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 yields design rules — when to average models, when to keep them
separate, how much real data suffices — and a first controlled test showing that measured functional
conflict, not weight distance, predicts when merging fails.

## 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. We take as settled that training on model output is genetic drift (model collapse). 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, with a critical
real-data fraction far below one but a per-capability floor that leaves the rarest knowledge
unrescuable; recombination, where averaging parents' output distributions exactly cancels the benefit
of multiple parents 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 surviving the full function-preserving symmetry group tracks functional
conflict, that isolation did not emerge from compatible specialisation, and, in a controlled
predictive test, that pre-merge functional disagreement predicts merge damage where weight-geometry
baselines do not. We state precisely what is exact, what is measured, and what remains hypothesis.

---

## Introduction

The unit of AI progress is quietly changing. Multi-agent systems arrange many models across *space* —
specialists cooperating on a task. A newer axis is *time*: populations of models that persist across
generations, each new model built from older ones — specialised by fine-tuning, trained on data earlier
models generated, and, increasingly, produced by **model merging**, the direct combination of trained
weights (1, 2). The engineering literature describes this openly in evolutionary vocabulary —
"crossover," "mutation," "mate choice," populations of merging models that climb benchmarks (2–5) —
but as metaphor over search algorithms. 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.**

One half of the transfer is settled and is not our contribution. 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 now established from several directions (7–9); a closed-form
first-extinction law even places collapse onset at the Wright–Fisher first-extinction time (8). We cite
this literature as the diagnosis and build on it.

Our contribution is on the remedy side, and 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 weight-geometry baselines did not.

The correspondences we develop, summarised in Table 1: single-teacher retraining is **asexual
reproduction**, and the irreversible arm of its decay corresponds to **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 precisely why remedies must act before fixation-by-loss. 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 a form
of **directed sex** with no biological analogue; 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. Throughout, we report negative
and tempering results 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 that temper 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. The honest statement, used throughout: a real learner is 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 | Exact-model result; replicated in LLMs |
| Outbreeding depression under epistasis | Merging entangled skills harms offspring | Exact-model (NK landscapes); hypothesis at LLM scale |
| Mating systems / population structure | Who merges with whom (breadth of the parent pool) | Exact-model result; 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") | Exact-model result (jointly necessary with sex and diversity) |

## 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. The engineering
headline is the *magnitude*: a critical grounding fraction `g* ≈ 0.05` retains most diversity
indefinitely — real data is cheap insurance. But the same analysis yields a floor the field's
average-loss framing misses: an individual capability of rarity `p` survives only if the *absolute*
real-data budget satisfies `m·p ≳ 1`. Protecting the rarest knowledge is priced per item, at cost
`∝ 1/p`, and no affordable grounding fraction rescues the deepest tail — that requires recombination
(next section). In trained networks the *sign* of the grounding response transfers everywhere we
looked, with two honest 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. **Averaging is blending inheritance, and it cancels the benefit of multiple
parents:** when a child is refit to the *mean of its 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 K parents, so adding parents cannot help. An operator that
keeps, per item, its strongest source (which presupposes a verifier or oracle to say which) realises
the union. That statement is exact for those operators 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**: in language
models, union-preserving operators beat the weight-average in proportion to how far that average is
from the best attainable. 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, sex, and diversity are jointly necessary

Composing the operators closes the loop (Fig. 4). A finite population of agents evolves on a rugged NK
landscape, with selection acting on a grounded score — `g`·true-fitness + (1−g)·conformity to the
population's own consensus, the analogue of training on the crowd's output. A four-arm ablation
separates the failure modes: the **full** system (grounding + directed recombination +
diversity-preserving selection) climbs to near the global optimum while keeping its specialists;
remove *grounding* and the population converges confidently on an unfit consensus (self-consumption);
remove *sex* and it strands on local optima; remove *diversity* and it converges prematurely to a
worse answer. Each removal fails *differently* — the operators are jointly necessary, which is the
system-level claim the single-operator results build toward. 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 modulo the **complete**
function-preserving unit symmetry group of the architecture tested (permutation composed with per-unit
positive rescaling, for plain ReLU MLPs) and decomposed the barrier (Fig. 5B): two networks trained
from different initialisations on the *same* task have a barrier that 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 full group leaves
intact (0.502 → 0.497), with the merged model functionally dead — and this cannot be an alignment
failure, because the same aligner succeeded on the control. Sweeping conflict traces the cliff as
hybrid fitness, 0.97 → 0.03. Two scope notes: exact recovery of a permuted-and-rescaled copy validates
a special case rather than global optimality, so the removable share is a lower bound and the residual
an upper bound; and 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 sharpest honesty 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 all inference is condition-clustered) 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, and only these baselines were
tested. Two further results earn their place by tempering: 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)*

## 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 retains most
diversity — but price the rarest capabilities individually (`m·p ≳ 1`) and use recombination, not
grounding, to reach the deep tail. *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 the society result shows grounding, recombination, and diversity are jointly
necessary. *Before merging, measure functional conflict* — cheap, pre-merge, and in our controlled
setting predictive where weight distance was not; and expect specialisation alone to be merge-safe,
with conflicting conventions on shared circuitry as the thing to detect and avoid.

**What is borrowed and what is ours.** The diagnosis — collapse as drift — is prior art (6–9), as 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 jointly-necessary society; model speciation as a named, tested question, with the
coordinate-versus-functional decomposition under a complete symmetry group 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 complete unit
symmetry group for this class), 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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