactly. That equilibrium is
+(29–31), and the equilibrium diversity has a closed form our simulator matches exactly. That equilibrium is
 *smooth* in the grounding fraction (there is no phase transition in aggregate diversity), so the
 practical number is an operational threshold, and we define it as such: under the tested population
 size and Zipf source distribution, `g ≈ 0.05` retained most (≥95%) of equilibrium diversity
@@ -238,7 +182,7 @@ contrasting union operator (keep each item's strongest source, then renormalise,
 redistributes mass and presupposes a verifier or oracle to identify the strongest source) increases
 expected retention with K in all regimes in the minimal model. The practically important
 operators, *weight averaging* (a nonlinear network's weight-mean does not compute its parents'
-output-mean) and *routing among intact specialists* (58) (different storage and inference budgets from a
+output-mean) and *routing among intact specialists* (32) (different storage and inference budgets from a
 single child), are its empirical cousins, and the measured bridge is a *headroom rule*, stated qualitatively: in language models,
 union-preserving operators beat the weight-average where that average falls short of attainable
 performance, and add nothing where it does not (easy-versus-hard contrasts at two scales; a
@@ -246,15 +190,15 @@ quantitative form of the relationship is untested). On easy tasks a capable base
 add nothing; on hard tasks the average dilutes a fragile specialist below even the best single parent
 and routing wins by a wide margin (Fig. 6A–B).
 
-The generative payoff is the Fisher–Muller effect: recombination assembles, in one offspring,
+The generative payoff is the *Fisher–Muller effect* (33, 34): recombination assembles, in one offspring,
 complementary variants that arose in different lineages, producing a genotype fitter than any parent.
 In the multi-locus model, sexual merging of decorrelated specialists climbs to the global optimum, a
 genotype no parent held, while the best single parent and the blended average both plateau below
 (Fig. 2). In real language models the signature replicates under seed replication: merges of three
-LoRA (59) specialists beat every parent overall (decisively at 7B: 0.87 vs 0.77), and on the sharper
+LoRA (35) specialists beat every parent overall (decisively at 7B: 0.87 vs 0.77), and on the sharper
 worst-family metric the merged models are the only ones competent everywhere, in every seed (Fig. 6A).
 
-Sex has risks and, for AI, an unfair advantage, both quantified on rugged (epistatic) NK landscapes (60)
+Sex has risks and, for AI, an unfair advantage, both quantified on rugged (epistatic) NK landscapes (36)
 (Fig. 3). When skills are entangled, blind recombination produces offspring *below* their parents
 (outbreeding depression), worsening with ruggedness, and the optimal recombination rate shrinks as
 entanglement grows. But an engineered population can do what biology cannot: recombine unbounded
@@ -283,7 +227,7 @@ the analogue of scoring models by the crowd's approval (the fitness channel). Th
 ideas, since both couple the lineage to a non-drifting external signal, but they are different operators,
 and we name them separately. In the tested society (a finite agent population on a rugged NK
 landscape), a four-arm ablation separates the failure modes: the full system (grounded evaluation +
-directed recombination + diversity-preserving selection (61)) climbs to near the global optimum while
+directed recombination + diversity-preserving selection (37)) climbs to near the global optimum while
 keeping its specialists; removing grounded evaluation converges the population confidently on an unfit
 consensus (self-consumption); removing recombination strands it on local optima; removing diversity
 converges it prematurely to a worse answer. Each removal fails differently; the three implementations
@@ -296,18 +240,18 @@ language-model scale this composed loop remains unbuilt; it is the paper's large
 ### The limit of sex: model speciation
 
 Recombination presupposes compatible parents. In biology, lineages pushed far enough apart become
-separate species (reproductive isolation) through Bateson–Dobzhansky–Muller incompatibilities:
+separate species (*reproductive isolation*) through Bateson–Dobzhansky–Muller incompatibilities (38, 39):
 changes harmless on their own background but deleterious in combination. A merged model is exactly the
 exposed hybrid. We built the analytic model (Fig. 5A): hybrid fitness tracks the parents while
 compatible, then peels off and crashes below the ancestor; the isolation cliff arrives earlier the
 denser the incompatibilities; and the incompatibility *count* snowballs quadratically with divergence
-(52). We note that a super-linear count does not by itself entail a sharp performance cliff without
+(39). We note that a super-linear count does not by itself entail a sharp performance cliff without
 the count-to-effect-size link, which the analytic model supplies under its assumptions and any neural
 test must establish separately.
 
 In trained networks, the claim must survive a known alternative: merge barriers between independently
-trained networks are famously *coordinate artefacts*, removable by re-aligning hidden units (62);
+trained networks are famously *coordinate artefacts*, removable by re-aligning hidden units (40);
-richer symmetry groups remove more (63), with known failures beyond the shared-data regime (64). We therefore aligned under the composition of
+richer symmetry groups remove more (41), with known failures beyond the shared-data regime (42). We therefore aligned under the composition of
 permutation matching and exact per-unit rescaling (the unit symmetry group of plain ReLU MLPs, as the
 search space) and decomposed the barrier (Fig. 5 C and D): two networks trained from different
 initialisations on the *same* task have a barrier that this alignment removes essentially entirely
@@ -330,7 +274,7 @@ catastrophically-forgetting specialists (parents ≈ 0.50, merge ≈ 0.955, a su
 rescue). The same double result appears at the language-model tier (Fig. 5 E and F): conflicting conventions
 produce *function-specific* hybrid breakdown (the merge scores below both parents on the conflicted
 function, while a budget-controlled design shows the disjoint skills merge unharmed), and over-training
-disjoint specialists 1→12 epochs (cf. the merging literature's expert-duration effect; 65) produces
+disjoint specialists 1→12 epochs (cf. the merging literature's expert-duration effect; 43) produces
 no isolation at all — the merge improves. Across every tier
 tested, isolation had to be provoked by functional conflict; specialisation alone did not speciate
 — a bound on the analogy that sharpens the design rule: what breaks merging is conflicting conventions
@@ -350,7 +294,7 @@ divergence with zero conflict). Before merging, six predictors are computed: *co
 functional conflict* (bilateral confident disagreement on probes drawn blind to where conflict lives —
 a proposed proxy for merge-relevant interactions, motivated by the observation that raw disagreement
 counts harmless complementation, one parent merely ignorant, as conflict), raw disagreement, gradient
-alignment at the shared base (56), LoRA-delta cosine and distance, and a cross-task performance
+alignment at the shared base (44), LoRA-delta cosine and distance, and a cross-task performance
 baseline. The pre-registered outcome is the merge penalty against oracle parent potential (the
 hybrid-load analogue), also reported against best- and mean-parent references because the predictor
 ordering is sensitive to that choice.
@@ -407,15 +351,29 @@ tested weight-distance baselines were not; and *do not treat divergence or speci
 evidence of incompatibility* — in every regime we tested, what broke merging was conflicting
 conventions on shared circuitry, which is the thing to detect.
 
-**What this offers continual learning.** Read into the field where these results most directly land:
+**Continual learning at the population scale.** Within a single network, the discipline's remedies
-(i) a first-principles account of the *replay ratio*: the field's constants (≈1%, 5%, 25%; 28, 29)
+for forgetting are this framework's operators writ small. Rehearsal and replay of stored data (26, 27)
-acquire an equilibrium theory and a sharper prediction, that the required fraction is set by the
+is grounded inheritance within one lineage, and the replay fractions the field settled on empirically,
-rarest capability one refuses to lose (the `1 − e^{−m·p}` law) rather than by average loss, which is
+on the order of 1% for instruction tuning (45) and 5% to 25% by distribution-shift strength in
-testable against published replay sweeps; (ii) a *failure theory for generative replay*:
+continual pretraining (46), sit where the minimal model's operational threshold lies.
+*Pseudo-rehearsal*, the replay of a network's own generated samples, proposed as a cure in 1995 (47)
+and revived as generative replay (48), is precisely the ungrounded null studied here: immigration from
+a drifting source, benign for one hop, compounding over generations, with verifier-filtering (29, 49)
+converting it back into grounding. Parameter isolation (50), including frozen-base adapters, which forget far
+less (51), is engineered decorrelation; complementary-learning-systems consolidation (52–54) is the periodic
+adapter-into-base merge; the recent turn to merging as a continual-learning mechanism (55–58) applies
+recombination within one lineage over time, where this paper applies it across lineages; and the
+observation that rare examples and long-tail knowledge are forgotten first (59–61) is tail extinction
+seen one model at a time. The mechanisms differ (forgetting is largely deterministic interference,
+collapse is sampling drift) but the victims and the remedies coincide, and to our knowledge no prior
+work carries population-genetic formalism into continual learning. Read into that field, the results
+offer: (i) an equilibrium theory for the replay ratio, with the sharper prediction that the required
+fraction is set by the rarest capability one refuses to lose (the `1 − e^{−m·p}` law) rather than by
+average loss, testable against published replay sweeps; (ii) a *failure theory for generative replay*:
 self-generated rehearsal is safe for short horizons and compounds into collapse across generations
-unless verifier-filtered back into grounding (30–33); (iii) *pre-merge interference
+unless verifier-filtered back into grounding (29, 47–49); (iii) *pre-merge interference
 prediction with a mechanism*: where the current state of the art fits regressions over candidate
-metrics (56), the functional-conflict measure arrives at a convergent signal from principle and comes
+metrics (44), the functional-conflict measure arrives at a convergent signal from principle and comes
 with an operator prescription — when conflict is high, do not average; route or breed-and-screen;
 (iv) a candidate *decision rule for the consolidate-versus-stay-modular question* that currently
 splits the field's practice (keep adapters separate vs merge them; 54–58): union-preserving operators
@@ -423,14 +381,16 @@ where headroom exists, fusion where the base composes, consolidation as the slow
 *tail monitoring as the leading indicator*: continual-learning evaluation that averages over
 capabilities hides exactly the losses that drift theory says come first and, past a threshold, become
 irreversible. On that last point we note the standing objection that apparent forgetting can be
-skewed task-inference over latent capability rather than erasure (66); our irreversibility results
+skewed task-inference over latent capability rather than erasure (62); our irreversibility results
 concern oracle-measured behavioural distributions, and distinguishing latent from extinct capability
 at language-model scale is an open experiment whose outcome would be decisive for both readings.
 
 **What is borrowed and what is ours.** The collapse-as-drift diagnosis is established prior work
 (21–25); so are the empirical facts that merges can beat parents, that decorrelated parents merge better, and that
-naive averaging loses to interference-aware or routed merges (4, 54, 55), that model populations can
+naive averaging loses to interference-aware or routed merges (4, 63, 64), that model populations can
-climb (5, 8–10), and that merge success admits ML-native predictors (56, 57). Ours is the framework-level
+climb (5, 8–10), and that merge success admits ML-native predictors (44, 65), correlational where this framework
+supplies mechanism; the reading of sex as an algorithm for mixability in the theory of computation
+(66) anticipated the transfer before model merging existed. Ours is the framework-level
 synthesis — inheritance, diversity, and compatibility as managed quantities — together with: the
 conservation law for blending inheritance and its operator boundaries; the per-item grounding floor;
 the society ablation with its complementary failure modes; model speciation as a named, tested question, with the
@@ -463,7 +423,7 @@ our tested regimes found, freely recombinable in the absence of conflicting conv
 ecosystem evolves as one interbreeding population, and the levers that matter are grounding budgets
 priced per rare capability and diversity preserved deliberately. If instead long-horizon
 specialisation at scale begins to produce emergent incompatibility, as the expert-training-duration
-observations hint (65) and our small-scale null does not rule out, then lineages will begin to
+observations hint (43) and our small-scale null does not rule out, then lineages will begin to
 speciate, and the ecosystem's future is a set of diverging species connected by routing rather than by
 merging. Which of the two it will be is measurable now, with the pre-merge conflict instruments this
 paper tested.
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