paper: claim-narrowing revision from the external review

The review's core instruments adopted: the interpretation/explanation/
prediction ladder is now explicit in §1 (with the decisive pre-merge
epistasis-prediction test stated as the open bar, not claimed); identity
claims scoped (WF exact only in the minimal model, with the
learning-kernel deviation cited against ourselves; Muller's ratchet
scoped to the irreversible arm — recombination reassembles only what
survives); "nobody has / none imports / theory outrun" removed;
merge-don't-average given explicit operator boundaries (output-mean vs
weight-average vs routing vs max-with-oracle; budgets; oracle; capacity
handoff to speciation); a "what these experiments do and do not
establish" scope block added to the speciation section (conflict floor
is information-theoretic, not genetic; epistasis-cliff + snowball =
hypotheses at the neural tier; emergent DMIs = flagship hypothesis,
bounded by our null); "control theory" -> "framework" (subtitle
included); §3/§11 overstatements fixed (frozen core != frozen behaviour;
Baldwin echo, not identity; operational vs archival irreversibility);
claims-at-a-glance table (status/assumptions/evidence/limits) added to
§13. Reviewer's framing sentence adopted as the stated core
contribution. Accessible version calibrated to match. md2tex gains pipe-
table support; PDF rebuilds clean (22 pp). Lessons recorded.

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

paper/arxiv/body.tex

@@ -7,9 +7,9 @@ This paper sits at the meeting point of three fields, and it is written so that
 \item \textbf{Distillation} \emph{(machine learning)} --- training a fresh ``student'' model on the outputs of one or more ``teacher'' models, so the student ends up knowing a compressed version of what they knew.
 \item \textbf{Model merging} \emph{(machine learning)} --- combining several trained models directly, at the level of their weights, into one --- no retraining. (Think of it as breeding two models rather than teaching a third.)
 \item \textbf{Genetic drift} \emph{(population genetics)} --- the random loss of rare variants that happens in any finite population simply because not everyone leaves offspring. It is the neutral, no-selection baseline of evolution.
-\item \textbf{Wright--Fisher process} \emph{(population genetics)} --- the standard mathematical model of drift. We will claim, and show, that generational model-training \emph{is} this process, not merely like it.
+\item \textbf{Wright--Fisher process} \emph{(population genetics)} --- the standard mathematical model of drift. Our minimal model of knowledge transmission \emph{is} this process exactly; a real trained network is this process plus a measurable, architecture-specific bias we quantify.
 \item \textbf{Recombination / sexual reproduction} \emph{(biology)} --- making an offspring by combining pieces from more than one parent, rather than copying a single parent (which is \emph{asexual} reproduction).
-\item \textbf{Muller's ratchet} \emph{(population genetics)} --- the way an asexual lineage, one that never recombines, accumulates damage it can never undo. It is, we will argue, the same thing as model collapse.
+\item \textbf{Muller's ratchet} \emph{(population genetics)} --- the way an asexual lineage, one that never recombines, accumulates damage it can never undo. We will argue it is the right lens for the \emph{irreversible} part of model collapse --- the capabilities that, once lost from every parent, no merging can rebuild.
 \item \textbf{Catastrophic forgetting} \emph{(machine learning / neuroscience)} --- a neural network overwriting what it knew when it learns something new.
 \end{itemize}
 
@@ -19,11 +19,11 @@ We have tried to keep the big picture legible on every page, and to be candid ab
 
 \section*{Abstract}
 
-AI is turning from single frozen models to \textbf{populations of agents} that persist, specialise, and are increasingly \emph{recombined} into new models --- a shift visible in multi-agent societies, population-based self-improvement, and the explosion of \textbf{model merging}. The field is doing this with the vocabulary of evolution --- ``crossover,'' ``mutation,'' ``mate choice,'' ``offspring that beat their parents'' --- but as loose metaphor draped over search algorithms. This paper argues that the right theory is already written, in the branch of biology that studies exactly this: the \textbf{evolution of sex}. Ninety years of population genetics say precisely when reproducing a population by \emph{recombination} beats copying, when it backfires, and how to do it better --- and, read as a control theory, it tells an engineer how to keep a society of models learning across generations instead of decaying.
+AI is turning from single frozen models to \textbf{populations of agents} that persist, specialise, and are increasingly \emph{recombined} into new models --- a shift visible in multi-agent societies, population-based self-improvement, and the explosion of \textbf{model merging}. The field is doing this with the vocabulary of evolution --- ``crossover,'' ``mutation,'' ``mate choice,'' ``offspring that beat their parents'' --- but as loose metaphor draped over search algorithms. This paper argues that a rich, quantitative body of applicable theory already exists in the branch of biology that studies exactly this: the \textbf{evolution of sex}. Ninety years of population genetics analyse when reproducing a population by \emph{recombination} beats copying, when it backfires, and how to do it better --- and, read as an engineering framework, it supplies overlooked variables and testable design rules for keeping a society of models learning across generations instead of decaying. The underlying shift of perspective is the contribution we most want to land: \textbf{treat multigenerational model populations as systems whose inheritance, diversity, and compatibility must be managed --- not merely as collections of models to optimise.}
 
-We take one diagnosis as settled and cite it as such: training each generation on the last is \textbf{genetic drift}, and the resulting \textbf{model collapse} is the loss of rare variants a finite population always suffers (the Wright--Fisher process; formalised for language models by Shumailov et al., 2024, and Riis, 2026). We claim none of that. Our contribution is the other half --- the \textbf{cure}, and its assembly into a theory with predictions. Single-teacher copying is \textbf{asexual} reproduction, and asexual lineages decay by \textbf{Muller's ratchet}, which \emph{is} model collapse; the remedy nature found is \textbf{sex}. A society of models should reproduce sexually --- each new model \textbf{recombined from several complementary parents} (which the field already does, as \emph{model merging}), selection \textbf{anchored to a reality that can say no} (not to the consensus of other models), and diversity actively \textbf{preserved}. With those three ingredients a lineage does not merely avoid collapse; it \textbf{climbs} --- producing models fitter than any ancestor (the \textbf{Fisher--Muller effect}) while each specialty is re-earned and exceeded.
+We take one diagnosis as settled and cite it as such: training each generation on the last is \textbf{genetic drift}, and the resulting \textbf{model collapse} is the loss of rare variants a finite population always suffers (the Wright--Fisher process; formalised for language models by Shumailov et al., 2024, and Riis, 2026). We claim none of that. Our contribution is on the remedy side. Single- teacher copying is \textbf{asexual} reproduction, and the irreversible arm of its decay corresponds to \textbf{Muller's ratchet} (a correspondence we state with its scope, not as identity); the remedy biology found for the ratchet is \textbf{sex}. A society of models should reproduce sexually --- each new model \textbf{recombined from several complementary parents} (which the field already does, as \emph{model merging}), selection \textbf{anchored to a reality that can say no} (not to the consensus of other models), and diversity actively \textbf{preserved}. In our models --- from closed-form to trained networks to a language-model prototype --- those three ingredients together let a lineage not merely avoid collapse but \textbf{climb}, producing models fitter than any ancestor (the \textbf{Fisher--Muller effect}) while each specialty is re-earned and exceeded; whether the full recipe holds at frontier scale is the open question the framework is built to test.
 
-From the geneticist's apparatus we extract falsifiable, load-bearing claims the merging literature has not: (i) \textbf{``merge, don't average''} --- recombination preserves the union of what parents kept, while averaging (a ``model soup'') is \emph{blending inheritance} that mathematically cancels the benefit; (ii) \textbf{offspring can exceed every parent} (Fisher--Muller), the real argument for sex in model societies; (iii) on \textbf{rugged, epistatic} task landscapes, blind recombination causes \textbf{outbreeding depression}, yielding a design rule --- \emph{merge freely when skills are additive, sparingly and with selection when entangled, and route rather than blend under overlap}; (iv) \textbf{grounding is immigration} from a non-drifting reality, giving a critical real-data fraction far below one; and (v) --- the sharpest new prediction --- sex has a \textbf{limit}: as two models diverge they undergo \textbf{speciation}, a merge-compatibility cliff (compatible \(\rightarrow\) outbreeding depression \(\rightarrow\) hybrid inviability) whose onset is set by divergence \emph{and} epistasis via \textbf{Bateson--Dobzhansky--Muller incompatibilities}, and whose damage grows \emph{super-linearly} (the Orr--Turelli snowball). We introduce and model this ``model speciation'' directly, and confirm it in real trained weights: a merge barrier that survives alignment under the \emph{full} function-preserving symmetry group of the network (not just Git Re-Basin permutations), rising with functional conflict while hybrid fitness falls to inviability --- with an honest converse we pre-registered and found: absent conflicting training signals, divergently-specialised lineages of shared ancestry developed \emph{no} isolation at any divergence tested, the merge instead \emph{rescuing} the forgetting specialists. Isolation must be provoked by conflict; specialisation alone did not speciate. AI also has an advantage biology lacks: \textbf{directed sex} --- unbounded parents, chosen mates, and offspring screened before they are kept --- which converts recombination from a gamble into a reliable engine and has no biological analogue.
+From the geneticist's apparatus we extract falsifiable, load-bearing claims (each stated with its operator and scope in the text): (i) \textbf{``merge, don't average''} --- a conservation result: refitting a child to the \emph{mean of its parents' output distributions} conserves rare-capability mass at the single-parent level, so adding parents cannot help, while union-preserving operators realise the gain --- exact in the minimal model, with its weight-space image verified as the headroom rule below; (ii) \textbf{offspring can exceed every parent} (Fisher--Muller), the real argument for sex in model societies; (iii) on \textbf{rugged, epistatic} task landscapes, blind recombination causes \textbf{outbreeding depression}, yielding a design rule --- \emph{merge freely when skills are additive, sparingly and with selection when entangled, and route rather than blend under overlap}; (iv) \textbf{grounding is immigration} from a non-drifting reality, giving a critical real-data fraction far below one; and (v) --- the sharpest new prediction --- sex has a \textbf{limit}: as two models diverge they undergo \textbf{speciation}, a merge-compatibility cliff (compatible \(\rightarrow\) outbreeding depression \(\rightarrow\) hybrid inviability) whose onset is set by divergence \emph{and} epistasis via \textbf{Bateson--Dobzhansky--Muller incompatibilities}, and whose damage grows \emph{super-linearly} (the Orr--Turelli snowball). We introduce and model this ``model speciation'' directly, and confirm it in real trained weights: a merge barrier that survives alignment under the \emph{full} function-preserving symmetry group of the network (not just Git Re-Basin permutations), rising with functional conflict while hybrid fitness falls to inviability --- with an honest converse we pre-registered and found: absent conflicting training signals, divergently-specialised lineages of shared ancestry developed \emph{no} isolation at any divergence tested, the merge instead \emph{rescuing} the forgetting specialists. Isolation must be provoked by conflict; specialisation alone did not speciate. AI also has an advantage biology lacks: \textbf{directed sex} --- unbounded parents, chosen mates, and offspring screened before they are kept --- which converts recombination from a gamble into a reliable engine and has no biological analogue.
 
 We support the argument with \textbf{minimal, reproducible models} --- a closed-form-exact account of drift and grounding, the same effects in small trained networks and an MNIST image generator, a real-weight demonstration of the speciation cliff (a Git Re-Basin residual that survives neuron alignment), and evolutionary simulations of the whole society --- and a first \textbf{language-model prototype}: merging LoRA-specialised Qwen models (to 7B on a GPU cluster) yields a generalist that beats every specialist parent, with the sharp headroom condition under which ``merge, don't average'' bites. The scope is honest: these are existence proofs and design rules; the \emph{whole grounded society} on a large language model is the open step. We position the work carefully against the crowded 2025--2026 landscape of evolutionary-AI and merging methods --- conceding what they own and marking, precisely, what a genuine population-genetics of sex adds.
 
@@ -39,7 +39,7 @@ The unit that matters is therefore the \textbf{generation}, and the event that m
 
 This axis is suddenly crowded. By 2026 several groups build \textbf{populations of models or agents that improve across generations}: societies of independently-specialised models that self-improve for more rounds than a single agent (Multiagent Finetuning --- Subramaniam et al., 2025); open-ended archives of self-rewriting coding agents (the Darwin--Gödel Machine --- Zhang et al., 2025); groups that evolve by sharing experience across branches (Weng et al., 2026); persistent agent \emph{ecologies} with reproduction and cumulative culture (TerraLingua --- 2026). In parallel, \textbf{model merging} has become a small industry with an overtly evolutionary vocabulary: crossover-mutation-selection over LLM populations (GENOME --- 2025), niching and ``mate choice'' (Sakana's M2N2 --- 2025), and evolutionary search over merge recipes (Akiba et al., \emph{Nature Mach. Intell.} 2024/25).
 
-We are candid about the consequence. Three things we do \textbf{not} claim. First, that collapse is Wright--Fisher drift: formalised independently (Riis, 2026; Shumailov et al., 2024), sharpened to a closed-form first-extinction law whose onset coincides with collapse (Benati et al., 2025) and to a quantitative-trait-genetics account for diffusion models (Yoon et al., ICLR 2025), and conceded here. Second, the bare empirical facts that a merged model can beat its parents, that decorrelated parents merge better, and that naive averaging is inferior to sign- or routing-based merges (TIES, DARE, mixture-of-experts routing): all established. Third, that merge success can be \emph{predicted at all}: machine-learning-native predictors exist, from interpretable pairwise metrics (gradient alignment --- Zhou et al., 2026) to capacity/rate-distortion accounts of ``merging collapse'' (2026); what they lack, and we supply, is the \emph{mechanism} --- when and why the failure is a coordinate artefact versus genuine functional incompatibility, and what moves the cliff. What is genuinely unoccupied --- and what a geneticist is placed to supply --- is a \textbf{theory} rather than a search heuristic. The nearest precursor is a theory-of-computation tradition reading sex as an algorithm for \emph{mixability} (Livnat \& Papadimitriou, 2016), pre-dating model merging and never applied to it. Every one of the works above uses evolution as \emph{metaphor over an optimiser}; none imports the predictive apparatus of the evolution of sex. Nobody has stated the \textbf{``merge, don't average'' conservation law}, derived \textbf{offspring-exceed-parents as Fisher--Muller}, predicted \textbf{outbreeding depression on rugged task landscapes}, framed \textbf{grounding as migration--drift balance} with a critical fraction, or connected \textbf{reproductive isolation} to when two models can be merged at all. An evolutionary algorithm that \emph{finds} a super-parent is evidence for the theory, not a substitute for it --- the way CMA-ES existing does not make fitness-landscape theory redundant. This paper supplies the theory the tinkering has outrun, and states what it predicts and where it would fail.
+We are candid about the consequence. Three things we do \textbf{not} claim. First, that collapse is Wright--Fisher drift: formalised independently (Riis, 2026; Shumailov et al., 2024), sharpened to a closed-form first-extinction law whose onset coincides with collapse (Benati et al., 2025) and to a quantitative-trait-genetics account for diffusion models (Yoon et al., ICLR 2025), and conceded here. Second, the bare empirical facts that a merged model can beat its parents, that decorrelated parents merge better, and that naive averaging is inferior to sign- or routing-based merges (TIES, DARE, mixture-of-experts routing): all established. Third, that merge success can be \emph{predicted at all}: machine-learning-native predictors exist, from interpretable pairwise metrics (gradient alignment --- Zhou et al., 2026) to capacity/rate-distortion accounts of ``merging collapse'' (2026); what they lack, and we supply, is the \emph{mechanism} --- when and why the failure is a coordinate artefact versus genuine functional incompatibility, and what moves the cliff. What a geneticist is placed to supply is a \textbf{framework} rather than a search heuristic. The nearest precursor is a theory-of-computation tradition reading sex as an algorithm for \emph{mixability} (Livnat \& Papadimitriou, 2016), pre-dating model merging; the works above use evolution chiefly as vocabulary over an optimiser, and --- to our knowledge --- the quantitative apparatus of the evolution of sex (Fisher--Muller, outbreeding depression, migration--drift balance, reproductive isolation) has not previously been carried over as more than metaphor. We are also candid about what \emph{kind} of contribution each of our claims is, because three different things are easily conflated: \textbf{interpretation} (an existing result is usefully understood in these terms --- e.g., merged offspring beating their parents as Fisher--Muller), \textbf{explanation} (the transferred mechanism accounts for observations existing accounts leave open --- e.g., which merge failures are coordinate artefacts and which are functional), and \textbf{prediction} (the framework forecasts an unmeasured outcome and improves a design decision --- e.g., an epistasis measure taken \emph{before} merging that beats geometry-based predictors of merge success). This paper is strongest on the first, makes concrete progress on the second, and states the third as its open, decisive test --- proposed here with pre-registered falsifiers, not claimed as done. The organising shift we argue for is prior to any single mechanism: \textbf{treat multigenerational model populations as systems whose inheritance, diversity, and compatibility must be managed --- not merely as collections of models to optimise.}
 
 \section*{2. Why today's models cannot do this}
 
@@ -51,7 +51,7 @@ There is a real reason for the freeze. Updating a neural network on new informat
 
 The individual model needs two properties.
 
-\textbf{It must not catastrophically forget.} Instead of overwriting its core as it learns, it keeps that core frozen and only \emph{readable}, and carves each new skill into freshly-added capacity beside it. In machine learning this is called \emph{parameter isolation} (progressive networks --- Rusu et al., 2016; prune-and-freeze --- Mallya \& Lazebnik, 2018; and, most practically, \textbf{LoRA} and other small trainable ``patches'' bolted onto a frozen model --- Hu et al., 2021). If the core is never altered, forgetting it is not merely unlikely but structurally impossible. This is what lets a model accumulate a coherent working life of expertise --- the kind of stable knowledge worth passing on.
+\textbf{It must not catastrophically forget.} Instead of overwriting its core as it learns, it keeps that core frozen and only \emph{readable}, and carves each new skill into freshly-added capacity beside it. In machine learning this is called \emph{parameter isolation} (progressive networks --- Rusu et al., 2016; prune-and-freeze --- Mallya \& Lazebnik, 2018; and, most practically, \textbf{LoRA} and other small trainable ``patches'' bolted onto a frozen model --- Hu et al., 2021). If the core is never altered, its \emph{parameters} cannot be forgotten --- though a precise reader should note the system's \emph{behaviour} can still shift while adapters are active, so the guarantee is of a recoverable core, not of unchanging conduct. This is what lets a model accumulate a coherent working life of expertise --- the kind of stable knowledge worth passing on.
 
 The brain offers a partial blueprint. \emph{Complementary Learning Systems} theory (McClelland, McNaughton \& O'Reilly, 1995) --- itself a response to the forgetting problem --- describes two subsystems: a \textbf{fast} store (the hippocampus) that grabs an experience in one shot, and a \textbf{slow} store (the neocortex) that integrates regularities gradually without disruption. We do not lean on any particular account of how the brain moves knowledge between them; the architecture needs only that \emph{some} periodic \textbf{offline consolidation} step exists, moving knowledge from the fast store to the slow one when the system is idle. The machine version is clean regardless: the prompt is working memory, an external database is the fast episodic store, the trained weights are the slow store, and consolidation migrates the first into the last.
 
@@ -71,9 +71,9 @@ Suppose a mature model simply teaches a fresh one --- distillation, one teacher
 
 \textbf{The machine-learning statement.} Training each generation on the previous generation's outputs is the recipe for \textbf{model collapse}: the model forgets the improbable, loses the \emph{tail} of the distribution (the rare cases) first, and drifts toward its own most common output (Shumailov et al., 2024). Worse for us, the very rule that makes distillation useful --- \emph{keep the general, drop the idiosyncratic} --- \textbf{is} tail-deletion by design. The operation that would power a cultural ratchet and the operation that drives model collapse are the same act.
 
-\textbf{The population-genetics statement (the same thing).} Represent a model's knowledge as a distribution over discrete ``items'' --- capabilities, facts, modes of behaviour. One generation is: \emph{draw a finite sample from the parent, and refit the child to it.} That finite-sampling step is \textbf{mathematically identical} to \textbf{genetic drift} --- the random loss of rare variants in a finite population --- described by the century-old \textbf{Wright--Fisher} model (Wright, 1931; Fisher, 1930). This is not an analogy we find pretty; it is the same equations, and we use them as an exact check on our simulations (the first of the minimal models below). Rare items go extinct first, roughly ten times faster than common ones, precisely as drift predicts.
+\textbf{The population-genetics statement (the same thing, for the minimal model).} Represent a model's knowledge as a distribution over discrete ``items'' --- capabilities, facts, modes of behaviour. One generation is: \emph{draw a finite sample from the parent, and refit the child to it.} In this \textbf{minimal inheritance model} the finite-sampling step is \textbf{exactly} genetic drift --- the random loss of rare variants in a finite population --- described by the century-old \textbf{Wright--Fisher} model (Wright, 1931; Fisher, 1930): the same equations, which we use as closed-form checks on our simulations. Rare items go extinct first, roughly ten times faster than common ones, precisely as drift predicts. \textbf{The boundary of the identity matters, and we measured it:} real neural training adds approximation, optimisation noise, and inductive bias on top of sampling, and when we fit trained networks against the exact drift null they deviate in \emph{opposite, architecture-specific directions} --- a smoothing recurrent model resists collapse (it keeps spurious variants alive), a sharpening image generator accelerates it (our learning-kernel result, below). So the honest statement is: the minimal inheritance model is exactly Wright--Fisher; a real learner is Wright--Fisher \emph{plus a signed, measurable estimator-bias operator} --- and the drift signs (rare-first loss, the grounding response) survive that operator in every architecture we tested.
 
-And single-teacher copying is \textbf{asexual reproduction} --- cloning one parent. Nature already knows what happens to an asexual lineage that never recombines: it accumulates damage it can never repair, a one-way decline geneticists call \textbf{Muller's ratchet} (Muller, 1964). \emph{Muller's ratchet is model collapse.} Naming it that way is not decoration; it tells us where the cure is, because biology solved this problem.
+And single-teacher copying is \textbf{asexual reproduction} --- cloning one parent. Nature already knows what happens to an asexual lineage that never recombines: it accumulates damage it can never repair, a one-way decline geneticists call \textbf{Muller's ratchet} (Muller, 1964). We use the ratchet as the \emph{organising correspondence} for model collapse, with its scope stated: strictly, the ratchet is the stochastic loss of the least-degraded class under recurring deleterious change in an asexual population, so it maps onto the \emph{irreversible} component of capability loss (once every copy of a rare capability is gone from all parents and sources, no recombination can rebuild it) rather than onto every form of degradation. That is exactly why the correspondence is useful rather than decorative: it says the cure must act \emph{before} fixation-by-loss --- keep complementary variants alive somewhere in the population --- because recombination can only reassemble what still survives. Biology solved this problem, and its solution is the subject of this paper.
 
 Two ingredients turn the collapse operation into a climb. Both are things nature does.
 
@@ -87,6 +87,8 @@ Geneticists call this the \textbf{Fisher--Muller effect} (Fisher, 1930; Muller,
 
 This is no longer only a simulation. In a first language-model prototype --- LoRA specialists on disjoint task families, recombined and judged by an exact verifier --- a merge of three specialist Qwen models (7B, on a GPU cluster) \textbf{beats every single specialist}, overall and on every family: the Fisher--Muller effect, in real weights. The same prototype pins down \emph{when} the finer ``inherit the union, don't average'' rule actually bites. Keeping each parent whole and \textbf{routing} each input to the right one beats the tail-thinning average --- but only when the task is hard enough to leave room to lose: on easy tasks a strong model's plain average is already at the ceiling, so the crude soup is fine, whereas on hard tasks the average dilutes a hard-won specialist so badly it falls below even the best single parent, and routing wins by a wide margin. The rule is therefore precise: \textbf{the union beats the average in exact proportion to how far the average is from the best attainable} --- a caveat that sharpens rather than weakens the claim, and that a practitioner needs before spending compute on the fancier operator.
 
+\textbf{The operator boundaries (stated, because ``merge, don't average'' is not one claim but a family).} Four different operators travel under these words, and the conservation result belongs to exactly one of them. What is \emph{derived} is this: when a pupil's knowledge is refit to the \textbf{mean of the parents' output distributions}, the expected mass on any rare item is conserved at the single-parent level --- in the rare-item regime the 1/K dilution of averaging exactly cancels the union gain of having K parents --- so adding parents cannot help; whereas an operator that keeps, per item, its \textbf{strongest source} realises the union. That statement is exact in the minimal model, and it presupposes an oracle (or verifier) able to say which source is strongest. The two operators the LLM prototype tests --- \textbf{weight averaging} (a nonlinear network's weight-mean does not compute the mean of its parents' outputs) and \textbf{routing among intact specialists} (which keeps K models' storage and an input classifier, a different parameter and inference budget from one fixed-size child) --- are \emph{empirical cousins} of the two sides of that law, not instances of it. The headroom rule above is precisely the empirical bridge: it says when the weight-average behaves like the diluting mean (hard tasks, weak base) and when a capable base absorbs the dilution (easy tasks). And all of it operates within a capacity boundary: when parental capabilities genuinely cannot coexist in the child's capacity, no operator preserves the union --- that regime is the subject of the speciation section below.
+
 Three results keep this honest, and all are results, not hand-waving.
 
 \emph{Sex can backfire.} When the parents' skills are not cleanly separable but \textbf{entangled} --- when the value of one capability depends on which others are present (geneticists call this \textbf{epistasis}) --- blindly recombining two good models can produce a \emph{worse} child, because recombination breaks up a combination that only worked as a whole. Biologists call this \textbf{outbreeding depression}, and we reproduce it: on ``rugged'' (highly entangled) problems, naive merging drops offspring below their parents, and the more you mix the worse it gets. The design rule that falls out is simple: \emph{merge freely when skills are complementary; merge sparingly, and carefully, when they are entangled.}
@@ -133,6 +135,8 @@ This is where a geneticist's lens earns its keep. The machine-learning literatur
 
 \textbf{And its honest converse: speciation must be provoked; it did not emerge.} A true Dobzhansky--Muller incompatibility is \emph{emergent} --- each lineage's changes harmless alone, incompatible only in combination --- whereas the conflict condition above \emph{imposes} contradiction. So we pre-registered the emergent test: fork two children from a shared base and let them diverge with \textbf{no conflicting training signal anywhere} --- one pair as complementary class specialists (one child trains only on digits 0--4, the other only on 5--9), one pair with divergent input conventions (views shifted in opposite directions) --- out to divergences 6.4\(\times\) the base training. The result is the second pre-registered reading, and it sharpens the theory's scope rather than confirming its most dramatic form: the residual barrier is \textbf{0.000 at every divergence in both conditions}, and far from failing, the merge \emph{rescues} the two specialists --- each parent decays toward \textasciitilde{}0.50 on the full task (catastrophically forgetting the classes it no longer sees) while the merged model holds \textasciitilde{}0.95 throughout, a sustained Fisher--Muller rescue at zero barrier. In real weights, at least in this regime of shared ancestry and compatible tasks, \textbf{reproductive isolation requires functional conflict; it does not arise spontaneously from divergent specialisation.} The design rule sharpens accordingly: \emph{merge freely across divergently-specialised lineages of shared ancestry --- what speciates model populations is conflicting conventions, not specialisation per se.} Whether long-horizon over-specialisation erodes mergeability at language-model scale --- as the empirical merging literature hints (experts trained longer merge worse under averaging) --- is exactly the next tier's question, and the theory now makes the prediction crisp: it should depend on whether extended training induces \emph{conflicting conventions on shared circuitry}, not on divergence time itself.
 
+\textbf{What these experiments do and do not establish.} Stated at exactly the strength of the evidence: they establish that \emph{some merge failures reflect incompatible functional requirements rather than a mismatch of coordinates} --- a residual that survives the full unit-symmetry group of the architecture tested, rises with functional conflict, and is absent under compatible specialisation. Three qualifiers. First, the impossibility at the heart of the conflict condition --- one deterministic model cannot satisfy two contradictory answer conventions --- is information-theoretic and needs no population genetics; what the genetic frame adds is \emph{structure around it}: which divergences generate such conflicts, the prediction that epistasis rather than distance sets the cliff's position, and the snowball's super-linear onset --- the latter two verified so far only in the analytic model, and therefore carried as \textbf{hypotheses at the neural tier, not results}. Second, our alignment removes the symmetries we enumerate for this architecture class; richer transformation families for other architectures could reapportion removable vs residual, though not below the conflict floor. Third, ``unmergeable'' here means by aligned linear interpolation of weights --- a barrier to that operator does not preclude every conceivable recombination method (routing, for one, sidesteps it by not blending). Emergent Dobzhansky--Muller incompatibilities in real weights remain the flagship \emph{hypothesis} of this programme: our tested regimes found none, which bounds where they can live --- longer horizons, shifted data distributions, capacity pressure --- and the decisive experiment (predicting merge success \emph{before} merging from an operational epistasis measure, against geometry- and gradient-based predictors) is posed in the closing section.
+
 One question remains, and the rest of the paper is largely about it: recombination combines what the parents kept --- but \emph{who decides what each parent keeps, and which offspring are worth keeping?}
 
 \section*{6. The second inheritance: letting ``what is worth keeping'' evolve}
@@ -192,13 +196,13 @@ Every new model is a pristine base plus an inherited \textbf{soft} delta --- the
 
 The fix is the same operation, one level up. When a lineage's acquired knowledge has proven stable across enough generations, \textbf{re-mint the base}: distil the accumulated soft inheritance into the \emph{weights} of a fresh foundation-scale model --- a new base born already \emph{natively knowing} what took many generations to acquire in patches. The soft budget resets; the next epoch begins from a richer floor. What was hard-won and \emph{learned} becomes cheap and \emph{innate}.
 
-This has a precise name, and it is not Lamarck's. Knowledge that is acquired and re-learned every generation, and --- once reliably present for long enough --- becomes part of the innate endowment so that it need no longer be re-learned, is the \textbf{Baldwin effect} (Baldwin, 1896; and its clean computational demonstration, Hinton \& Nowlan, 1987). It is the valve between the two substrates: the soft, learned patches, and the hard base weights every model is born with.
+The pattern \textbf{echoes the Baldwin effect} (Baldwin, 1896; its clean computational demonstration is Hinton \& Nowlan, 1987): knowledge acquired and re-learned every generation eventually becoming part of the innate endowment. We use the echo advisedly --- Baldwin's mechanism is \emph{selection} favouring genotypes that learn the trait ever more easily, whereas re-minting is direct distillation, a deliberate engineering shortcut through the same soft-to-innate valve. The valve is the point: two substrates, the soft learned patches and the hard base weights every model is born with, with a controlled passage between them.
 
 Three honest riders, because re-minting is the most consequential step in the scheme:
 
 \begin{itemize}
 \item \textbf{Cost.} This is the one step that re-pays part of the pre-training bill, breaking §10's cheapness \emph{locally}. It is bearable only because it is \emph{rare}, amortised over many cheap generations, and is continued training from the lineage's own rich outputs rather than a de-novo run.
-\item \textbf{Irreversibility.} Until now, one thing was always recoverable --- the original pristine base, whose lost tails could be restored just by reloading the file. Bake the current lineage into new immutable weights and that escape hatch closes: if the lineage had been quietly collapsing, re-minting \emph{fixes the collapse in place} and discards the one uncollapsed reference that could have diagnosed it. In our minimal models this is exactly what happens, and a cheap safeguard prevents it: \textbf{re-mint only while the lineage is demonstrably diverse and healthy}, never as a rescue for a line already drifting. It is the sharpest instance of the human seat of §9 --- choosing what no future generation will think to question.
+\item \textbf{Irreversibility (of the lineage, not the archive).} A digital system can, of course, keep every old base on disk --- nothing forces deletion, and archives should be kept. The irreversibility is \emph{operational}: once the lineage's production base, training mixtures, and selection all run downstream of the re-minted weights, a quiet collapse baked into them propagates to every descendant, and the archived ancestor helps only if some process still compares against it --- which nothing in the loop does by default. In our minimal models a collapsed-then-re-minted lineage locks in its loss exactly this way, and a cheap safeguard prevents it: \textbf{re-mint only while the lineage is demonstrably diverse and healthy} (and keep an audit that diffs against the archived ancestor), never as a rescue for a line already drifting. It is the sharpest instance of the human seat of §9 --- choosing what no future generation will think to question.
 \item \textbf{Speciation.} A re-minting is a founder event. Different laboratories, re-basing on different criteria, will mint divergent bases; the lineage branches. This is not a defect but \emph{adaptive radiation}, and it is exactly what open weights make possible. The society grows not as one heavy trunk but as a branching tree of bases.
 \end{itemize}
 
@@ -236,9 +240,29 @@ The previous drafts of this paper promised a ``companion paper'' that \emph{woul
 \item \emph{The whole society, and why every part is needed.} In a population evolving on a ``reality'' landscape, the full system --- grounding + sexual recombination + preserved diversity --- climbs to the top while keeping its specialists. Remove \emph{grounding} and it collapses into a confident, wrong consensus (a direct analogue of training on the internet's growing crowd of AI-generated text); remove \emph{sex} and it gets stuck; remove \emph{diversity} and it converges too fast to a worse answer. Each removal fails differently; only the whole system climbs. This is the closest thing we have to a test of the actual thesis, rather than of the borrowed scaffolding around it.
 \end{itemize}
 
+\subsection*{The claims at a glance: status, assumptions, evidence, limits}
+
+Because a perspective of this breadth risks blurring what is proved, what is measured, and what is proposed, here is the ledger of the load-bearing claims --- each labelled \textbf{exact} (closed-form in the minimal model), \textbf{empirical} (measured in trained systems), or \textbf{hypothesis} (stated with a falsifier, not yet established):
+
+\medskip\noindent\begin{center}\footnotesize
+\begin{tabular}{p{0.184\textwidth} p{0.184\textwidth} p{0.184\textwidth} p{0.184\textwidth} p{0.184\textwidth}}
+\hline
+Claim & Status & Key assumptions & Evidence & Known limits \\ \hline
+Collapse = Wright--Fisher drift (minimal model) & Exact (diagnosis conceded to prior work) & Knowledge = categorical distribution; refit = resample & Closed forms reproduced to <0.5\% & Real learners add a signed, architecture-specific estimator bias (measured) \\[3pt]
+Grounding = immigration; critical real-data fraction ≪ 1 & Exact + empirical sign & Fresh samples from a fixed, non-drifting truth & Exact \texttt{H\_eq}; \texttt{g*\(\approx\)0.048}; sign holds in RNN/MLP/VAE and on MNIST & Deepest tail unrescuable at feasible budgets (\texttt{m ∼ 1/p}); sharp threshold softens in trained nets \\[3pt]
+``Merge, don't average'' conservation & Exact \textbf{for the output-mean operator} & Rare-item regime; an oracle/verifier identifies the strongest source & E4 closed form + simulation; neural reproduction & Weight-averaging and routing are empirical cousins, not instances; budgets differ; bridge = the headroom rule \\[3pt]
+Offspring exceed every parent (Fisher--Muller) & Interpretation + empirical & Complementary (decorrelated) parents; verifiable fitness & E8 analytic; 7B LoRA merge beats every specialist on every family & LLM tier: 3 lexically-distinct families; multi-seed replication in progress \\[3pt]
+Outbreeding depression on rugged landscapes; operator design rule & Exact-model result; hypothesis at LLM scale & NK epistasis stands in for skill entanglement & E9--E10; directed selection rescues & Not yet mapped onto a real task-entanglement measure \\[3pt]
+Optimal mate-pool breadth shrinks with ruggedness & Exact-model result; hypothesis for merging populations & Ring population, local selection & E14 & Phenomenon known to island-model evolutionary computation; our contribution is the mapping and the diversity/mean decomposition \\[3pt]
+Merge failure decomposes into coordinate artefact + functional residual & Empirical (MLP tier; LLM tier in progress) & Alignment enumerates the architecture's unit symmetries & Full-symmetry residual \(\approx\) 0 (compatible) vs \(\approx\) naive (conflict); cliff in hybrid fitness & Scoped to aligned linear interpolation; conflict floor is information-theoretic, not genetic \\[3pt]
+Epistasis (not divergence) sets the cliff; snowball onset & Exact-model result; \textbf{hypothesis} at the neural tier & BDM incompatibility structure & E12 & The decisive pre-merge prediction test is proposed, not run \\[3pt]
+Emergent speciation without conflict & \textbf{Not observed} (pre-registered) & Shared ancestry, compatible tasks, tested divergences & E13b: residual 0.000; merge rescues specialists & Bounds the hypothesis; longer horizons/distribution shift/capacity pressure untested \\[3pt]
+Grounding + sex + diversity jointly necessary & Exact-model result; hypothesis at LLM scale & Conformity stands in for self-consumption & E11 four-arm ablation, each arm failing distinctly & The full grounded LLM society is unbuilt \\[3pt]
+\hline\end{tabular}\end{center}\medskip
+
 \textbf{What is borrowed, and what is ours.} We are deliberate about the ledger, because the surrounding literature is crowded and a reader deserves to know exactly where the line falls. \textbf{Conceded as prior art:} (a) \emph{model collapse is genetic drift} --- derived independently and cleanly (Riis, 2026; the Wright--Fisher collapse literature following Shumailov et al., 2024; the closed-form first-extinction law of Benati et al., 2025; the quantitative-trait account of Yoon et al., 2025); (b) the empirical facts that a merged model can \emph{beat its parents}, that \emph{decorrelated} parents merge better, and that \emph{naive averaging is inferior} to sign-reconciled or routed merges (model soups, TIES, DARE, mixture-of-experts routing); (c) that a \emph{population} of merging or self-improving models can climb (GENOME, M2N2, Multiagent Finetuning, the Darwin--Gödel Machine); (d) that merge success has machine-learning-native \emph{predictors} --- interpretable pairwise metrics (Zhou et al., 2026), capacity/rate-distortion accounts of merging collapse (Cao et al., 2026), and stability/scaling analyses of multi-task degradation; and (e) that verifier-screened synthetic data can avert collapse (Yi et al., 2025) --- the statistical cousin of our grounding operator. We claim none of these.
 
-\textbf{Ours} is the theory those results have outrun: a \textbf{population-genetics of sex} applied to model societies, which is \emph{generative} where the incumbents are empirical. Concretely --- the \textbf{``merge, don't average'' conservation law} (recombination preserves the union; blending inheritance cancels it), derived not observed; \textbf{Fisher--Muller} named and used to explain \emph{why} offspring exceed parents; \textbf{outbreeding depression on rugged/epistatic landscapes}, which turns ``when does merging help vs hurt'' from a thing you must run a search to discover into a thing the landscape's ruggedness \emph{predicts}, with the operator-choice design rule that follows (average / union-route / directed-select); \textbf{grounding as migration--drift balance}, giving a critical real-data fraction and a phase boundary a closed self-consuming loop cannot have; \textbf{directed sex} as the distinctly-AI advantage (unbounded parents, offspring preview, mate choice); and the \textbf{integrated society} whose four operators are shown \emph{jointly necessary}. The value-add over the machine-learning-native merge theory is that ours predicts \emph{which operator to use and when it will backfire}, not merely how fast quality decays. And it opens --- and begins to occupy --- a question nobody has framed: \textbf{model speciation}, the population-genetics of \emph{reproductive isolation} (Bateson--Dobzhansky--Muller incompatibilities) as the account of \emph{when two models are too diverged to be merged at all}. We model it explicitly (§5), predicting the compatible \(\rightarrow\) outbreeding-depression \(\rightarrow\) inviability curve, its super-linear (snowball) onset, and its control by epistasis rather than divergence alone --- the one place the merge literature has phenomena (Pari et al., 2024; Zhou et al., 2026) but no theory --- and we confirm it in real trained weights, where a merge barrier survives alignment under the \emph{full} function-preserving symmetry group (not only Re-Basin permutations) as a residual, functional reproductive isolation with an information-theoretic floor --- together with the pre-registered emergent converse: absent conflicting training signals, divergently-specialised lineages of shared ancestry showed \emph{no} isolation at any divergence tested, the merge instead rescuing the forgetting specialists (isolation must be provoked; specialisation alone did not speciate). In one sentence: the field agrees on the disease and tinkers at the cure with evolutionary metaphors; we bring the evolutionary \emph{theory}, and it makes falsifiable predictions --- a merge-compatibility cliff among them --- that the metaphors do not.
+\textbf{Ours} is the framework those results invite: a \textbf{population-genetics of sex} applied to model societies, generative where the incumbents are empirical. Concretely --- the \textbf{``merge, don't average'' conservation law} (recombination preserves the union; blending inheritance cancels it), derived not observed; \textbf{Fisher--Muller} named and used to explain \emph{why} offspring exceed parents; \textbf{outbreeding depression on rugged/epistatic landscapes}, which turns ``when does merging help vs hurt'' from a thing you must run a search to discover into a thing the landscape's ruggedness \emph{predicts}, with the operator-choice design rule that follows (average / union-route / directed-select); \textbf{grounding as migration--drift balance}, giving a critical real-data fraction and a phase boundary a closed self-consuming loop cannot have; \textbf{directed sex} as the distinctly-AI advantage (unbounded parents, offspring preview, mate choice); and the \textbf{integrated society} whose four operators are shown \emph{jointly necessary}. The value-add over the machine-learning-native merge theory is that ours predicts \emph{which operator to use and when it will backfire}, not merely how fast quality decays. And it opens --- and begins to occupy --- a question nobody has framed: \textbf{model speciation}, the population-genetics of \emph{reproductive isolation} (Bateson--Dobzhansky--Muller incompatibilities) as the account of \emph{when two models are too diverged to be merged at all}. We model it explicitly (§5), predicting the compatible \(\rightarrow\) outbreeding-depression \(\rightarrow\) inviability curve, its super-linear (snowball) onset, and its control by epistasis rather than divergence alone --- the one place the merge literature has phenomena (Pari et al., 2024; Zhou et al., 2026) but no theory --- and we confirm it in real trained weights, where a merge barrier survives alignment under the \emph{full} function-preserving symmetry group (not only Re-Basin permutations) as a residual, functional reproductive isolation with an information-theoretic floor --- together with the pre-registered emergent converse: absent conflicting training signals, divergently-specialised lineages of shared ancestry showed \emph{no} isolation at any divergence tested, the merge instead rescuing the forgetting specialists (isolation must be provoked; specialisation alone did not speciate). In one sentence: the field agrees on the disease and tinkers at the cure with evolutionary metaphors; we bring the evolutionary \emph{theory}, and it makes falsifiable predictions --- a merge-compatibility cliff among them --- that the metaphors do not.
 
 \textbf{What is still open --- honestly.} The old hole (what to select) we fill in kind: don't design the selector, evolve it. But the hole has \emph{moved}, not closed, and the new one is harder: \textbf{the fitness function} --- what reality-anchored measure selects for \emph{truth} without also selecting for \emph{persuasion}, given that in our own species the two have been at war for the whole history of ideas. Alongside it: the \textbf{institutions} that let contemporaries correct one another before error is inherited (§8), which we do not solve; and the \textbf{calibration} of everything the results left as knobs --- how many parents, how complementary, at what ratio of inherited-to-real data, and how healthy a lineage must be before its knowledge is safe to make irreversibly innate. These are, at least, \emph{measurable} --- which is the difference between an open problem and a hole. And the largest gap of all: the \emph{recombination} claims now hold in real language models, but the \emph{society} --- the grounded, diversity-preserving, continually reproducing loop --- does not yet. The real test is to build that whole system out of actual open-weight language models, and see whether all the signs survive contact with a system too big to write down. The operators, checked; the living society, next.
 

paper/arxiv/md2tex.py

@@ -121,10 +121,28 @@ def convert(text: str) -> str:
         while fig_queue:
             out.append(fig_queue.pop(0)); out.append("")
 
+    def emit_table(block: list[str], out: list[str]) -> None:
+        """Pipe table -> small-font tabular with wrapped paragraph columns (full text width)."""
+        rows = [[c.strip() for c in line.strip().strip("|").split("|")] for line in block]
+        header, body = rows[0], [r for r in rows[2:]]        # rows[1] is the |---| separator
+        n = len(header)
+        widths = " ".join([f"p{{{0.92 / n:.3f}\\textwidth}}"] * n)
+        out.append("\\medskip\\noindent\\begin{center}\\footnotesize")   # non-floating: stays in place
+        out.append(f"\\begin{{tabular}}{{{widths}}}")
+        out.append("\\hline")
+        out.append(" & ".join(inline(c) for c in header) + " \\\\ \\hline")
+        for r in body:
+            r = (r + [""] * n)[:n]
+            out.append(" & ".join(inline(c) for c in r) + " \\\\[3pt]")
+        out.append("\\hline\\end{tabular}\\end{center}\\medskip")
+        out.append("")
+
     out: list[str] = []
     for block in blocks:
         first = block[0].strip()
-        if first == "---" and len(block) == 1:
+        if first.startswith("|") and len(block) >= 2 and set(block[1].strip()) <= set("|-: "):
+            emit_table(block, out)
+        elif first == "---" and len(block) == 1:
             out.append("\\medskip\\hrule\\medskip"); out.append("")
         elif first.startswith("## "):
             out.append(f"\\section*{{{inline(first[3:])}}}"); out.append("")