From 6f1f8bf172ca7e1f1149f276bb1b2e5cf6a1b240 Mon Sep 17 00:00:00 2001 From: Giorgio Gilestro Date: Mon, 7 Sep 2026 09:00:56 +0100 Subject: [PATCH] references: citation-order renumbering, PNAS style, full verification All 66 references renumbered to first-appearance order (programmatically verified: in-text sequence = 1..66 = list order; ranges expanded, remapped, recompressed) and rewritten in PNAS style (initials-first authors with the >5 -> et-al rule, sentence-case titles, abbreviated italic venues, bold volumes, year-at-end, arXiv [Preprint] + 10.48550 DOIs). Correctness: 47 arXiv ids batch-verified against the arXiv API (title/first-author/year); caught and fixed an authorless GENOME entry (Y. Zhang et al.), "Sakana AI" -> J. Abrantes et al., a wrong Kotha id (2310.05719, a different paper -> 2309.10105), Nemotron's corporate author, and Liang's truncated title. Also: six load-bearing refs that lost their in-text anchors during the restructure re-anchored (NK, QD, Pari, LoRA, Sharma, Kozodoi), one real mis-citation fixed (Self-Instruct credited to Multiagent-Finetuning; new ref added), and four figure captions in build.py brought up to third-review calibration (operational grounding threshold; first-order conservation; complementary-contributions society; permutation-and-rescaling alignment). 20-pp rebuild clean. Co-Authored-By: Claude Fable 5 Claude-Session: https://claude.ai/code/session_01BkRLcc18rwT2Lysu6PbG7v --- paper/pnas/body.tex | 169 +++++++++++----------- paper/pnas/build.py | 22 +-- paper/pnas/main.md | 218 +++++++++++++++-------------- paper/pnas/main.pdf | Bin 646880 -> 650117 bytes tasks/workorder-pnas-submission.md | 14 +- 5 files changed, 218 insertions(+), 205 deletions(-) diff --git a/paper/pnas/body.tex b/paper/pnas/body.tex index 98c5b42..fbe0613 100644 --- a/paper/pnas/body.tex +++ b/paper/pnas/body.tex @@ -10,17 +10,17 @@ AI development increasingly resembles a population process. Models are specialis \section*{Introduction} -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). +Machine learning has become a population-scale phenomenon. Public repositories host millions of models (Hugging Face alone grew past three million by 2026), and these are not independent creations: the overwhelming majority are fine-tunes, distillations, or merges of a small number of foundation models, forming large family trees whose lineage structure, inherited traits, and mutation dynamics are already being mapped with explicitly phylogenetic methods (1--3). This population also reproduces. Weight-space \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 (4--7), and the engineering literature describes it in evolutionary vocabulary: ``crossover,'' ``mutation,'' ``mate choice,'' populations of merging models that climb benchmarks (5, 8--10). -The generations are coupled through data as well as through weights. Successive models increasingly learn from model output rather than from fresh human experience: frontier alignment pipelines are now predominantly synthetic (over 98\% in documented cases; 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 generations are coupled through data as well as through weights. Successive models increasingly learn from model output rather than from fresh human experience: frontier alignment pipelines are now predominantly synthetic (over 98\% in documented cases; 11, 12), self-generated instruction data seeds whole lineages of descendants (13), a large and growing share of the public web is machine-generated or machine-translated text (14, 15), and the stock of human text is projected to be exhausted by frontier training within this decade (16). Meanwhile persistent multi-agent systems and emerging agent economies put many interacting models into sustained contact (17--20). A population whose members inherit from one another, recombine, and retransmit under these conditions is an evolving population in the technical sense. The claim of this paper is that the vocabulary should be given its mathematics: \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 (21). 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 (22--24), including a closed-form first-extinction law placing collapse onset at the Wright--Fisher first-extinction time (23), and that was anticipated, before deep learning, in an analysis of sequential inference chains as generalised genetic drift (25). 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. -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. +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; 26, 27), 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 (28) and 5\% to 25\% by distribution-shift strength in continual pretraining (29), sit where the minimal model's operational grounding threshold lies, a correspondence for which the framework supplies the missing theory (equilibrium diversity, and a per-capability survival law). \textbf{Pseudo-rehearsal}, the replay of the network's own generated samples, proposed as a cure in 1995 (30) and revived as generative replay (31), is this paper's ungrounded null: immigration from a drifting source, benign for one hop and compounding into collapse over generations; verifier-filtering (32, 33) converts it back into grounding. \textbf{Parameter isolation} (34, and frozen-base adapters, which forget far less; 35) is the engineered decorrelation our specialists use; \textbf{complementary-learning-systems consolidation} (36--38) is our periodic adapter-into-base merge; the recent turn to \textbf{merging as a continual-learning mechanism} (39--42) applies recombination within one lineage over time, where we apply it across lineages; and the observation that \textbf{rare examples and long-tail knowledge are forgotten first} (43--45) is tail-allele extinction observed one model at a time. One distinction is kept explicit throughout: catastrophic forgetting is largely deterministic interference from shifted training, whereas collapse is stochastic sampling drift; the two phenomena share their victims, the rare, and their remedies, but not their mechanism. To our knowledge, no prior work carries population-genetic formalism into continual learning itself; that bridge (replay as immigration with a survival law, merging as recombination with a compatibility criterion, consolidation as the slow store of a two-speed memory) is where this framework may matter most. We are explicit about what kind of contribution each claim is, distinguishing \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 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. +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} (46): once every copy of a rare capability is gone from all parents and sources, no recombination can rebuild it, which is why remedies must act before fixation-by-loss (a consequence- level correspondence: the minimal model lacks the ratchet's recurrent deleterious-mutation mechanism, so irreversible loss alone does not identify that specific mechanism). Injecting verified real data is \textbf{immigration} from a non-drifting source (32, 47, 48). Model merging is \textbf{recombination}, and its central payoff, a merged model exceeding every parent, is the \textbf{Fisher--Muller effect} (49, 50). 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 (51, 52) supplies the structure. The nearest precursor to this programme reads sex as an algorithm for mixability in the theory of computation (53), pre-dating model merging; the model-merging literature itself has strong empirical operators (4, 54, 55) and emerging merge-success predictors (56, 57), to which our delta is mechanism: \emph{when and why} failure is coordinate versus functional, and what moves the boundary. We support the framework at three tiers of evidence, in ascending realism and descending exactness: a \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. @@ -56,21 +56,21 @@ In the minimal model, grounding from a fixed real source is immigration into a d \begin{figure*}[p]\centering % fig1 \includegraphics[width=\textwidth,height=0.63\textheight,keepaspectratio]{figs/fig1_E2.pdf}\par\smallskip \includegraphics[width=\textwidth,height=0.63\textheight,keepaspectratio]{figs/fig1_mnist_montage.pdf}\par\smallskip -\caption{Collapse is drift; grounding is immigration. (A, top) The grounding phase response in the minimal model: a critical real-data fraction $g^*\!\approx\!0.05$ retains most diversity indefinitely, while tail survival obeys the per-item floor $m\,p \gtrsim 1$. (B, bottom) The same signs on real images: a convolutional VAE retrained each generation on its own output collapses to a single blurred mode (rows: generations), while $\sim$10\% grounding holds all thirty class$\times$style modes.}\label{fig1} +\caption{Collapse is drift; grounding is immigration. (A, top) The grounding response in the minimal model: an operational threshold $g\!\approx\!0.05$ retained most equilibrium diversity in the tested setting (the equilibrium is smooth in $g$), while per-item observation obeys $1-e^{-m p}$. (B, bottom) The same signs on real images: a convolutional VAE retrained each generation on its own output collapses to a single blurred mode (rows: generations), while $\sim$10\% grounding holds all thirty class$\times$style modes.}\label{fig1} \end{figure*} \subsection*{Recombination: a conservation law, its operators, and offspring that exceed every parent} -The largest returns from the transfer concern merging. We begin with a result about the most common operator, stated with its assumptions. \textbf{Proposition (blending inheritance, rare-item regime).} Let K parents independently retain a rare item (mass \texttt{p} when retained), and let the child draw \texttt{n} samples either from one parent chosen at random or from the \emph{mean of the parents' output distributions}. Expected item mass is identical under the two schemes; and in the rare-item regime \texttt{n\(\cdot\)p/K \(\ll\) 1}, where per-item survival is first-order in sampled mass, expected \emph{survival} is also identical: the 1/K dilution of averaging cancels the K-parent union gain to first order, so in this regime adding parents through the output-mean does not increase expected tail retention. Two boundaries: outside that regime, survival is a convex function of mixed mass, so the variance reduction from averaging can \emph{reduce} extinction relative to a randomly chosen single parent; the cancellation is a first-order result about rare items, not a universal impossibility; and the contrasting union operator (keep each item's strongest source, then renormalise, which itself redistributes mass and presupposes a verifier or oracle to identify the strongest source) increases expected retention with K in all regimes in the minimal model. The practically important operators, \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} (58) (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 (59) specialists beat every parent overall (decisively at 7B: 0.87 vs 0.77), and on the sharper worst-family metric the merged models are the only ones competent everywhere, in every seed (Fig. 6A). -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 (60) (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 \includegraphics[width=\textwidth,height=0.63\textheight,keepaspectratio]{figs/fig2_E8.pdf}\par\smallskip -\caption{Recombination: the conservation law and the Fisher--Muller effect. (A, top) Refitting a child to the mean of its parents' output distributions conserves rare-item mass at single-parent level regardless of parent count (blending inheritance); a strongest-source (union) operator realises the multi-parent gain. (B, bottom) Multi-locus recombination of decorrelated specialists assembles a genotype fitter than any parent, climbing to the optimum as parents are added, while the best single parent and the blended average plateau below.}\label{fig2} +\caption{Recombination: blending inheritance and the Fisher--Muller effect. (A, top) Refitting a child to the mean of its parents' output distributions conserves expected rare-item mass at the single-parent level, cancelling the multi-parent gain to first order in the rare-item regime; a strongest-source (union) operator, with renormalisation and an oracle, realises it. (B, bottom) Multi-locus recombination of decorrelated specialists assembles a genotype fitter than any parent, climbing to the optimum as parents are added, while the best single parent and the blended average plateau below.}\label{fig2} \end{figure*} \begin{figure*}[p]\centering % fig3 @@ -82,31 +82,31 @@ Sex has risks and, for AI, an unfair advantage, both quantified on rugged (epist \subsection*{The society: grounding, recombination, and diversity make complementary contributions} -Composing the operators (Fig. 4) requires one definitional distinction first. In the inheritance model, grounding is \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. +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 (61)) climbs to near the global optimum while keeping its specialists; removing grounded evaluation converges the population confidently on an unfit consensus (self-consumption); removing recombination strands it on local optima; removing diversity converges it prematurely to a worse answer. Each removal fails differently; the three implementations make complementary contributions \emph{under the tested conditions}; general joint necessity is not established (alternative mutation, restart, archive, or selection schemes could alter the picture). At language-model scale this composed loop remains unbuilt; it is the paper's largest stated gap. \begin{figure*}[p]\centering % fig4 \includegraphics[width=\textwidth,height=0.98\textheight,keepaspectratio]{figs/fig4_E11.pdf}\par\smallskip -\caption{The society: grounding, sex, and diversity are jointly necessary. A finite agent population on a rugged NK landscape under a grounded selection score. Four-arm ablation: the full system climbs to near the global optimum; removing grounding collapses the population onto a confident, unfit consensus (self-consumption); removing recombination strands it on local optima; removing diversity converges it prematurely. Each ablation fails differently.}\label{fig4} +\caption{The society: grounded evaluation, recombination, and diversity preservation make complementary contributions in the tested model. A finite agent population on a rugged NK landscape, with selection weighting true fitness against conformity. Four-arm ablation: the full system climbs to near the global optimum; removing grounding collapses the population onto a confident, unfit consensus (self-consumption); removing recombination strands it on local optima; removing diversity converges it prematurely. Each ablation fails differently.}\label{fig4} \end{figure*} \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). We note that a super-linear count does not by itself entail a sharp performance cliff without the count-to-effect-size link, which the analytic model supplies under its assumptions and any neural test must establish separately. +Recombination presupposes compatible parents. In biology, lineages pushed far enough apart become separate species (\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 (52). We note that a super-linear count does not by itself entail a sharp performance cliff without the count-to-effect-size link, which the analytic model supplies under its assumptions and any neural test must establish separately. -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 (62); richer symmetry groups remove more (63), with known failures beyond the shared-data regime (64). We therefore aligned under the composition of permutation matching and exact per-unit rescaling (the unit symmetry group of plain ReLU MLPs, as the search space) and decomposed the barrier (Fig. 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}, 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. +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 (cf. the merging literature's expert-duration effect; 65) 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 \includegraphics[width=\textwidth,height=0.42\textheight,keepaspectratio]{figs/fig5_speciation_real.pdf}\par\smallskip \includegraphics[width=\textwidth,height=0.42\textheight,keepaspectratio]{figs/fig5_llm_speciation.pdf}\par\smallskip -\caption{Model speciation across three tiers. (A, top) Analytic: hybrid fitness traces compatible $\rightarrow$ outbreeding depression $\rightarrow$ inviability; the cliff arrives earlier the denser the incompatibilities; incompatibility count snowballs with divergence. (B, middle) Trained MLPs: the merge barrier decomposed under the complete unit symmetry group --- same-task/different-init barriers are coordinate artefacts (removed by alignment); conflicting-task barriers survive in full, with hybrid fitness falling 0.97 $\rightarrow$ 0.03; divergence without conflict produced no isolation, the merge instead rescuing the forgetting specialists. (C, bottom) Language models: conflicting conventions produce function-specific hybrid breakdown; over-training disjoint specialists produces none --- at every tier tested, isolation had to be provoked by functional conflict.}\label{fig5} +\caption{Model speciation across three tiers. (A, top) Analytic: hybrid fitness traces compatible $\rightarrow$ outbreeding depression $\rightarrow$ inviability; the cliff arrives earlier the denser the incompatibilities; incompatibility count snowballs with divergence. (B, middle) Trained MLPs: the merge barrier decomposed under permutation-and-rescaling alignment --- same-task/different-init barriers are coordinate artefacts (removed by alignment); conflicting-task barriers survive in full, with hybrid fitness falling 0.97 $\rightarrow$ 0.03; divergence without conflict produced no isolation, the merge instead rescuing the forgetting specialists. (C, bottom) Language models: conflicting conventions produce function-specific hybrid breakdown; over-training disjoint specialists produces none --- at every tier tested, isolation had to be provoked by functional conflict.}\label{fig5} \end{figure*} \subsection*{A controlled predictive test: functional conflict, measured pre-merge, predicts merge damage} -The framework's prediction-level claim was put to a designed test (Fig. 6C). Thirty-nine parent pairs (13 conditions \(\times\) 3 seeds; rows are not independent --- parents share task-data seeds across conditions, so inference is condition-clustered, and because shared seeds also couple rows \emph{across} conditions we report per-seed and leave-one-seed-out sensitivity alongside) span three axes decorrelated by construction: \emph{conflict} (contradictory conventions on shared prompts, private budgets fixed), \emph{compatible overlap} (the same shared prompts under the same convention --- overlap and volume without conflict), and \emph{duration} (weight divergence with zero conflict). Before merging, six predictors are computed: \textbf{confidence-weighted functional conflict} (bilateral confident disagreement on probes drawn blind to where conflict lives --- a proposed proxy for merge-relevant interactions, motivated by the observation that raw disagreement counts harmless complementation, one parent merely ignorant, as conflict), raw disagreement, gradient alignment at the shared base (21), LoRA-delta cosine and distance, and a cross-task performance baseline. The pre-registered outcome is the merge penalty against oracle parent potential (the hybrid-load analogue), also reported against best- and mean-parent references because the predictor ordering is sensitive to that choice. +The framework's prediction-level claim was put to a designed test (Fig. 6C). Thirty-nine parent pairs (13 conditions \(\times\) 3 seeds; rows are not independent --- parents share task-data seeds across conditions, so inference is condition-clustered, and because shared seeds also couple rows \emph{across} conditions we report per-seed and leave-one-seed-out sensitivity alongside) span three axes decorrelated by construction: \emph{conflict} (contradictory conventions on shared prompts, private budgets fixed), \emph{compatible overlap} (the same shared prompts under the same convention --- overlap and volume without conflict), and \emph{duration} (weight divergence with zero conflict). Before merging, six predictors are computed: \textbf{confidence-weighted functional conflict} (bilateral confident disagreement on probes drawn blind to where conflict lives --- a proposed proxy for merge-relevant interactions, motivated by the observation that raw disagreement counts harmless complementation, one parent merely ignorant, as conflict), raw disagreement, gradient alignment at the shared base (56), LoRA-delta cosine and distance, and a cross-task performance baseline. The pre-registered outcome is the merge penalty against oracle parent potential (the hybrid-load analogue), also reported against best- and mean-parent references because the predictor ordering is sensitive to that choice. The supported conclusion, stated conditionally: \textbf{across this controlled grid, pre-merge functional disagreement predicted merge penalties (clustered bootstrap CIs excluding zero; held-out leave-one-condition-out \(\rho\) \(\approx\) 0.35--0.40), whereas LoRA-delta cosine and L2 showed no statistically detectable association; gradient alignment carried intermediate signal.} Head-to-head predictor differences are not individually significant at this sample size; only these baselines were tested; and with three seeds, uncertainty about seed generalisation remains substantial --- though the seed sensitivity favours the functional measures (per-seed \(\rho\) stable at +0.37 to +0.53 in each seed alone, geometry \(\approx\) 0 in every seed, gradient alignment seed-unstable at −0.11 to −0.55). Two further results bound the claim: the initial two-axis grid's best predictor was delta-cosine (\(\rho\) = +0.60) --- an overlap artefact that the compatible-overlap control was added to expose, and did (collapse to +0.03); and the pre-registered internal prediction that confidence weighting would beat raw disagreement \textbf{failed} (they are statistically indistinguishable as rank predictors), so the present evidence favours functional disagreement generally, not the DMI-specific refinement. The framework motivated the measurement and the controls; their success does not validate the specifically population-genetic mechanism. Whether the prediction improves a budget-matched operator choice, and whether it generalises to unfamiliar conflict structures and real task pairs, are the experiment's open front. @@ -137,9 +137,9 @@ Predictive test & 13 conditions \(\times\) 3 seeds (0.5B) & Merge penalty vs ora \textbf{Design rules.} As engineering guidance, the results reduce to rules that an operator of a model population can apply. \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 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 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\%; 28, 29) acquire an equilibrium theory and a sharper prediction, that the required fraction is set by the rarest capability one refuses to lose (the \texttt{1 − e\textasciicircum{}{−m\(\cdot\)p}} law) rather than by average loss, which is testable against published replay sweeps; (ii) a \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 (30--33); (iii) \textbf{pre-merge interference prediction with a mechanism}: where the current state of the art fits regressions over candidate metrics (56), the functional-conflict measure arrives at a convergent signal from principle and comes with an operator prescription --- when conflict is high, do not average; route or breed-and-screen; (iv) a candidate \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 (66); our irreversibility results concern oracle-measured behavioural distributions, and distinguishing latent from extinct capability at language-model scale is an open experiment whose outcome would be decisive for both readings. -\textbf{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. +\textbf{What is borrowed and what is ours.} The diagnosis --- collapse as drift --- was published first by others and we cite it so (21--24), 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 (4, 54, 55), that model populations can climb (5, 8--10), and that merge success admits ML-native predictors (56, 57). Ours is the framework-level synthesis --- inheritance, diversity, and compatibility as managed quantities --- together with: the conservation law for blending inheritance and its operator boundaries; the per-item grounding floor; the society ablation with its complementary failure modes; model speciation as a named, tested question, with the coordinate-versus-functional decomposition under permutation-and-rescaling alignment and the emergent null that bounds it; and the controlled predictive test with its controls. We claim the framework generated these measurements and experiments; we do not claim their outcomes validate a uniquely population-genetic mechanism, and one refinement it proposed was not supported. \textbf{Limits and open problems.} The demonstrations are deliberately small: exact where small is a virtue, sign-level and seed-replicated at the language-model tier, on constructed task families with a trivially separable router and one model lineage (Qwen, 0.5B--7B). The composed society has not been built at language-model scale. The predictive test's next bars, in order of value: generalisation to \emph{unfamiliar} conflict structures and real task pairs; a demonstrably better \emph{budget-matched} merging decision; then scale replication. Beyond engineering, the framework's hardest open problem is the fitness function itself: selection optimises what is measured, and for knowledge systems the persuasive and the true compete --- grounding against a reality that can refuse is the only anchor we trust, and institutionalising that anchor (verification, replication, and challenge among models) is the society-level problem we pose but do not solve. What biology receives in return is a new model system: populations of learners where every genotype, environment, and mating decision is observable and manipulable --- where the evolution of sex can be studied with interventions (unbounded parents, offspring preview, directed mating) that no living system permits. @@ -156,70 +156,71 @@ Predictive test & 13 conditions \(\times\) 3 seeds (0.5B) & Merge penalty vs ora \section*{References} \begin{enumerate} -\item Yadav P, Tam D, Choshen L, Raffel C, Bansal M (2023) TIES-Merging: resolving interference when merging models. \emph{NeurIPS}. arXiv:2306.01708. -\item Akiba T, Shing M, Tang Y, Sun Q, Ha D (2025) Evolutionary optimization of model merging recipes. \emph{Nat Mach Intell} 7:195--204. -\item GENOME: Nature-inspired population-based evolution of large language models (2025). arXiv:2503.01155. -\item Sakana AI (2025) Competition and attraction improve model fusion (M2N2). \emph{GECCO}. arXiv:2508.16204. -\item Subramaniam V, Du Y, Tenenbaum JB, Torralba A, Li S, Mordatch I (2025) Multiagent finetuning: self-improvement with diverse reasoning chains. arXiv:2501.05707. -\item Shumailov I, et al. 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Raghunathan, Understanding catastrophic forgetting in language models via implicit inference. \emph{Int. Conf. Learn. Represent.} (2024). https://doi.org/10.48550/arXiv.2309.10105. \end{enumerate} diff --git a/paper/pnas/build.py b/paper/pnas/build.py index a22b835..cb21107 100644 --- a/paper/pnas/build.py +++ b/paper/pnas/build.py @@ -23,17 +23,18 @@ OUT = HERE / "body.tex" # figure name -> (list of source PDFs (stacked top->bottom), caption) FIGURES: dict[str, tuple[list[str], str]] = { "fig1": (["results/E2/E2.pdf", "results/mnist_collapse/mnist_montage.pdf"], - "Collapse is drift; grounding is immigration. (A, top) The grounding phase response in the " - "minimal model: a critical real-data fraction $g^*\\!\\approx\\!0.05$ retains most diversity " - "indefinitely, while tail survival obeys the per-item floor $m\\,p \\gtrsim 1$. (B, bottom) " + "Collapse is drift; grounding is immigration. (A, top) The grounding response in the minimal " + "model: an operational threshold $g\\!\\approx\\!0.05$ retained most equilibrium diversity in " + "the tested setting (the equilibrium is smooth in $g$), while per-item observation obeys " + "$1-e^{-m p}$. (B, bottom) " "The same signs on real images: a convolutional VAE retrained each generation on its own " "output collapses to a single blurred mode (rows: generations), while $\\sim$10\\% grounding " "holds all thirty class$\\times$style modes."), "fig2": (["results/E4/E4.pdf", "results/E8/E8.pdf"], - "Recombination: the conservation law and the Fisher--Muller effect. (A, top) Refitting a child " - "to the mean of its parents' output distributions conserves rare-item mass at single-parent " - "level regardless of parent count (blending inheritance); a strongest-source (union) operator " - "realises the multi-parent gain. (B, bottom) Multi-locus recombination of decorrelated " + "Recombination: blending inheritance and the Fisher--Muller effect. (A, top) Refitting a child " + "to the mean of its parents' output distributions conserves expected rare-item mass at the " + "single-parent level, cancelling the multi-parent gain to first order in the rare-item " + "regime; a strongest-source (union) operator, with renormalisation and an oracle, realises it. (B, bottom) Multi-locus recombination of decorrelated " "specialists assembles a genotype fitter than any parent, climbing to the optimum as parents " "are added, while the best single parent and the blended average plateau below."), "fig3": (["results/E9/E9.pdf", "results/E10/E10.pdf", "results/E14/E14.pdf"], @@ -45,8 +46,9 @@ FIGURES: dict[str, tuple[list[str], str]] = { "(promiscuous) mixing maximises the population mean but monotonically destroys diversity; the " "champion-optimal mate-pool breadth narrows as the landscape roughens."), "fig4": (["results/E11/E11.pdf"], - "The society: grounding, sex, and diversity are jointly necessary. A finite agent population " - "on a rugged NK landscape under a grounded selection score. Four-arm ablation: the full system " + "The society: grounded evaluation, recombination, and diversity preservation make complementary " + "contributions in the tested model. A finite agent population on a rugged NK landscape, with " + "selection weighting true fitness against conformity. Four-arm ablation: the full system " "climbs to near the global optimum; removing grounding collapses the population onto a " "confident, unfit consensus (self-consumption); removing recombination strands it on local " "optima; removing diversity converges it prematurely. Each ablation fails differently."), @@ -55,7 +57,7 @@ FIGURES: dict[str, tuple[list[str], str]] = { "Model speciation across three tiers. (A, top) Analytic: hybrid fitness traces compatible " "$\\rightarrow$ outbreeding depression $\\rightarrow$ inviability; the cliff arrives earlier " "the denser the incompatibilities; incompatibility count snowballs with divergence. (B, " - "middle) Trained MLPs: the merge barrier decomposed under the complete unit symmetry group --- " + "middle) Trained MLPs: the merge barrier decomposed under permutation-and-rescaling alignment --- " "same-task/different-init barriers are coordinate artefacts (removed by alignment); " "conflicting-task barriers survive in full, with hybrid fitness falling 0.97 $\\rightarrow$ " "0.03; divergence without conflict produced no isolation, the merge instead rescuing the " diff --git a/paper/pnas/main.md b/paper/pnas/main.md index de14e7d..ac84097 100644 --- a/paper/pnas/main.md +++ b/paper/pnas/main.md @@ -46,20 +46,20 @@ Machine learning has become a population-scale phenomenon. Public repositories h 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). +and mutation dynamics are already being mapped with explicitly phylogenetic methods (1–3). 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 +hybrid checkpoints, including leaderboard-topping ones (4–7), and the engineering literature describes it in evolutionary vocabulary: "crossover," "mutation," "mate choice," populations of -merging models that climb benchmarks (2–5). +merging models that climb benchmarks (5, 8–10). The generations are coupled through data as well as through weights. Successive models increasingly learn from model output rather than from fresh human experience: frontier alignment pipelines are now -predominantly synthetic (over 98% in documented cases; 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 +predominantly synthetic (over 98% in documented cases; 11, 12), self-generated instruction data +seeds whole lineages of descendants (13), a large and growing share of the public web is +machine-generated or machine-translated text (14, 15), and the stock of human text is projected to be +exhausted by frontier training within this decade (16). Meanwhile persistent multi-agent systems and +emerging agent economies put many interacting models into sustained contact (17–20). A population whose members inherit from one another, recombine, and retransmit under these conditions is an evolving population in the technical sense. The claim of this paper is that the vocabulary should be given its mathematics: **multigenerational model populations are systems whose inheritance, @@ -69,12 +69,12 @@ 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 -drifts toward its own most common behaviour (6). That this is the mathematics of **genetic drift** in +drifts toward its own most common behaviour (21). 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 +one that has been derived in parallel from several other directions (22–24), including a closed-form +first-extinction law placing collapse onset at the Wright–Fisher first-extinction time (23), 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 +genetic drift (25). 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, @@ -83,21 +83,21 @@ framework from closed forms to trained networks to language models. In machine learning's own terms, the problem this frame addresses is the field's oldest, **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 +overwrites prior knowledge (catastrophic forgetting; 26, 27), and the discipline's remedies are, one by one, the population operators of this paper in single-model form: **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 +on the order of 1% for instruction tuning (28) and 5% to 25% by distribution-shift strength in +continual pretraining (29), sit where the minimal model's operational grounding threshold lies, a correspondence for which the framework supplies the missing theory (equilibrium diversity, and a per-capability survival law). **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 +samples, proposed as a cure in 1995 (30) and revived as generative replay (31), is this paper's ungrounded null: immigration from a drifting source, benign for one hop and compounding into -collapse over generations; verifier-filtering (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 +collapse over generations; verifier-filtering (32, 33) converts it back into grounding. +**Parameter isolation** (34, and frozen-base adapters, which forget far less; 35) is the engineered +decorrelation our specialists use; **complementary-learning-systems consolidation** (36–38) is our periodic adapter-into-base merge; the recent turn to **merging as a continual-learning mechanism** -(55–58) applies recombination within one lineage over time, where we apply it across lineages; and -the observation that **rare examples and long-tail knowledge are forgotten first** (59–61) is +(39–42) applies recombination within one lineage over time, where we apply it across lineages; and +the observation that **rare examples and long-tail knowledge are forgotten first** (43–45) is tail-allele extinction observed one model at a time. One distinction is kept explicit throughout: catastrophic forgetting is largely deterministic interference from shifted training, whereas collapse is stochastic sampling drift; the two phenomena share their victims, the rare, and their remedies, but not their mechanism. To our knowledge, no prior work carries population-genetic formalism into continual @@ -116,19 +116,19 @@ 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** (46): once every copy of a rare capability is gone from all parents and sources, no recombination can rebuild it, which is why remedies must act before fixation-by-loss (a consequence- level correspondence: the minimal model lacks the ratchet's recurrent deleterious-mutation mechanism, so irreversible loss alone does not identify that specific mechanism). Injecting verified real data is -**immigration** from a non-drifting source (11–13). Model merging is **recombination**, and its -central payoff, a merged model exceeding every parent, is the **Fisher–Muller effect** (14, 15). +**immigration** from a non-drifting source (32, 47, 48). Model merging is **recombination**, and its +central payoff, a merged model exceeding every parent, is the **Fisher–Muller effect** (49, 50). Merging entangled skills courts **outbreeding depression**; screening many candidate merges is engineered recombination with unusually flexible parent choice and pre-deployment screening (we use the shorthand **directed sex**); restricting who merges with whom is **population structure**. Merging's hard limit, models too diverged in function to combine, is **reproductive -isolation**, for which the Bateson–Dobzhansky–Muller theory of incompatibilities (16, 17) supplies the +isolation**, for which the Bateson–Dobzhansky–Muller theory of incompatibilities (51, 52) supplies the structure. The nearest precursor to this programme reads sex as an algorithm for mixability in the -theory of computation (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 +theory of computation (53), pre-dating model merging; the model-merging literature itself has strong +empirical operators (4, 54, 55) and emerging merge-success predictors (56, 57), to which our delta is mechanism: *when and why* failure is coordinate versus functional, and what moves the boundary. We support the framework at three tiers of evidence, in ascending realism and descending exactness: a @@ -223,7 +223,7 @@ contrasting union operator (keep each item's strongest source, then renormalise, redistributes mass and presupposes a verifier or oracle to identify the strongest source) increases expected retention with K in all regimes in the minimal model. The practically important operators, **weight averaging** (a nonlinear network's weight-mean does not compute its parents' -output-mean) and **routing among intact specialists** (different storage and inference budgets from a +output-mean) and **routing among intact specialists** (58) (different storage and inference budgets from a single child), are its empirical cousins, and the measured bridge is a **headroom rule**, stated qualitatively: in language models, union-preserving operators beat the weight-average where that average falls short of attainable performance, and add nothing where it does not (easy-versus-hard contrasts at two scales; a @@ -236,10 +236,10 @@ complementary variants that arose in different lineages, producing a genotype fi 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 +LoRA (59) specialists beat every parent overall (decisively at 7B: 0.87 vs 0.77), and on the sharper worst-family metric the merged models are the only ones competent everywhere, in every seed (Fig. 6A). -Sex has risks and, for AI, an unfair advantage, both quantified on rugged (epistatic) NK landscapes +Sex has risks and, for AI, an unfair advantage, both quantified on rugged (epistatic) NK landscapes (60) (Fig. 3). When skills are entangled, blind recombination produces offspring *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 @@ -268,7 +268,7 @@ the analogue of scoring models by the crowd's approval (the fitness channel). Th ideas, since both couple the lineage to a non-drifting external signal, but they are different operators, and we name them separately. In the tested society (a finite agent population on a rugged NK landscape), a four-arm ablation separates the failure modes: the full system (grounded evaluation + -directed recombination + diversity-preserving selection) climbs to near the global optimum while +directed recombination + diversity-preserving selection (61)) climbs to near the global optimum while keeping its specialists; removing grounded evaluation converges the population confidently on an unfit consensus (self-consumption); removing recombination strands it on local optima; removing diversity converges it prematurely to a worse answer. Each removal fails differently; the three implementations @@ -286,13 +286,13 @@ changes harmless on their own background but deleterious in combination. A merge 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). We note that a super-linear count does not by itself entail a sharp performance cliff without +(52). We note that a super-linear count does not by itself entail a sharp performance cliff without the count-to-effect-size link, which the analytic model supplies under its assumptions and any neural test must establish separately. In trained networks, the claim must survive a known alternative: merge barriers between independently -trained networks are famously *coordinate artefacts*, removable by re-aligning hidden units (23), and -richer symmetry groups remove more (24). We therefore aligned under the composition of +trained networks are famously *coordinate artefacts*, removable by re-aligning hidden units (62); +richer symmetry groups remove more (63), with known failures beyond the shared-data regime (64). We therefore aligned under the composition of permutation matching and exact per-unit rescaling (the unit symmetry group of plain ReLU MLPs, as the search space) and decomposed the barrier (Fig. 5B): two networks trained from different initialisations on the *same* task have a barrier that this alignment removes essentially entirely @@ -315,7 +315,8 @@ catastrophically-forgetting specialists (parents ≈ 0.50, merge ≈ 0.955, a su 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 -disjoint specialists 1→12 epochs produces no isolation at all — the merge improves. Across every tier +disjoint specialists 1→12 epochs (cf. the merging literature's expert-duration effect; 65) produces +no isolation at all — the merge improves. Across every tier tested, **isolation had to be provoked by functional conflict; specialisation alone did not speciate** — a bound on the analogy that sharpens the design rule: what breaks merging is conflicting conventions on shared circuitry, not divergence per se. @@ -334,7 +335,7 @@ divergence with zero conflict). Before merging, six predictors are computed: **c functional conflict** (bilateral confident disagreement on probes drawn blind to where conflict lives — a proposed proxy for merge-relevant interactions, motivated by the observation that raw disagreement counts harmless complementation, one parent merely ignorant, as conflict), raw disagreement, gradient -alignment at the shared base (21), LoRA-delta cosine and distance, and a cross-task performance +alignment at the shared base (56), LoRA-delta cosine and distance, and a cross-task performance baseline. The pre-registered outcome is the merge penalty against oracle parent potential (the hybrid-load analogue), also reported against best- and mean-parent references because the predictor ordering is sensitive to that choice. @@ -392,14 +393,14 @@ 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 field's constants (≈1%, 5%, 25%; 52, 53) +(i) a first-principles account of the **replay ratio**: the field's constants (≈1%, 5%, 25%; 28, 29) acquire an equilibrium theory and a sharper prediction, that the required fraction is set by the rarest capability one refuses to lose (the `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 +unless verifier-filtered back into grounding (30–33); (iii) **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 +metrics (56), the functional-conflict measure arrives at a convergent signal from principle and comes with an operator prescription — when conflict is high, do not average; route or breed-and-screen; (iv) a candidate **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 @@ -407,15 +408,15 @@ where headroom exists, fusion where the base composes, consolidation as the slow **tail monitoring as the leading indicator**: continual-learning evaluation that averages over capabilities hides exactly the losses that drift theory says come first and, past a threshold, become irreversible. On that last point we note the standing objection that apparent forgetting can be -skewed task-inference over latent capability rather than erasure (64); our irreversibility results +skewed task-inference over latent capability rather than erasure (66); our irreversibility results concern oracle-measured behavioural distributions, and distinguishing latent from extinct capability at language-model scale is an open experiment whose outcome would be decisive for both readings. **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 +others and we cite it so (21–24), 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 +naive averaging loses to interference-aware or routed merges (4, 54, 55), that model populations can +climb (5, 8–10), and that merge success admits ML-native predictors (56, 57). Ours is the framework-level synthesis — inheritance, diversity, and compatibility as managed quantities — together with: the conservation law for blending inheritance and its operator boundaries; the per-item grounding floor; the society ablation with its complementary failure modes; model speciation as a named, tested question, with the @@ -474,68 +475,69 @@ publication; every figure in this paper regenerates from committed artifacts wit ## References -1. Yadav P, Tam D, Choshen L, Raffel C, Bansal M (2023) TIES-Merging: resolving interference when merging models. *NeurIPS*. arXiv:2306.01708. -2. Akiba T, Shing M, Tang Y, Sun Q, Ha D (2025) Evolutionary optimization of model merging recipes. *Nat Mach Intell* 7:195–204. -3. 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md2tex + tectonic at that point) - [x] Downplaying removed: convergence framing (diagnosis reached independently; Riis/Benati/Yoon + Crutchfield&Whalen 2012 cited for priority of publication; convergence = corroboration; the full arc owned). Applied to PNAS draft + v6 abstract. -- [x] References 30 → 65 (all verified by the lit agents; append-only numbering — citation-order - renumbering is a submission-time mechanical fix, with refs 31–65 to be reordered by first - appearance when reflowing to pnas.cls). +- [x] References: DONE fully (2026-08-11) — 66 entries renumbered to first-appearance order + (verified programmatically: in-text order = 1..66 = list order), reformatted to PNAS style + (initials-first, sentence case, abbreviated italic journals, bold volumes, year-at-end, + arXiv [Preprint] + DOI), and verified: all 47 arXiv ids batch-checked against the API + (caught + fixed: authorless GENOME -> Y. Zhang et al.; "Sakana AI" -> J. Abrantes et al.; + wrong Kotha id 2310.05719 -> 2309.10105; Nemotron corporate author; Liang full title; + >5-author entries to et-al form). Six orphaned refs re-anchored in text (NK, QD, Pari, LoRA, + Sharma, Kozodoi) and one mis-citation fixed (Self-Instruct, new ref, was credited to + Multiagent-Finetuning). Figure captions in build.py brought to third-review calibration + (operational threshold; first-order conservation; complementary-contributions society; + permutation-and-rescaling alignment). - [ ] v6 long-form: sync the premise/CL sections if GG wants the long document to match (currently only the convergence fix is propagated).