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references: citation-order renumbering, PNAS style, full verification

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All 66 references renumbered to first-appearance order (programmatically
verified: in-text sequence = 1..66 = list order; ranges expanded,
remapped, recompressed) and rewritten in PNAS style (initials-first
authors with the >5 -> et-al rule, sentence-case titles, abbreviated
italic venues, bold volumes, year-at-end, arXiv [Preprint] + 10.48550
DOIs). Correctness: 47 arXiv ids batch-verified against the arXiv API
(title/first-author/year); caught and fixed an authorless GENOME entry
(Y. Zhang et al.), "Sakana AI" -> J. Abrantes et al., a wrong Kotha id
(2310.05719, a different paper -> 2309.10105), Nemotron's corporate
author, and Liang's truncated title. Also: six load-bearing refs that
lost their in-text anchors during the restructure re-anchored (NK, QD,
Pari, LoRA, Sharma, Kozodoi), one real mis-citation fixed
(Self-Instruct credited to Multiagent-Finetuning; new ref added), and
four figure captions in build.py brought up to third-review calibration
(operational grounding threshold; first-order conservation;
complementary-contributions society; permutation-and-rescaling
alignment). 20-pp rebuild clean.

Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01BkRLcc18rwT2Lysu6PbG7v
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@ -46,20 +46,20 @@ Machine learning has become a population-scale phenomenon. Public repositories h
millions of models (Hugging Face alone grew past three million by 2026), and these are not
independent creations: the overwhelming majority are fine-tunes, distillations, or merges of a small
number of foundation models, forming large family trees whose lineage structure, inherited traits,
and mutation dynamics are already being mapped with explicitly phylogenetic methods (31–33).
and mutation dynamics are already being mapped with explicitly phylogenetic methods (1–3).
This population also reproduces. Weight-space **model merging**, the direct combination of trained
parents into a new model, is mainstream community practice with standard tooling and thousands of
hybrid checkpoints, including leaderboard-topping ones (1, 2, 37, 38), and the engineering literature
hybrid checkpoints, including leaderboard-topping ones (4–7), and the engineering literature
describes it in evolutionary vocabulary: "crossover," "mutation," "mate choice," populations of
merging models that climb benchmarks (2–5).
merging models that climb benchmarks (5, 8–10).
The generations are coupled through data as well as through weights. Successive models increasingly
learn from model output rather than from fresh human experience: frontier alignment pipelines are now
predominantly synthetic (over 98% in documented cases; 43, 44), self-generated instruction data
seeds whole lineages of descendants (5), a large and growing share of the public web is
machine-generated or machine-translated text (35, 36), and the stock of human text is projected to be
exhausted by frontier training within this decade (34). Meanwhile persistent multi-agent systems and
emerging agent economies put many interacting models into sustained contact (39–42). A population
predominantly synthetic (over 98% in documented cases; 11, 12), self-generated instruction data
seeds whole lineages of descendants (13), a large and growing share of the public web is
machine-generated or machine-translated text (14, 15), and the stock of human text is projected to be
exhausted by frontier training within this decade (16). Meanwhile persistent multi-agent systems and
emerging agent economies put many interacting models into sustained contact (17–20). A population
whose members inherit from one another, recombine, and retransmit under these conditions is an
evolving population in the technical sense. The claim of this paper is that the vocabulary should be
given its mathematics: **multigenerational model populations are systems whose inheritance,
@ -69,12 +69,12 @@ transfers as a quantitative framework.**
The frame's entry point is the diagnosis. Training each generation of a model on the previous
generation's output degrades it (*model collapse*): rare capabilities vanish first and the lineage
drifts toward its own most common behaviour (6). That this is the mathematics of **genetic drift** in
drifts toward its own most common behaviour (21). That this is the mathematics of **genetic drift** in
a finite population is a conclusion we reached independently in building the present framework, and
one that has been derived in parallel from several other directions (7–9), including a closed-form
first-extinction law placing collapse onset at the Wright–Fisher first-extinction time (8), and that
one that has been derived in parallel from several other directions (22–24), including a closed-form
first-extinction law placing collapse onset at the Wright–Fisher first-extinction time (23), and that
was anticipated, before deep learning, in an analysis of sequential inference chains as generalised
genetic drift (63). We cite these works for priority of publication on the diagnosis and read the
genetic drift (25). We cite these works for priority of publication on the diagnosis and read the
convergence, independent arrivals at the same population-genetic account by different routes and in
different decades, as corroboration that the frame is the natural one. What none of that parallel work develops, and what this paper is about, is the
structure the diagnosis opens: the full arc from drift through its remedies (immigration,
@ -83,21 +83,21 @@ framework from closed forms to trained networks to language models.
In machine learning's own terms, the problem this frame addresses is the field's oldest,
**continual learning**, reappearing one level up. Within a single network, sequential learning
overwrites prior knowledge (catastrophic forgetting; 45, 46), and the discipline's remedies are, one
overwrites prior knowledge (catastrophic forgetting; 26, 27), and the discipline's remedies are, one
by one, the population operators of this paper in single-model form: **rehearsal and replay** of past
data is grounding's within-lineage counterpart, and the field's empirically settled replay fractions,
on the order of 1% for instruction tuning (53) and 5% to 25% by distribution-shift strength in
continual pretraining (52), sit where the minimal model's operational grounding threshold lies, a
on the order of 1% for instruction tuning (28) and 5% to 25% by distribution-shift strength in
continual pretraining (29), sit where the minimal model's operational grounding threshold lies, a
correspondence for which the framework supplies the missing theory (equilibrium diversity, and a
per-capability survival law). **Pseudo-rehearsal**, the replay of the network's own generated
samples, proposed as a cure in 1995 (47) and revived as generative replay (48), is this paper's
samples, proposed as a cure in 1995 (30) and revived as generative replay (31), is this paper's
ungrounded null: immigration from a drifting source, benign for one hop and compounding into
collapse over generations; verifier-filtering (12, 62) converts it back into grounding.
**Parameter isolation** (65, and frozen-base adapters, which forget far less; 54) is the engineered
decorrelation our specialists use; **complementary-learning-systems consolidation** (49–51) is our
collapse over generations; verifier-filtering (32, 33) converts it back into grounding.
**Parameter isolation** (34, and frozen-base adapters, which forget far less; 35) is the engineered
decorrelation our specialists use; **complementary-learning-systems consolidation** (36–38) is our
periodic adapter-into-base merge; the recent turn to **merging as a continual-learning mechanism**
(55–58) applies recombination within one lineage over time, where we apply it across lineages; and
the observation that **rare examples and long-tail knowledge are forgotten first** (59–61) is
(39–42) applies recombination within one lineage over time, where we apply it across lineages; and
the observation that **rare examples and long-tail knowledge are forgotten first** (43–45) is
tail-allele extinction observed one model at a time. One distinction is kept explicit throughout:
catastrophic forgetting is largely deterministic interference from shifted training, whereas collapse
is stochastic sampling drift; the two phenomena share their victims, the rare, and their remedies, but not their mechanism. To our knowledge, no prior work carries population-genetic formalism into continual
@ -116,19 +116,19 @@ no detectable association.
The correspondences we develop, summarised in Table 1: single-teacher retraining is **asexual
reproduction**, and the irreversible arm of its decay shares the defining consequence of **Muller's
ratchet** (10): once every copy of a rare capability is gone from all parents and sources, no
ratchet** (46): once every copy of a rare capability is gone from all parents and sources, no
recombination can rebuild it, which is why remedies must act before fixation-by-loss (a consequence-
level correspondence: the minimal model lacks the ratchet's recurrent deleterious-mutation mechanism,
so irreversible loss alone does not identify that specific mechanism). Injecting verified real data is
**immigration** from a non-drifting source (11–13). Model merging is **recombination**, and its
central payoff, a merged model exceeding every parent, is the **Fisher–Muller effect** (14, 15).
**immigration** from a non-drifting source (32, 47, 48). Model merging is **recombination**, and its
central payoff, a merged model exceeding every parent, is the **Fisher–Muller effect** (49, 50).
Merging entangled skills courts **outbreeding depression**; screening many candidate merges is
engineered recombination with unusually flexible parent choice and pre-deployment screening (we use
the shorthand **directed sex**); restricting who merges with whom is **population structure**. Merging's hard limit, models too diverged in function to combine, is **reproductive
isolation**, for which the Bateson–Dobzhansky–Muller theory of incompatibilities (16, 17) supplies the
isolation**, for which the Bateson–Dobzhansky–Muller theory of incompatibilities (51, 52) supplies the
structure. The nearest precursor to this programme reads sex as an algorithm for mixability in the
theory of computation (18), pre-dating model merging; the model-merging literature itself has strong
empirical operators (1, 19, 20) and emerging merge-success predictors (21, 22), to which our delta is
theory of computation (53), pre-dating model merging; the model-merging literature itself has strong
empirical operators (4, 54, 55) and emerging merge-success predictors (56, 57), to which our delta is
mechanism: *when and why* failure is coordinate versus functional, and what moves the boundary.
We support the framework at three tiers of evidence, in ascending realism and descending exactness: a
@ -223,7 +223,7 @@ contrasting union operator (keep each item's strongest source, then renormalise,
redistributes mass and presupposes a verifier or oracle to identify the strongest source) increases
expected retention with K in all regimes in the minimal model. The practically important
operators, **weight averaging** (a nonlinear network's weight-mean does not compute its parents'
output-mean) and **routing among intact specialists** (different storage and inference budgets from a
output-mean) and **routing among intact specialists** (58) (different storage and inference budgets from a
single child), are its empirical cousins, and the measured bridge is a **headroom rule**, stated qualitatively: in language models,
union-preserving operators beat the weight-average where that average falls short of attainable
performance, and add nothing where it does not (easy-versus-hard contrasts at two scales; a
@ -236,10 +236,10 @@ complementary variants that arose in different lineages, producing a genotype fi
In the multi-locus model, sexual merging of decorrelated specialists climbs to the global optimum, a
genotype no parent held, while the best single parent and the blended average both plateau below
(Fig. 2). In real language models the signature replicates under seed replication: merges of three
LoRA specialists beat every parent overall (decisively at 7B: 0.87 vs 0.77), and on the sharper
LoRA (59) specialists beat every parent overall (decisively at 7B: 0.87 vs 0.77), and on the sharper
worst-family metric the merged models are the only ones competent everywhere, in every seed (Fig. 6A).
Sex has risks and, for AI, an unfair advantage, both quantified on rugged (epistatic) NK landscapes
Sex has risks and, for AI, an unfair advantage, both quantified on rugged (epistatic) NK landscapes (60)
(Fig. 3). When skills are entangled, blind recombination produces offspring *below* their parents
(**outbreeding depression**), worsening with ruggedness, and the optimal recombination rate shrinks as
entanglement grows. But an engineered population can do what biology cannot: recombine unbounded
@ -268,7 +268,7 @@ the analogue of scoring models by the crowd's approval (the fitness channel). Th
ideas, since both couple the lineage to a non-drifting external signal, but they are different operators,
and we name them separately. In the tested society (a finite agent population on a rugged NK
landscape), a four-arm ablation separates the failure modes: the full system (grounded evaluation +
directed recombination + diversity-preserving selection) climbs to near the global optimum while
directed recombination + diversity-preserving selection (61)) climbs to near the global optimum while
keeping its specialists; removing grounded evaluation converges the population confidently on an unfit
consensus (self-consumption); removing recombination strands it on local optima; removing diversity
converges it prematurely to a worse answer. Each removal fails differently; the three implementations
@ -286,13 +286,13 @@ changes harmless on their own background but deleterious in combination. A merge
exposed hybrid. We built the analytic model (Fig. 5A): hybrid fitness tracks the parents while
compatible, then peels off and crashes below the ancestor; the isolation cliff arrives earlier the
denser the incompatibilities; and the incompatibility *count* snowballs quadratically with divergence
(17). We note that a super-linear count does not by itself entail a sharp performance cliff without
(52). We note that a super-linear count does not by itself entail a sharp performance cliff without
the count-to-effect-size link, which the analytic model supplies under its assumptions and any neural
test must establish separately.
In trained networks, the claim must survive a known alternative: merge barriers between independently
trained networks are famously *coordinate artefacts*, removable by re-aligning hidden units (23), and
richer symmetry groups remove more (24). We therefore aligned under the composition of
trained networks are famously *coordinate artefacts*, removable by re-aligning hidden units (62);
richer symmetry groups remove more (63), with known failures beyond the shared-data regime (64). We therefore aligned under the composition of
permutation matching and exact per-unit rescaling (the unit symmetry group of plain ReLU MLPs, as the
search space) and decomposed the barrier (Fig. 5B): two networks trained from different
initialisations on the *same* task have a barrier that this alignment removes essentially entirely
@ -315,7 +315,8 @@ catastrophically-forgetting specialists (parents ≈ 0.50, merge ≈ 0.955, a su
rescue). The same double result appears at the language-model tier (Fig. 5C): conflicting conventions
produce **function-specific** hybrid breakdown (the merge scores below both parents on the conflicted
function, while a budget-controlled design shows the disjoint skills merge unharmed), and over-training
disjoint specialists 1→12 epochs produces no isolation at all — the merge improves. Across every tier
disjoint specialists 1→12 epochs (cf. the merging literature's expert-duration effect; 65) produces
no isolation at all — the merge improves. Across every tier
tested, **isolation had to be provoked by functional conflict; specialisation alone did not speciate**
— a bound on the analogy that sharpens the design rule: what breaks merging is conflicting conventions
on shared circuitry, not divergence per se.
@ -334,7 +335,7 @@ divergence with zero conflict). Before merging, six predictors are computed: **c
functional conflict** (bilateral confident disagreement on probes drawn blind to where conflict lives —
a proposed proxy for merge-relevant interactions, motivated by the observation that raw disagreement
counts harmless complementation, one parent merely ignorant, as conflict), raw disagreement, gradient
alignment at the shared base (21), LoRA-delta cosine and distance, and a cross-task performance
alignment at the shared base (56), LoRA-delta cosine and distance, and a cross-task performance
baseline. The pre-registered outcome is the merge penalty against oracle parent potential (the
hybrid-load analogue), also reported against best- and mean-parent references because the predictor
ordering is sensitive to that choice.
@ -392,14 +393,14 @@ evidence of incompatibility* — in every regime we tested, what broke merging w
conventions on shared circuitry, which is the thing to detect.
**What this offers continual learning.** Read into the field where these results most directly land:
(i) a first-principles account of the **replay ratio**: the field's constants (≈1%, 5%, 25%; 52, 53)
(i) a first-principles account of the **replay ratio**: the field's constants (≈1%, 5%, 25%; 28, 29)
acquire an equilibrium theory and a sharper prediction, that the required fraction is set by the
rarest capability one refuses to lose (the `1 − e^{−m·p}` law) rather than by average loss, which is
testable against published replay sweeps; (ii) a **failure theory for generative replay**:
self-generated rehearsal is safe for short horizons and compounds into collapse across generations
unless verifier-filtered back into grounding (47, 48, 12, 62); (iii) **pre-merge interference
unless verifier-filtered back into grounding (30–33); (iii) **pre-merge interference
prediction with a mechanism**: where the current state of the art fits regressions over candidate
metrics (21), the functional-conflict measure arrives at a convergent signal from principle and comes
metrics (56), the functional-conflict measure arrives at a convergent signal from principle and comes
with an operator prescription — when conflict is high, do not average; route or breed-and-screen;
(iv) a candidate **decision rule for the consolidate-versus-stay-modular question** that currently
splits the field's practice (keep adapters separate vs merge them; 54–58): union-preserving operators
@ -407,15 +408,15 @@ where headroom exists, fusion where the base composes, consolidation as the slow
**tail monitoring as the leading indicator**: continual-learning evaluation that averages over
capabilities hides exactly the losses that drift theory says come first and, past a threshold, become
irreversible. On that last point we note the standing objection that apparent forgetting can be
skewed task-inference over latent capability rather than erasure (64); our irreversibility results
skewed task-inference over latent capability rather than erasure (66); our irreversibility results
concern oracle-measured behavioural distributions, and distinguishing latent from extinct capability
at language-model scale is an open experiment whose outcome would be decisive for both readings.
**What is borrowed and what is ours.** The diagnosis — collapse as drift — was published first by
others and we cite it so (6–9), while noting the derivations are independent and convergent; prior art
others and we cite it so (21–24), while noting the derivations are independent and convergent; prior art
in the strict sense are the empirical facts that merges can beat parents, that decorrelated parents merge better, and that
naive averaging loses to interference-aware or routed merges (1, 19, 20), that model populations can
climb (2–5), and that merge success admits ML-native predictors (21, 22). Ours is the framework-level
naive averaging loses to interference-aware or routed merges (4, 54, 55), that model populations can
climb (5, 8–10), and that merge success admits ML-native predictors (56, 57). Ours is the framework-level
synthesis — inheritance, diversity, and compatibility as managed quantities — together with: the
conservation law for blending inheritance and its operator boundaries; the per-item grounding floor;
the society ablation with its complementary failure modes; model speciation as a named, tested question, with the
@ -474,68 +475,69 @@ publication; every figure in this paper regenerates from committed artifacts wit
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1. B. Laufer, H. Oderinwale, J. Kleinberg, Anatomy of a machine learning ecosystem: 2 million models on Hugging Face. arXiv [Preprint] (2025). https://doi.org/10.48550/arXiv.2508.06811.
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