From 543cbe3e45c76d5c99bdeeccfabcffd5965f3c8f Mon Sep 17 00:00:00 2001 From: Giorgio Gilestro Date: Sun, 6 Sep 2026 22:50:24 +0100 Subject: [PATCH] format compliance + de-claudism pass MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit PNAS hard limits now met: title 120/135 chars; Significance rewritten to 117/120 words (plain register, carries the CL frame); Abstract rewritten to 241/250. Style pass over the whole manuscript per GG: em-dashes cut 94 -> 20 in the body (appositives to commas/parentheses, trailing clauses to colons/semicolons), tic phrases removed (quietly/sprawling/ no-longer-metaphorical/pays-for-itself/deserves-its/whatever-one-thinks/ celebrated/we-think and kin), rhetorical framings flattened to plain statements. Main text 4,809 words + 456 table words + 65 refs; estimated ~10 PNAS pages with the six composed figures (within the 12-page hard max; above the 6-page preference — trim options noted in work order). Co-Authored-By: Claude Fable 5 Claude-Session: https://claude.ai/code/session_01BkRLcc18rwT2Lysu6PbG7v --- paper/pnas/body.tex | 40 ++++----- paper/pnas/main.md | 209 +++++++++++++++++++++----------------------- paper/pnas/main.pdf | Bin 647768 -> 646880 bytes 3 files changed, 121 insertions(+), 128 deletions(-) diff --git a/paper/pnas/body.tex b/paper/pnas/body.tex index 1551242..98c5b42 100644 --- a/paper/pnas/body.tex +++ b/paper/pnas/body.tex @@ -1,34 +1,34 @@ \section*{Significance statement} -Artificial intelligence is shifting from single, frozen models to populations of models that specialise, are retrained on each other's output, and are combined (``merged'') into new models. Trained on their own output, model lineages degenerate --- a process already recognised as the mathematics of genetic drift. This paper imports the other half of population genetics: the biology of sexual reproduction. It treats model merging as recombination, real data as immigration, and merge failure as reproductive isolation, and tests each correspondence in simulations, small neural networks, and language models. The framework recasts machine learning's oldest problem --- continual learning without forgetting --- at the population scale, and yields design rules: when to average models, when to keep them separate, how much real data suffices (a theory for the field's empirical replay fractions), and a controlled small-model test in which pre-merge functional disagreement predicted merge damage, motivating further comparison with weight-space measures. +Artificial intelligence increasingly consists of populations of models rather than single systems. Models are fine-tuned from common ancestors, trained on data that earlier models generated, and combined by weight merging. These practices couple model generations the way reproduction couples biological generations, and they raise the same question: how does a population retain and accumulate abilities over time? We transfer the population genetics of sexual reproduction to this setting and test it in simulations, small neural networks, and language models. The framework recasts continual learning at the population scale and yields design rules: how much real data retraining requires, when to combine models, when to keep them separate, and how to anticipate a failed combination before making it. \section*{Abstract} -AI development increasingly resembles a population process: models are specialised, retrained on model output, and recombined by weight merging, with an openly evolutionary vocabulary but little use of evolutionary theory. Here we treat multigenerational model populations as systems whose inheritance, diversity, and compatibility must be managed, and transfer the quantitative apparatus of the evolution of sex. Its entry point --- training on model output is genetic drift, and model collapse is its signature --- we developed independently, and parallel work has now formalised the same diagnosis from several directions, a convergence we read as evidence for the frame rather than as a shared discovery to be divided. In a minimal inheritance model that is exactly Wright--Fisher --- and measurably Wright--Fisher-plus-bias in trained networks --- we derive and test the remedies: grounding as immigration, where a real-data fraction far below one retained most equilibrium diversity in the tested settings, with a per-capability observation floor that makes the rarest knowledge expensive under unstratified sampling; recombination, where refitting to the mean of parents' output distributions cancels the multi-parent gain to first order in the rare-item regime while union-preserving operators realise it; the Fisher--Muller effect, with merged language-model specialists exceeding every parent in replicated experiments; outbreeding depression on rugged task landscapes, converted into reliable gains by directed, offspring-screened recombination; and population structure, where the optimal mating breadth shrinks as skills entangle. Sex has a limit: we introduce model speciation --- merge failure as reproductive isolation --- and show in trained networks that a merge barrier remaining after permutation-and-rescaling alignment tracks functional conflict, that isolation did not emerge from compatible specialisation, and, in a controlled predictive test, that pre-merge functional disagreement predicted merge damage while the tested weight-geometry baselines showed no detectable association. We state precisely what is exact, what is measured, and what remains hypothesis. +AI development increasingly resembles a population process. Models are specialised, retrained on model output, and recombined by weight merging, and the practice is described in evolutionary vocabulary with little use of evolutionary theory. We treat multigenerational model populations as systems whose inheritance, diversity, and compatibility must be managed, and we transfer the quantitative framework of the evolution of sex. Its starting point, that training on model output is genetic drift and model collapse its signature, we reached independently; parallel work has formalised the same diagnosis, a convergence we take as support for the frame. In a minimal inheritance model that is exactly Wright--Fisher, and measurably Wright--Fisher plus estimator bias in trained networks, we derive and test remedies. Grounding acts as immigration: a real-data fraction far below one retained most equilibrium diversity, with a per-capability observation floor that makes the rarest knowledge expensive under unstratified sampling. Refitting a child to the mean of its parents' output distributions cancels the multi-parent gain to first order in the rare-item regime; union-preserving operators realise it. Merged language-model specialists exceeded every parent in replicated experiments. Blind recombination fails on rugged task landscapes; screening candidate offspring restores the gain. The optimal mating breadth narrows as skills entangle. Finally, we introduce model speciation: a merge barrier remaining after permutation-and-rescaling alignment tracks functional conflict, isolation did not emerge from compatible specialisation, and in a controlled test pre-merge functional disagreement predicted merge damage while weight-geometry baselines showed no detectable association. \medskip\hrule\medskip \section*{Introduction} -Machine learning has quietly become a population-scale phenomenon. Public repositories now 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 sprawling family trees whose lineage structure, inherited traits, and mutation dynamics are already being mapped with explicitly phylogenetic methods (31--33). Reproduction in this population is no longer metaphorical. Weight-space \textbf{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 (1, 2, 37, 38), and the engineering literature describes it in openly evolutionary vocabulary: ``crossover,'' ``mutation,'' ``mate choice,'' populations of merging models that climb benchmarks (2--5). +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 (31--33). This population also reproduces. Weight-space \textbf{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 (1, 2, 37, 38), and the engineering literature describes it in evolutionary vocabulary: ``crossover,'' ``mutation,'' ``mate choice,'' populations of merging models that climb benchmarks (2--5). -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 (43, 44) --- self-generated instruction data seeds whole lineages of descendants (5), a large and growing share of the public web is machine-generated or machine-translated text (35, 36), and the stock of human text is projected to be exhausted by frontier training within this decade (34). Meanwhile persistent multi-agent systems and emerging agent economies put many interacting models into sustained contact (39--42). A population whose members inherit from one another, recombine, and retransmit under these conditions is an evolving population in the technical sense, whatever one thinks of the metaphors. The organising claim of this paper is that the vocabulary deserves its mathematics: \textbf{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; 43, 44), self-generated instruction data seeds whole lineages of descendants (5), a large and growing share of the public web is machine-generated or machine-translated text (35, 36), and the stock of human text is projected to be exhausted by frontier training within this decade (34). Meanwhile persistent multi-agent systems and emerging agent economies put many interacting models into sustained contact (39--42). 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: \textbf{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 frame's entry point is the diagnosis. 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 (6). That this is the mathematics of \textbf{genetic drift} in a finite population is a conclusion we reached independently in building the present framework, and one that has been derived in parallel from several other directions (7--9), including a closed-form first-extinction law placing collapse onset at the Wright--Fisher first-extinction time (8) --- and that was anticipated, before deep learning, in an analysis of sequential inference chains as generalised genetic drift (63). We cite these works for priority of publication on the diagnosis and read the convergence --- independent arrivals at the same population-genetic account by different routes and in different decades --- as corroboration that the frame is the natural one. What none of that parallel work develops, and what this paper is about, is the structure the diagnosis opens: the full arc from drift through its remedies (immigration, recombination, selection, population structure) to its limit (reproductive isolation), carried as one framework from closed forms to trained networks to language models. +The frame's entry point is the diagnosis. 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 (6). That this is the mathematics of \textbf{genetic drift} in a finite population is a conclusion we reached independently in building the present framework, and one that has been derived in parallel from several other directions (7--9), including a closed-form first-extinction law placing collapse onset at the Wright--Fisher first-extinction time (8), and that was anticipated, before deep learning, in an analysis of sequential inference chains as generalised genetic drift (63). We cite these works for priority of publication on the diagnosis and read the convergence, independent arrivals at the same population-genetic account by different routes and in different decades, as corroboration that the frame is the natural one. What none of that parallel work develops, and what this paper is about, is the structure the diagnosis opens: the full arc from drift through its remedies (immigration, recombination, selection, population structure) to its limit (reproductive isolation), carried as one framework from closed forms to trained networks to language models. -Seen from machine learning's own history, the problem this frame addresses is the field's oldest --- \textbf{continual learning} --- reappearing one level up. Within a single network, sequential learning overwrites prior knowledge (catastrophic forgetting; 45, 46), and the discipline's remedies are, one by one, the population operators of this paper in single-model form: \textbf{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 (53), 5\% for weak and 25\% for strong distribution shift in continual pretraining (52) --- sit exactly 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). \textbf{Pseudo-rehearsal} --- replaying the network's own generated samples, proposed as a cure in 1995 (47) and revived as generative replay (48) --- is precisely this paper's ungrounded null: immigration from a drifting source, benign for one hop and compounding into collapse over generations, with verifier-filtering (12, 62) as what converts it back into grounding. \textbf{Parameter isolation} (65, and frozen-base adapters, which forget far less; 54) is the engineered decorrelation our specialists use; \textbf{complementary-learning-systems consolidation} (49--51) is our periodic adapter-into-base merge; the recent turn to \textbf{merging as a continual-learning mechanism} (55--58) applies recombination within one lineage over time, where we apply it across lineages; and the observation that \textbf{rare examples and long-tail knowledge are forgotten first} (59--61) 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 phenomena share their victims (the rare) and their remedies, 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. +In machine learning's own terms, the problem this frame addresses is the field's oldest, \textbf{continual learning}, reappearing one level up. Within a single network, sequential learning overwrites prior knowledge (catastrophic forgetting; 45, 46), and the discipline's remedies are, one by one, the population operators of this paper in single-model form: \textbf{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 (53) and 5\% to 25\% by distribution-shift strength in continual pretraining (52), 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). \textbf{Pseudo-rehearsal}, the replay of the network's own generated samples, proposed as a cure in 1995 (47) and revived as generative replay (48), is this paper's ungrounded null: immigration from a drifting source, benign for one hop and compounding into collapse over generations; verifier-filtering (12, 62) converts it back into grounding. \textbf{Parameter isolation} (65, and frozen-base adapters, which forget far less; 54) is the engineered decorrelation our specialists use; \textbf{complementary-learning-systems consolidation} (49--51) is our periodic adapter-into-base merge; the recent turn to \textbf{merging as a continual-learning mechanism} (55--58) applies recombination within one lineage over time, where we apply it across lineages; and the observation that \textbf{rare examples and long-tail knowledge are forgotten first} (59--61) 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 \textbf{interpretation} (an existing result understood in population-genetic terms), \textbf{explanation} (the transferred mechanism accounts for observations existing accounts leave open), and \textbf{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. +We are explicit about what kind of contribution each claim is, distinguishing \textbf{interpretation} (an existing result understood in population-genetic terms), \textbf{explanation} (the transferred mechanism accounts for observations existing accounts leave open), and \textbf{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. -The correspondences we develop, summarised in Table 1: single-teacher retraining is \textbf{asexual reproduction}, and the irreversible arm of its decay shares the defining consequence of \textbf{Muller's ratchet} (10) --- 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 \textbf{immigration} from a non-drifting source (11--13). Model merging is \textbf{recombination}, and its celebrated payoff --- a merged model exceeding every parent --- is the \textbf{Fisher--Muller effect} (14, 15). Merging entangled skills courts \textbf{outbreeding depression}; screening many candidate merges is engineered recombination with unusually flexible parent choice and pre-deployment screening (we use the shorthand \textbf{directed sex}); restricting who merges with whom is \textbf{population structure}. And merging's hard limit --- models too diverged in function to combine --- is \textbf{reproductive isolation}, for which the Bateson--Dobzhansky--Muller theory of incompatibilities (16, 17) supplies the structure. The nearest precursor to this programme reads sex as an algorithm for mixability in the theory of computation (18), pre-dating model merging; the model-merging literature itself has strong empirical operators (1, 19, 20) and emerging merge-success predictors (21, 22), to which our delta is mechanism: \emph{when and why} failure is coordinate versus functional, and what moves the boundary. +The correspondences we develop, summarised in Table 1: single-teacher retraining is \textbf{asexual reproduction}, and the irreversible arm of its decay shares the defining consequence of \textbf{Muller's ratchet} (10): 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 \textbf{immigration} from a non-drifting source (11--13). Model merging is \textbf{recombination}, and its central payoff, a merged model exceeding every parent, is the \textbf{Fisher--Muller effect} (14, 15). Merging entangled skills courts \textbf{outbreeding depression}; screening many candidate merges is engineered recombination with unusually flexible parent choice and pre-deployment screening (we use the shorthand \textbf{directed sex}); restricting who merges with whom is \textbf{population structure}. Merging's hard limit, models too diverged in function to combine, is \textbf{reproductive isolation}, for which the Bateson--Dobzhansky--Muller theory of incompatibilities (16, 17) supplies the structure. The nearest precursor to this programme reads sex as an algorithm for mixability in the theory of computation (18), pre-dating model merging; the model-merging literature itself has strong empirical operators (1, 19, 20) and emerging merge-success predictors (21, 22), 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 \textbf{minimal analytic model} validated against closed forms to a fraction of a percent; \textbf{small trained networks} (MLPs, recurrent networks, an MNIST image generator) where the operators are measured in real weights; and \textbf{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*. In this minimal inheritance model the resampling step \textbf{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. +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*. In this minimal inheritance model the resampling step \textbf{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 \textbf{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 \textbf{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). \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. @@ -51,7 +51,7 @@ 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 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). \begin{figure*}[p]\centering % fig1 \includegraphics[width=\textwidth,height=0.63\textheight,keepaspectratio]{figs/fig1_E2.pdf}\par\smallskip @@ -61,11 +61,11 @@ 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} -Merging is where the evolution-of-sex apparatus pays for itself, beginning with a result about the obvious 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 --- \textbf{weight averaging} (a nonlinear network's weight-mean does not compute its parents' output-mean) and \textbf{routing among intact specialists} (different storage and inference budgets from a single child) --- are its empirical cousins, and the measured bridge is a \textbf{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, \textbf{weight averaging} (a nonlinear network's weight-mean does not compute its parents' output-mean) and \textbf{routing among intact specialists} (different storage and inference budgets from a single child), are its empirical cousins, and the measured bridge is a \textbf{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 \textbf{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 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 \textbf{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 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 (Fig. 3). When skills are entangled, blind recombination produces offspring \emph{below} their parents --- \textbf{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 \textbf{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 (Fig. 3). When skills are entangled, blind recombination produces offspring \emph{below} their parents (\textbf{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 \textbf{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,height=0.63\textheight,keepaspectratio]{figs/fig2_E4.pdf}\par\smallskip @@ -82,7 +82,7 @@ Sex has risks and, for AI, an unfair advantage --- both quantified on rugged (ep \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 \textbf{grounded inheritance}: external samples added to the reproduction process (the data channel). In the society model, grounding is \textbf{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 --- 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) 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 \textbf{grounded inheritance}: external samples added to the reproduction process (the data channel). In the society model, grounding is \textbf{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) 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,height=0.98\textheight,keepaspectratio]{figs/fig4_E11.pdf}\par\smallskip @@ -91,11 +91,11 @@ 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 --- \textbf{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 (17) --- noting 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 (\textbf{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 (17). 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 (23), and richer symmetry groups remove more (24). 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. 5B): 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 (23), and richer symmetry groups remove more (24). 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. 5B): 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 \textbf{emergent test}: true BDM incompatibilities are emergent (each lineage's changes harmless alone), so we let children diverge with \emph{no conflicting signal anywhere} --- complementary class specialists, and divergent input conventions --- to 6.4\(\times\) the base training. \textbf{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. 5C): conflicting conventions produce \textbf{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 produces no isolation at all --- the merge improves. Across every tier tested, \textbf{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 \textbf{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. \textbf{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. 5C): conflicting conventions produce \textbf{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 produces no isolation at all --- the merge improves. Across every tier tested, \textbf{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,height=0.42\textheight,keepaspectratio]{figs/fig5_E12.pdf}\par\smallskip @@ -135,9 +135,9 @@ Predictive test & 13 conditions \(\times\) 3 seeds (0.5B) & Merge penalty vs ora \section*{Discussion} -\textbf{Design rules.} Read as engineering, the results compress into rules an operator of a model population can apply. \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{Design rules.} As engineering guidance, the results reduce to rules that an operator of a model population can apply. \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 \textbf{replay ratio} --- the folklore constants (\(\approx\)1\%, 5\%, 25\%; 52, 53) 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), not by average loss --- directly testable against published replay sweeps; (ii) a \textbf{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 (47, 48, 12, 62); (iii) \textbf{pre-merge interference prediction with a mechanism}: where the current state of the art fits regressions over candidate metrics (21), 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 \textbf{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) \textbf{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 (64); our irreversibility results concern oracle-measured behavioural distributions, and distinguishing latent from extinct capability at language-model scale is an open and, we think, decisive experiment for both readings. +\textbf{What this offers continual learning.} Read into the field where these results most directly land: (i) a first-principles account of the \textbf{replay ratio}: the field's constants (\(\approx\)1\%, 5\%, 25\%; 52, 53) 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 \textbf{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 (47, 48, 12, 62); (iii) \textbf{pre-merge interference prediction with a mechanism}: where the current state of the art fits regressions over candidate metrics (21), 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 \textbf{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) \textbf{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 (64); 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 diagnosis --- collapse as drift --- was published first by others and we cite it so (6--9), while noting the derivations are independent and convergent; prior art in the strict sense 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 (1, 19, 20), that model populations can climb (2--5), and that merge success admits ML-native predictors (21, 22). 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. diff --git a/paper/pnas/main.md b/paper/pnas/main.md index b9fce83..de14e7d 100644 --- a/paper/pnas/main.md +++ b/paper/pnas/main.md @@ -6,99 +6,93 @@ ## Significance statement -Artificial intelligence is shifting from single, frozen models to populations of models that -specialise, are retrained on each other's output, and are combined ("merged") into new models. Trained -on their own output, model lineages degenerate — a process already recognised as the mathematics of -genetic drift. This paper imports the other half of population genetics: the biology of sexual -reproduction. It treats model merging as recombination, real data as immigration, and merge failure as -reproductive isolation, and tests each correspondence in simulations, small neural networks, and -language models. The framework recasts machine learning's oldest problem — continual learning without -forgetting — at the population scale, and yields design rules: when to average models, when to keep -them separate, how much real data suffices (a theory for the field's empirical replay fractions), and -a controlled small-model test in which pre-merge functional disagreement predicted merge damage, -motivating further comparison with weight-space measures. +Artificial intelligence increasingly consists of populations of models rather than single systems. +Models are fine-tuned from common ancestors, trained on data that earlier models generated, and +combined by weight merging. These practices couple model generations the way reproduction couples +biological generations, and they raise the same question: how does a population retain and +accumulate abilities over time? We transfer the population genetics of sexual reproduction to this +setting and test it in simulations, small neural networks, and language models. The framework +recasts continual learning at the population scale and yields design rules: how much real data +retraining requires, when to combine models, when to keep them separate, and how to anticipate a +failed combination before making it. ## Abstract -AI development increasingly resembles a population process: models are specialised, retrained on model -output, and recombined by weight merging, with an openly evolutionary vocabulary but little use of -evolutionary theory. Here we treat multigenerational model populations as systems whose inheritance, -diversity, and compatibility must be managed, and transfer the quantitative apparatus of the evolution -of sex. Its entry point — training on model output is genetic drift, and model collapse is its -signature — we developed independently, and parallel work has now formalised the same diagnosis from -several directions, a convergence we read as evidence for the frame rather than as a shared discovery -to be divided. In a -minimal inheritance model that is exactly Wright–Fisher — and measurably Wright–Fisher-plus-bias in -trained networks — we derive and test the remedies: grounding as immigration, where a real-data -fraction far below one retained most equilibrium diversity in the tested settings, with a -per-capability observation floor that makes the rarest knowledge expensive under unstratified -sampling; recombination, where refitting to the mean of parents' output distributions cancels the -multi-parent gain to first order in the rare-item regime while union-preserving operators realise it; -the Fisher–Muller effect, with merged language-model specialists exceeding every parent in replicated -experiments; outbreeding depression on rugged task landscapes, converted into reliable gains by -directed, offspring-screened recombination; and population structure, where the optimal mating breadth -shrinks as skills entangle. Sex has a limit: we introduce model speciation — merge failure as -reproductive isolation — and show in trained networks that a merge barrier remaining after -permutation-and-rescaling alignment tracks functional conflict, that isolation did not emerge from -compatible specialisation, and, in a controlled predictive test, that pre-merge functional -disagreement predicted merge damage while the tested weight-geometry baselines showed no detectable -association. We state precisely what is exact, what is measured, and what remains hypothesis. +AI development increasingly resembles a population process. Models are specialised, retrained on +model output, and recombined by weight merging, and the practice is described in evolutionary +vocabulary with little use of evolutionary theory. We treat multigenerational model populations as +systems whose inheritance, diversity, and compatibility must be managed, and we transfer the +quantitative framework of the evolution of sex. Its starting point, that training on model output is +genetic drift and model collapse its signature, we reached independently; parallel work has +formalised the same diagnosis, a convergence we take as support for the frame. In a minimal +inheritance model that is exactly Wright–Fisher, and measurably Wright–Fisher plus estimator bias in +trained networks, we derive and test remedies. Grounding acts as immigration: a real-data fraction +far below one retained most equilibrium diversity, with a per-capability observation floor that +makes the rarest knowledge expensive under unstratified sampling. Refitting a child to the mean of +its parents' output distributions cancels the multi-parent gain to first order in the rare-item +regime; union-preserving operators realise it. Merged language-model specialists exceeded every +parent in replicated experiments. Blind recombination fails on rugged task landscapes; screening +candidate offspring restores the gain. The optimal mating breadth narrows as skills entangle. +Finally, we introduce model speciation: a merge barrier remaining after permutation-and-rescaling +alignment tracks functional conflict, isolation did not emerge from compatible specialisation, and +in a controlled test pre-merge functional disagreement predicted merge damage while weight-geometry +baselines showed no detectable association. --- ## Introduction -Machine learning has quietly become a population-scale phenomenon. Public repositories now host -millions of models — Hugging Face alone grew past three million by 2026 — and these are not +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 sprawling family trees whose lineage structure, inherited traits, +number of foundation models, forming large family trees whose lineage structure, inherited traits, and mutation dynamics are already being mapped with explicitly phylogenetic methods (31–33). -Reproduction in this population is no longer metaphorical. Weight-space **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 (1, 2, 37, 38), and -the engineering literature describes it in openly evolutionary vocabulary: "crossover," "mutation," -"mate choice," populations of merging models that climb benchmarks (2–5). +This population also reproduces. Weight-space **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 (1, 2, 37, 38), and the engineering literature +describes it in evolutionary vocabulary: "crossover," "mutation," "mate choice," populations of +merging models that climb benchmarks (2–5). 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 (43, 44) — self-generated instruction data +predominantly synthetic (over 98% in documented cases; 43, 44), self-generated instruction data seeds whole lineages of descendants (5), a large and growing share of the public web is machine-generated or machine-translated text (35, 36), and the stock of human text is projected to be exhausted by frontier training within this decade (34). Meanwhile persistent multi-agent systems and emerging agent economies put many interacting models into sustained contact (39–42). A population whose members inherit from one another, recombine, and retransmit under these conditions is an -evolving population in the technical sense, whatever one thinks of the metaphors. The organising claim -of this paper is that the vocabulary deserves 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. 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 frame's entry point is the diagnosis. Training each generation of a model on the previous -generation's output degrades it — *model collapse*: rare capabilities vanish first and the lineage +generation's output degrades it (*model collapse*): rare capabilities vanish first and the lineage drifts toward its own most common behaviour (6). That this is the mathematics of **genetic drift** in a finite population is a conclusion we reached independently in building the present framework, and one that has been derived in parallel from several other directions (7–9), including a closed-form -first-extinction law placing collapse onset at the Wright–Fisher first-extinction time (8) — and that +first-extinction law placing collapse onset at the Wright–Fisher first-extinction time (8), and that was anticipated, before deep learning, in an analysis of sequential inference chains as generalised genetic drift (63). We cite these works for priority of publication on the diagnosis and read the -convergence — independent arrivals at the same population-genetic account by different routes and in -different decades — as corroboration that the frame is the natural one. What none of that parallel work develops, and what this paper is about, is the +convergence, independent arrivals at the same population-genetic account by different routes and in +different decades, as corroboration that the frame is the natural one. What none of that parallel work develops, and what this paper is about, is the structure the diagnosis opens: the full arc from drift through its remedies (immigration, recombination, selection, population structure) to its limit (reproductive isolation), carried as one framework from closed forms to trained networks to language models. -Seen from machine learning's own history, the problem this frame addresses is the field's oldest — -**continual learning** — reappearing one level up. Within a single network, sequential learning +In machine learning's own terms, the problem this frame addresses is the field's oldest, +**continual learning**, reappearing one level up. Within a single network, sequential learning overwrites prior knowledge (catastrophic forgetting; 45, 46), and the discipline's remedies are, one by one, the population operators of this paper in single-model form: **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 (53), 5% for weak and 25% for strong distribution shift in -continual pretraining (52) — sit exactly 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). **Pseudo-rehearsal** — replaying the network's own generated -samples, proposed as a cure in 1995 (47) and revived as generative replay (48) — is precisely this -paper's ungrounded null: immigration from a drifting source, benign for one hop and compounding into -collapse over generations, with verifier-filtering (12, 62) as what converts it back into grounding. +data is grounding's within-lineage counterpart, and the field's empirically settled replay fractions, +on the order of 1% for instruction tuning (53) and 5% to 25% by distribution-shift strength in +continual pretraining (52), 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). **Pseudo-rehearsal**, the replay of the network's own generated +samples, proposed as a cure in 1995 (47) and revived as generative replay (48), is this paper's +ungrounded null: immigration from a drifting source, benign for one hop and compounding into +collapse over generations; verifier-filtering (12, 62) converts it back into grounding. **Parameter isolation** (65, and frozen-base adapters, which forget far less; 54) is the engineered decorrelation our specialists use; **complementary-learning-systems consolidation** (49–51) is our periodic adapter-into-base merge; the recent turn to **merging as a continual-learning mechanism** @@ -106,32 +100,31 @@ periodic adapter-into-base merge; the recent turn to **merging as a continual-le the observation that **rare examples and long-tail knowledge are forgotten first** (59–61) 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 phenomena share their victims (the rare) and their remedies, 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 +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 +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. 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** (10) — once every copy of a rare capability is gone from all parents and sources, no +ratchet** (10): 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 (11–13). Model merging is **recombination**, and its -celebrated payoff — a merged model exceeding every parent — is the **Fisher–Muller effect** (14, 15). +central payoff, a merged model exceeding every parent, is the **Fisher–Muller effect** (14, 15). 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**. And merging's hard limit — models too diverged in function to combine — is **reproductive +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 (16, 17) supplies the structure. The nearest precursor to this programme reads sex as an algorithm for mixability in the theory of computation (18), pre-dating model merging; the model-merging literature itself has strong @@ -148,11 +141,11 @@ sensitivity analyses on the predictive test. ## The minimal model, and where its exactness ends -Knowledge is modelled as a distribution `p_t` over `K` discrete items — capabilities, facts, modes of -behaviour — with a fixed true distribution `p*` whose rare tail carries the knowledge most at risk. +Knowledge is modelled as a distribution `p_t` over `K` discrete items (capabilities, facts, modes of +behaviour), with a fixed true distribution `p*` whose rare tail carries the knowledge most at risk. One generation is: *draw `n` samples from the parent's distribution, optionally mix in `m` verified real samples ("grounding", `g = m/(n+m)`), and refit the child*. In this minimal inheritance model the -resampling step **is** the Wright–Fisher process — the same equations, which we exploit as an +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 `E[H_t] = H_0(1 − 1/n)^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. @@ -163,7 +156,7 @@ exact drift null they deviate in *opposite, architecture-specific* directions: a network resists collapse (keeping spurious variants alive), while a sharpening image generator accelerates it. A one-parameter **learning kernel** (a smoothing knob and a sharpening knob on the refit) reproduces both. Throughout, a real learner is therefore treated as Wright–Fisher *plus a signed, measurable -estimator bias* — and the drift signs (rare-first loss; the grounding response) +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). @@ -191,7 +184,7 @@ known limits is SI Appendix, Table S1. 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 -*smooth* in the grounding fraction — there is no phase transition in aggregate diversity — so the +*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 indefinitely, with the required fraction depending on sample size, source distribution, and the @@ -199,7 +192,7 @@ chosen retention target (dependencies in SI). The engineering point survives the 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 `p` appears in a real-data batch of size `m` with probability `1 − e^{−m·p}`, so `m·p ≈ 1` marks roughly a -63% chance of one example per batch — a soft observation floor, with higher confidence priced +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 `∝ 1/p`; targeted @@ -209,29 +202,29 @@ looked, with two deviations, both traced to the estimator bias above: sharp thre 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 ~10% grounding holds all thirty (the trained model needs roughly twice the exact-operator -fraction — the measured price of the estimator bias). +fraction, the measured price of the estimator bias). *(FIG:fig1)* ### Recombination: a conservation law, its operators, and offspring that exceed every parent -Merging is where the evolution-of-sex apparatus pays for itself, beginning with a result about the -obvious operator, stated with its assumptions. **Proposition (blending inheritance, rare-item +The largest returns from the transfer concern merging. We begin with a result about the most common +operator, stated with its assumptions. **Proposition (blending inheritance, rare-item regime).** Let K parents independently retain a rare item (mass `p` when retained), and let the child draw `n` samples either from one parent chosen at random or from the *mean of the parents' output distributions*. Expected item mass is identical under the two schemes; and in the rare-item regime `n·p/K ≪ 1`, where per-item survival is first-order in sampled mass, expected *survival* is also -identical — the 1/K dilution of averaging cancels the K-parent union gain to first order, so in this +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 *reduce* extinction relative to a randomly chosen single parent — the +reduction from averaging can *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 +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 — **weight averaging** (a nonlinear network's weight-mean does not compute its parents' +operators, **weight averaging** (a nonlinear network's weight-mean does not compute its parents' output-mean) and **routing among intact specialists** (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, +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 quantitative form of the relationship is untested). On easy tasks a capable base's average is already at ceiling and refinements @@ -240,25 +233,25 @@ 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 +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 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 -(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 +Sex has risks and, for AI, an unfair advantage, both quantified on rugged (epistatic) NK landscapes +(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 parents, choose complementary mates, and *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 +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 *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 *champion* shifts from wide breadth on smooth -landscapes to intermediate breadth on rugged ones (Fig. 3C) — the mating-system phenomenon known to +landscapes to intermediate breadth on rugged ones (Fig. 3C), the mating-system phenomenon known to structured-population search, mapped onto merging populations. *(FIG:fig2)* @@ -270,15 +263,15 @@ structured-population search, mapped onto merging populations. Composing the operators (Fig. 4) requires one definitional distinction first. In the inheritance model, grounding is **grounded inheritance**: external samples added to the reproduction process (the data channel). In the society model, grounding is **grounded evaluation**: selection weights true -fitness against conformity to the population's own consensus — `g`·true-fitness + (1−g)·conformity — +fitness against conformity to the population's own consensus, `g`·true-fitness + (1−g)·conformity, the analogue of scoring models by the crowd's approval (the fitness channel). These are related design -ideas — both couple the lineage to a non-drifting external signal — but they are different operators, +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) 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 +converges it prematurely to a worse answer. Each removal fails differently; the three implementations make complementary contributions *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. @@ -288,12 +281,12 @@ 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: 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 -(17) — noting that a super-linear count does not by itself entail a sharp performance cliff without +(17). 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. @@ -303,22 +296,22 @@ richer symmetry groups remove more (24). We therefore aligned under the composit permutation matching and exact per-unit rescaling (the unit symmetry group of plain ReLU MLPs, as the search space) and decomposed the barrier (Fig. 5B): two networks trained from different initialisations on the *same* task have a barrier that this alignment removes essentially entirely -(residual ≈ 0.001, the aligned merge performing at parent level) — coordinate, not functional; two +(residual ≈ 0.001, the aligned merge performing at parent level): coordinate, not functional; two networks trained on *conflicting* label maps have a barrier the same alignment leaves largely unchanged (0.502 → 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 +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 → 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 *structure around it*: which divergences +information-theoretic (no single model can satisfy contradictory conventions; SI Appendix, +Proposition S2), with the framework's role being the *structure around it*: which divergences generate conflict, and what moves the cliff. The strongest constraint comes from the pre-registered **emergent test**: true BDM incompatibilities are emergent (each lineage's changes harmless alone), so we let children diverge with *no conflicting -signal anywhere* — complementary class specialists, and divergent input conventions — to 6.4× the base +signal anywhere*, using complementary class specialists and divergent input conventions, to 6.4× the base training. **No isolation emerged** (residual 0.000 throughout); instead the merge *rescued* the two -catastrophically-forgetting specialists (parents ≈ 0.50, merge ≈ 0.955 — a sustained Fisher–Muller +catastrophically-forgetting specialists (parents ≈ 0.50, merge ≈ 0.955, a sustained Fisher–Muller rescue). The same double result appears at the language-model tier (Fig. 5C): 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 @@ -382,7 +375,7 @@ are the experiment's open front. ## Discussion -**Design rules.** Read as engineering, the results compress into rules an operator of a model +**Design rules.** As engineering guidance, the results reduce to rules that an operator of a model population can apply. *Ground every generation* in verified reality — a few percent retained most diversity in our tested settings — but price the rarest capabilities individually (observation probability `1 − e^{−m·p}` per batch under unstratified sampling), consider targeted sampling for the deep tail, and use @@ -399,9 +392,9 @@ evidence of incompatibility* — in every regime we tested, what broke merging w 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: -(i) a first-principles account of the **replay ratio** — the folklore constants (≈1%, 5%, 25%; 52, 53) +(i) a first-principles account of the **replay ratio**: the field's constants (≈1%, 5%, 25%; 52, 53) acquire an equilibrium theory and a sharper prediction, that the required fraction is set by the -rarest capability one refuses to lose (the `1 − e^{−m·p}` law), not by average loss — directly +rarest capability one refuses to lose (the `1 − e^{−m·p}` law) rather than by average loss, which is 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 (47, 48, 12, 62); (iii) **pre-merge interference @@ -416,7 +409,7 @@ capabilities hides exactly the losses that drift theory says come first and, pas irreversible. On that last point we note the standing objection that apparent forgetting can be skewed task-inference over latent capability rather than erasure (64); our irreversibility results concern oracle-measured behavioural distributions, and distinguishing latent from extinct capability -at language-model scale is an open and, we think, decisive experiment for both readings. +at language-model scale is an open experiment whose outcome would be decisive for both readings. **What is borrowed and what is ours.** The diagnosis — collapse as drift — was published first by others and we cite it so (6–9), while noting the derivations are independent and convergent; prior art diff --git a/paper/pnas/main.pdf b/paper/pnas/main.pdf index 875be6e069a2e6a37d8555f7b9343ec8559b3abf..23a3857096454881874b5c3cfafd7bd6ddc348a0 100644 GIT binary patch delta 78765 zcmYhCV|ykFlx(ApZKq?~X2-T|+sPZ-wr$(CZQFM4nLA(RPwf5Fs#R-4lz*M)mN6LiJcB`khHT3?GH}-w)(z(hwghi`bus!&ROash`i!WBOe>z)xn+ 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