MachineSex/paper/pnas/main.md

# The evolution of sex for artificial intelligence: a population-genetic framework for multigenerational model populations

**Giorgio F. Gilestro** — Department of Life Sciences, Imperial College London. giorgio@gilest.ro

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## Significance statement

Artificial intelligence is shifting from single, frozen models to populations of models that
specialise, are retrained on each other's output, and are combined ("merged") into new models. Trained
on their own output, model lineages degenerate — a process already recognised as the mathematics of
genetic drift. This paper imports the other half of population genetics: the biology of sexual
reproduction. It treats model merging as recombination, real data as immigration, and merge failure as
reproductive isolation, and tests each correspondence in simulations, small neural networks, and
language models. The framework recasts machine learning's oldest problem — continual learning without
forgetting — at the population scale, and yields design rules: when to average models, when to keep
them separate, how much real data suffices (a theory for the field's empirical replay fractions), and
a controlled small-model test in which pre-merge functional disagreement predicted merge damage,
motivating further comparison with weight-space measures.

## Abstract

AI development increasingly resembles a population process: models are specialised, retrained on model
output, and recombined by weight merging, with an openly evolutionary vocabulary but little use of
evolutionary theory. Here we treat multigenerational model populations as systems whose inheritance,
diversity, and compatibility must be managed, and transfer the quantitative apparatus of the evolution
of sex. Its entry point — training on model output is genetic drift, and model collapse is its
signature — we developed independently, and parallel work has now formalised the same diagnosis from
several directions, a convergence we read as evidence for the frame rather than as a shared discovery
to be divided. In a
minimal inheritance model that is exactly Wright–Fisher — and measurably Wright–Fisher-plus-bias in
trained networks — we derive and test the remedies: grounding as immigration, where a real-data
fraction far below one retained most equilibrium diversity in the tested settings, with a
per-capability observation floor that makes the rarest knowledge expensive under unstratified
sampling; recombination, where refitting to the mean of parents' output distributions cancels the
multi-parent gain to first order in the rare-item regime while union-preserving operators realise it;
the Fisher–Muller effect, with merged language-model specialists exceeding every parent in replicated
experiments; outbreeding depression on rugged task landscapes, converted into reliable gains by
directed, offspring-screened recombination; and population structure, where the optimal mating breadth
shrinks as skills entangle. Sex has a limit: we introduce model speciation — merge failure as
reproductive isolation — and show in trained networks that a merge barrier remaining after
permutation-and-rescaling alignment tracks functional conflict, that isolation did not emerge from
compatible specialisation, and, in a controlled predictive test, that pre-merge functional
disagreement predicted merge damage while the tested weight-geometry baselines showed no detectable
association. We state precisely what is exact, what is measured, and what remains hypothesis.

---

## Introduction

Machine learning has quietly become a population-scale phenomenon. Public repositories now host
millions of models — Hugging Face alone grew past three million by 2026 — and these are not
independent creations: the overwhelming majority are fine-tunes, distillations, or merges of a small
number of foundation models, forming sprawling family trees whose lineage structure, inherited traits,
and mutation dynamics are already being mapped with explicitly phylogenetic methods (31–33).
Reproduction in this population is no longer metaphorical. Weight-space **model merging** — the direct
combination of trained parents into a new model — is mainstream community practice with standard
tooling and thousands of hybrid checkpoints, including leaderboard-topping ones (1, 2, 37, 38), and
the engineering literature describes it in openly evolutionary vocabulary: "crossover," "mutation,"
"mate choice," populations of merging models that climb benchmarks (2–5).

The generations are coupled through data as well as through weights. Successive models increasingly
learn from model output rather than from fresh human experience: frontier alignment pipelines are now
predominantly synthetic — over 98% in documented cases (43, 44) — self-generated instruction data
seeds whole lineages of descendants (5), a large and growing share of the public web is
machine-generated or machine-translated text (35, 36), and the stock of human text is projected to be
exhausted by frontier training within this decade (34). Meanwhile persistent multi-agent systems and
emerging agent economies put many interacting models into sustained contact (39–42). A population
whose members inherit from one another, recombine, and retransmit under these conditions is an
evolving population in the technical sense, whatever one thinks of the metaphors. The organising claim
of this paper is that the vocabulary deserves its mathematics: **multigenerational model populations
are systems whose inheritance, diversity, and compatibility must be managed — not merely collections
of models to optimise — and the branch of biology that studies exactly this problem, the population
genetics of the evolution of sex, transfers as a quantitative framework.**

The frame's entry point is the diagnosis. Training each generation of a model on the previous
generation's output degrades it — *model collapse*: rare capabilities vanish first and the lineage
drifts toward its own most common behaviour (6). That this is the mathematics of **genetic drift** in
a finite population is a conclusion we reached independently in building the present framework, and
one that has been derived in parallel from several other directions (7–9), including a closed-form
first-extinction law placing collapse onset at the Wright–Fisher first-extinction time (8) — and that
was anticipated, before deep learning, in an analysis of sequential inference chains as generalised
genetic drift (63). We cite these works for priority of publication on the diagnosis and read the
convergence — independent arrivals at the same population-genetic account by different routes and in
different decades — as corroboration that the frame is the natural one. What none of that parallel work develops, and what this paper is about, is the
structure the diagnosis opens: the full arc from drift through its remedies (immigration,
recombination, selection, population structure) to its limit (reproductive isolation), carried as one
framework from closed forms to trained networks to language models.

Seen from machine learning's own history, the problem this frame addresses is the field's oldest —
**continual learning** — reappearing one level up. Within a single network, sequential learning
overwrites prior knowledge (catastrophic forgetting; 45, 46), and the discipline's remedies are, one
by one, the population operators of this paper in single-model form: **rehearsal and replay** of past
data is grounding's within-lineage counterpart, and the field's empirically settled replay fractions —
on the order of 1% for instruction tuning (53), 5% for weak and 25% for strong distribution shift in
continual pretraining (52) — sit exactly where the minimal model's operational grounding threshold
lies, a correspondence for which the framework supplies the missing theory (equilibrium diversity,
and a per-capability survival law). **Pseudo-rehearsal** — replaying the network's own generated
samples, proposed as a cure in 1995 (47) and revived as generative replay (48) — is precisely this
paper's ungrounded null: immigration from a drifting source, benign for one hop and compounding into
collapse over generations, with verifier-filtering (12, 62) as what converts it back into grounding.
**Parameter isolation** (65, and frozen-base adapters, which forget far less; 54) is the engineered
decorrelation our specialists use; **complementary-learning-systems consolidation** (49–51) is our
periodic adapter-into-base merge; the recent turn to **merging as a continual-learning mechanism**
(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
tail-allele extinction observed one model at a time. One distinction is kept explicit throughout:
catastrophic forgetting is largely deterministic interference from shifted training, whereas collapse
is stochastic sampling drift — the phenomena share their victims (the rare) and their remedies, not
their mechanism. To our knowledge, no prior work carries population-genetic formalism into continual
learning itself; that bridge — replay as immigration with a survival law, merging as recombination
with a compatibility criterion, consolidation as the slow store of a two-speed memory — is where this
framework may matter most.

We are explicit about what kind of contribution each claim is, distinguishing **interpretation** (an existing result understood in population-genetic terms),
**explanation** (the transferred mechanism accounts for observations existing accounts leave open),
and **prediction** (the framework forecasts an unmeasured outcome). The paper is strongest on the
first; makes concrete progress on the second — separating merge failures that are coordinate artefacts
from those that are functional; and reports a first, bounded step on the third — a controlled
predictive test in which pre-merge functional-disagreement measures, chosen by the framework,
predicted merge damage on a constructed task grid while the tested weight-geometry baselines showed
no detectable association.

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
recombination can rebuild it, which is why remedies must act before fixation-by-loss (a consequence-
level correspondence: the minimal model lacks the ratchet's recurrent deleterious-mutation mechanism,
so irreversible loss alone does not identify that specific mechanism). Injecting verified real data is
**immigration** from a non-drifting source (11–13). Model merging is **recombination**, and its
celebrated payoff — a merged model exceeding every parent — is the **Fisher–Muller effect** (14, 15).
Merging entangled skills courts **outbreeding depression**; screening many candidate merges is
engineered recombination with unusually flexible parent choice and pre-deployment screening (we use
the shorthand **directed sex**); restricting who merges with whom is **population structure**. And merging's hard limit — models too diverged in function to combine — is **reproductive
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: *when and why* failure is coordinate versus functional, and what moves the boundary.

We support the framework at three tiers of evidence, in ascending realism and descending exactness: a
**minimal analytic model** validated against closed forms to a fraction of a percent; **small trained
networks** (MLPs, recurrent networks, an MNIST image generator) where the operators are measured in
real weights; and **language models** (LoRA-specialised Qwen models, 0.5B locally and 7B on a compute
cluster) where the claims are tested as signs under seed replication. Negative results are reported with the same prominence as confirmations; they include the failure of
an internal pre-registered prediction, a null on emergent speciation that bounds the analogy, and the
sensitivity analyses on the predictive test.

## The minimal model, and where its exactness ends

Knowledge is modelled as a distribution `p_t` over `K` discrete items — capabilities, facts, modes of
behaviour — with a fixed true distribution `p*` whose rare tail carries the knowledge most at risk.
One generation is: *draw `n` samples from the parent's distribution, optionally mix in `m` verified
real samples ("grounding", `g = m/(n+m)`), and refit the child*. In this minimal inheritance model the
resampling step **is** the Wright–Fisher process — the same equations, which we exploit as an
engineering gate: our simulator reproduces the classical closed forms (heterozygosity decay
`E[H_t] = H_0(1 − 1/n)^t`; the exact immigration–drift equilibrium; the closed-form multi-teacher
union) to within 0.5%, and these are standing tests in the codebase, not one-off checks.

The boundary of the exactness matters, and we measured it rather than assumed it. Real training adds
approximation, optimisation noise, and inductive bias, and when trained networks are fit against the
exact drift null they deviate in *opposite, architecture-specific* directions: a smoothing recurrent
network resists collapse (keeping spurious variants alive), while a sharpening image generator
accelerates it. A one-parameter **learning kernel** (a smoothing knob and a sharpening knob on the
refit) reproduces both. Throughout, a real learner is therefore treated as Wright–Fisher *plus a signed, measurable
estimator bias* — and the drift signs (rare-first loss; the grounding response)
survived that bias in every architecture we tested, including a convolutional VAE retrained on its own
generated digits, where the dry lineage collapses to a single blurred digit class while 10% grounding
holds all thirty modes (Fig. 1).

**Table 1.** The dictionary. Each correspondence is stated with the level of support it currently has
(exact = closed form in the minimal model; empirical = measured in trained systems; hypothesis =
stated with a falsifier, untested or unconfirmed). The full claim-by-claim ledger with assumptions and
known limits is SI Appendix, Table S1.

| Population genetics | Model populations | Support |
|---|---|---|
| Genetic drift in a finite population | Training on finite samples of model output | Exact (minimal model); signs in trained nets; diagnosis conceded to prior work |
| Immigration from a fixed source | Grounding with verified real data | Exact equilibrium; signs in RNN/MLP/VAE/MNIST |
| Muller's ratchet (asexual decay) | Irreversible arm of model collapse | Correspondence, scoped: applies to unrecoverable loss |
| Recombination / sexual reproduction | Model merging | Empirical at 0.5B–7B |
| Fisher–Muller effect | Merged specialists exceed every parent | Analytic model; replicated in LLMs |
| Outbreeding depression under epistasis | Merging entangled skills harms offspring | Analytic model (NK landscapes); hypothesis at LLM scale |
| Mating systems / population structure | Who merges with whom (breadth of the parent pool) | Analytic model; hypothesis for real populations |
| Reproductive isolation (BDM incompatibilities) | Merge failure from functional conflict | Empirical (MLP + LLM tiers, conflict-associated); emergent form not observed |
| Selection on a fitness function | Verifier-anchored selection ("reality that can say no") | Analytic model (complementary with recombination and diversity in the tested society) |

## Results

### Grounding is immigration: cheap, with a floor

In the minimal model, grounding from a fixed real source is immigration into a drifting population,
and the equilibrium diversity has a closed form our simulator matches exactly. That equilibrium is
*smooth* in the grounding fraction — there is no phase transition in aggregate diversity — so the
practical number is an operational threshold, and we define it as such: under the tested population
size and Zipf source distribution, `g ≈ 0.05` retained most (≥95%) of equilibrium diversity
indefinitely, with the required fraction depending on sample size, source distribution, and the
chosen retention target (dependencies in SI). The engineering point survives the definition: verified
real data is cheap insurance at fractions far below one. But the same analysis yields a floor the field's
average-loss framing misses: under unstratified sampling from the source, a capability of rarity `p`
appears in a real-data batch of size `m` with probability `1 − e^{−m·p}`, so `m·p ≈ 1` marks roughly a
63% chance of one example per batch — a soft observation floor, with higher confidence priced
accordingly, and with distinct consequences for continuous retention, stationary occupancy, and
reintroduction after loss (immigration can restore an absent item; SI separates these). Protecting the
rarest knowledge under unstratified grounding is therefore priced per item at cost `∝ 1/p`; targeted
or stratified sampling changes that cost, and recombination can recover rare capabilities *that are
still retained across complementary parents* (next section). In trained networks the *sign* of the grounding response transfers everywhere we
looked, with two deviations, both traced to the estimator bias above: sharp thresholds soften,
and support-counting metrics decouple from truth (forward-KL is the operative collapse metric for a
smoothing learner). On real images (Fig. 1B), dry self-training collapses a convolutional VAE to one
mode while ~10% grounding holds all thirty (the trained model needs roughly twice the exact-operator
fraction — the measured price of the estimator bias).

*(FIG:fig1)*

### Recombination: a conservation law, its operators, and offspring that exceed every parent

Merging is where the evolution-of-sex apparatus pays for itself, beginning with a result about the
obvious operator, stated with its assumptions. **Proposition (blending inheritance, rare-item
regime).** Let K parents independently retain a rare item (mass `p` when retained), and let the child
draw `n` samples either from one parent chosen at random or from the *mean of the parents' output
distributions*. Expected item mass is identical under the two schemes; and in the rare-item regime
`n·p/K ≪ 1`, where per-item survival is first-order in sampled mass, expected *survival* is also
identical — the 1/K dilution of averaging cancels the K-parent union gain to first order, so in this
regime adding parents through the output-mean does not increase expected tail retention. Two
boundaries: outside that regime, survival is a convex function of mixed mass, so the variance
reduction from averaging can *reduce* extinction relative to a randomly chosen single parent — the
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 — **weight averaging** (a nonlinear network's weight-mean does not compute its parents'
output-mean) and **routing among intact specialists** (different storage and inference budgets from a
single child) — are its empirical cousins, and the measured bridge is a **headroom rule**, stated qualitatively: in language models,
union-preserving operators beat the weight-average where that average falls short of attainable
performance, and add nothing where it does not (easy-versus-hard contrasts at two scales; a
quantitative form of the relationship is untested). On easy tasks a capable base's average is already at ceiling and refinements
add nothing; on hard tasks the average dilutes a fragile specialist below even the best single parent
and routing wins by a wide margin (Fig. 6A–B).

The generative payoff is the **Fisher–Muller effect**: recombination assembles, in one offspring,
complementary variants that arose in different lineages, producing a genotype fitter than any parent.
In the multi-locus model, sexual merging of decorrelated specialists climbs to the global optimum — a
genotype no parent held — while the best single parent and the blended average both plateau below
(Fig. 2). In real language models the signature replicates under seed replication: merges of three
LoRA specialists beat every parent overall (decisively at 7B: 0.87 vs 0.77), and on the sharper
worst-family metric the merged models are the only ones competent everywhere, in every seed (Fig. 6A).

Sex has risks and, for AI, an unfair advantage — both quantified on rugged (epistatic) NK landscapes
(Fig. 3). When skills are entangled, blind recombination produces offspring *below* their parents —
**outbreeding depression** — worsening with ruggedness, and the optimal recombination rate shrinks as
entanglement grows. But an engineered population can do what biology cannot: recombine unbounded
parents, choose complementary mates, and *screen many candidate offspring against a verifier before
keeping one*. This **directed sex** converts the outbreeding catastrophe into a reliable gain in the
model (tracking or exceeding the best parent at every ruggedness) and replicates as a sign in language
models: bred-and-screened merges beat the a-priori blend in every seed on headroom tasks — including
one seed where the blend failed catastrophically and selection was immune (Fig. 6A). Finally,
population *structure* is itself a knob: sweeping the mate-pool breadth from monogamous (local) to
promiscuous (panmictic) against ruggedness, wide mixing maximises the population mean while
monotonically destroying diversity, and the best *champion* shifts from wide breadth on smooth
landscapes to intermediate breadth on rugged ones (Fig. 3C) — the mating-system phenomenon known to
structured-population search, mapped onto merging populations.

*(FIG:fig2)*

*(FIG:fig3)*

### 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 **grounded inheritance**: external samples added to the reproduction process (the
data channel). In the society model, grounding is **grounded evaluation**: selection weights true
fitness against conformity to the population's own consensus — `g`·true-fitness + (1−g)·conformity —
the analogue of scoring models by the crowd's approval (the fitness channel). These are related design
ideas — both couple the lineage to a non-drifting external signal — but they are different operators,
and we name them separately. In the tested society (a finite agent population on a rugged NK
landscape), a four-arm ablation separates the failure modes: the full system (grounded evaluation +
directed recombination + diversity-preserving selection) climbs to near the global optimum while
keeping its specialists; removing grounded evaluation converges the population confidently on an unfit
consensus (self-consumption); removing recombination strands it on local optima; removing diversity
converges it prematurely to a worse answer. Each removal fails differently — the three implementations
make complementary contributions *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.

*(FIG:fig4)*

### The limit of sex: model speciation

Recombination presupposes compatible parents. In biology, lineages pushed far enough apart become
separate species — **reproductive isolation** — through Bateson–Dobzhansky–Muller incompatibilities:
changes harmless on their own background but deleterious in combination. A merged model is exactly the
exposed hybrid. We built the analytic model (Fig. 5A): hybrid fitness tracks the parents while
compatible, then peels off and crashes below the ancestor; the isolation cliff arrives earlier the
denser the incompatibilities; and the incompatibility *count* snowballs quadratically with divergence
(17) — noting that a super-linear count does not by itself entail a sharp performance cliff without
the count-to-effect-size link, which the analytic model supplies under its assumptions and any neural
test must establish separately.

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
permutation matching and exact per-unit rescaling (the unit symmetry group of plain ReLU MLPs, as the
search space) and decomposed the barrier (Fig. 5B): two networks trained from different
initialisations on the *same* task have a barrier that this alignment removes essentially entirely
(residual ≈ 0.001, the aligned merge performing at parent level) — coordinate, not functional; two
networks trained on *conflicting* label maps have a barrier the same alignment leaves largely
unchanged (0.502 → 0.497), with the merged model functionally dead. The tested alignment removes the
same-task barrier but leaves the conflict-associated barrier intact — supporting a functional-conflict
interpretation without proving optimal alignment: exact recovery of a permuted-and-rescaled copy
validates a special case, so the removable share is a lower bound and the residual an upper bound.
Sweeping conflict traces the cliff as hybrid fitness, 0.97 → 0.03. The conflict floor itself is
information-theoretic — no single model can satisfy contradictory conventions (SI Appendix,
Proposition S2) — with the framework's role being the *structure around it*: which divergences
generate conflict, and what moves the cliff.

The strongest constraint comes from the pre-registered **emergent test**: true BDM incompatibilities are
emergent (each lineage's changes harmless alone), so we let children diverge with *no conflicting
signal anywhere* — complementary class specialists, and divergent input conventions — to 6.4× the base
training. **No isolation emerged** (residual 0.000 throughout); instead the merge *rescued* the two
catastrophically-forgetting specialists (parents ≈ 0.50, merge ≈ 0.955 — a sustained Fisher–Muller
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
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.

*(FIG:fig5)*

### 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 × 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 *across* conditions we
report per-seed and leave-one-seed-out sensitivity alongside) span three axes decorrelated by construction: *conflict*
(contradictory conventions on shared prompts, private budgets fixed), *compatible overlap* (the same
shared prompts under the same convention — overlap and volume without conflict), and *duration* (weight
divergence with zero conflict). Before merging, six predictors are computed: **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 supported conclusion, stated conditionally: **across this controlled grid, pre-merge functional
disagreement predicted merge penalties (clustered bootstrap CIs excluding zero; held-out
leave-one-condition-out ρ ≈ 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 ρ stable at +0.37 to +0.53 in each seed alone,
geometry ≈ 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 (ρ = +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 **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.

*(FIG:fig6)*

**Table 2.** Headline quantitative results with sample sizes, uncertainty, and outcome definitions
(full per-experiment tables and falsifier status in SI Appendix and per-experiment documentation).

| Result | Setting / n | Outcome definition | Headline |
|---|---|---|---|
| Closed-form validation | Analytic tier; standing tests | Simulated vs closed-form H-decay, immigration equilibrium, multi-teacher union | Agreement < 0.5% |
| Grounding retention | Minimal model; 18+ replicates per point | Fraction of equilibrium diversity retained at grounding g (operational threshold) | g ≈ 0.05 retained ≥95% (tested setting); smooth in g |
| MNIST collapse & rescue | Conv-VAE, 4 replicates; frozen oracle (98.5% mode acc.) | Mode support / forward-KL over generations | Dry: 30→1 modes; 10% grounding: 30/30 held |
| Fisher–Muller in LLMs | 5 seeds (0.5B), fixed tests; single 7B run | Merged vs best-specialist accuracy (overall; worst family) | Ties 0.647±0.027 vs 0.592±0.009; 7B 0.87 vs 0.77 |
| Union vs blend (headroom) | 3 seeds (0.5B hard); single 7B-hard run | Paired per-seed ordering, routing vs weight-average | Routing > blend in 3/3 seeds; one catastrophic blend failure avoided |
| Speciation decomposition | MLPs, 3 replicates | LMC error barrier residual after permutation+rescaling alignment | Same-task 0.001; conflict 0.497 (naive 0.502) |
| Emergent isolation | MLPs 4 reps to 6.4× base training; LLM 1→12 epochs | Residual barrier; merged vs parent accuracy | 0.000 everywhere; merge rescues parents (≈0.955 vs ≈0.50) |
| Predictive test | 13 conditions × 3 seeds (0.5B) | Merge penalty vs oracle parent potential (pre-registered; ±: clustered 95% CI) | Functional ρ +0.45/+0.46, CI excl. 0; LOCO ρ ≈ 0.4; geometry n.s.; paired differences n.s. |

## Discussion

**Design rules.** Read as engineering, the results compress into rules an operator of a model
population can apply. *Ground every generation* in verified reality — a few percent retained most diversity in our tested
settings — but price the rarest capabilities individually (observation probability `1 − e^{−m·p}` per
batch under unstratified sampling), consider targeted sampling for the deep tail, and use
recombination to recover rare capabilities still retained across complementary parents. *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. *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. *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. *Before merging,
measure functional conflict* — cheap, pre-merge, and in our controlled setting predictive where the
tested weight-distance baselines were not; and *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.

**What this offers continual learning.** Read into the field where these results most directly land:
(i) a first-principles account of the **replay ratio** — the folklore constants (≈1%, 5%, 25%; 52, 53)
acquire an equilibrium theory and a sharper prediction, that the required fraction is set by the
rarest capability one refuses to lose (the `1 − e^{−m·p}` law), not by average loss — directly
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
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 **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)
**tail monitoring as the leading indicator**: continual-learning evaluation that averages over
capabilities hides exactly the losses that drift theory says come first and, past a threshold, become
irreversible. On that last point we note the standing objection that apparent forgetting can be
skewed task-inference over latent capability rather than erasure (64); our irreversibility results
concern oracle-measured behavioural distributions, and distinguishing latent from extinct capability
at language-model scale is an open and, we think, decisive experiment for both readings.

**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.

**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
*unfamiliar* conflict structures and real task pairs; a demonstrably better *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.

## Materials and Methods

**Analytic tier.** Pure NumPy/SciPy Wright–Fisher simulator over `K`-item distributions (knowledge as
`p_t`; Zipf-tailed truth `p*`; drift–grounding–refit generations), extended with a learning kernel
(smoothing/sharpening refit), multi-locus genotypes on additive and Kauffman NK landscapes, n-parent
crossover, and finite-population society loops. All parameters live in per-experiment YAML configs;
every run derives all randomness from one master seed (`SeedSequence.spawn`) and is bitwise
reproducible; scientific-validation tests assert the closed forms (heterozygosity decay, immigration
equilibrium, closed-form union) to <0.5% and run in CI with 151 further correctness tests.

**Neural tier.** Trained-network experiments realise the same abstractions with an exact oracle:
histogram/RNN/MLP/VAE generators on a synthetic mode universe (the histogram model reduces the harness
exactly to the analytic tier — the bridge gate), and a convolutional VAE on MNIST with a frozen CNN
oracle (98.5% mode accuracy; confusion matrix recorded as the measurement floor). Speciation
experiments fork no-BatchNorm MLPs (784–512–512–10) from a shared base, weight-average, and measure
linear-mode-connectivity error barriers before and after alignment; alignment composes deterministic
Git Re-Basin permutation matching with exact per-unit scale canonicalisation (the unit symmetry
group of this class, as the alignment search space; control recovery does not establish global
optimality), gated by exact recovery of a permuted-and-rescaled copy.

**Language-model tier.** LoRA specialists (rank 16) on procedurally generated task families with an
exact-match verifier, on frozen Qwen2.5-Instruct bases (0.5B on one 16 GB GPU; 7B on one L40S).
Operators: weight-space merges (soup/TIES via adapter arithmetic), per-input routing, and
Dirichlet-sampled offspring populations screened on held-out validation splits. Multi-seed protocols
fix the test sets and vary the training seed. The predictive test computes all predictors pre-merge
(generation confidence from token log-probabilities; base-model gradient cosines; exact r-space
LoRA-delta geometry) and evaluates merges on held-out tests; robust statistics (condition-clustered
bootstrap, paired predictor contrasts, leave-one-condition-out prediction, multi-reference outcomes)
are produced by a committed script. Statistical, per-seed reproducibility is documented for GPU tiers.

**Data and code availability.** All code, configs, seeds, results artifacts (with content hashes),
figures, and a one-command reproduction script will be deposited openly (repository + archived DOI) on
publication; every figure in this paper regenerates from committed artifacts without re-simulation.

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