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Accessibility pass: introduce every pop-gen term at first use, with real-world anchors

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The CS reader is now walked into the biology as it arrives: drift (surname
extinction, island alleles), Wright-Fisher and heterozygosity defined in
place (collision-probability reading), Muller's ratchet (Y-chromosome decay),
immigration (the one-migrant-per-generation rule of conservation management),
blending inheritance (Jenkin's 1867 swamping argument, which the Proposition
makes exact), Fisher-Muller (clonal competition vs assembly), genotype/locus,
epistasis and NK landscapes glossed, outbreeding depression (the Tatra ibex
restocking collapse), panmixia, BDM incompatibilities (mule sterility),
hybrid load. Three new literature anchors (Mills & Allendorf 1996; Jenkin
1867; Templeton 1986), all verified; references renumbered to
first-appearance order (now 72) and re-verified 1..72.

Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01BkRLcc18rwT2Lysu6PbG7v
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@ -69,7 +69,9 @@ as systems whose inheritance, diversity, and compatibility can be measured, pred
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 (21). That degradation is, mathematically, *genetic drift*, the loss of rare
variants that any finite population suffers when each generation is a finite sample of the last. The
variants that any finite population suffers when each generation is a finite sample of the last — the
same sampling accident by which rare surnames vanish from small villages and rare alleles (gene
variants) drift out of island populations with no selection against them. The
identification has been made repeatedly and independently: for sequential inference chains before deep
learning (22), for language-model text ecosystems (23), as a closed-form first-extinction law placing
collapse onset at the Wright–Fisher first-extinction time (24), and in quantitative-genetic form for
@ -78,8 +80,9 @@ points, marks population genetics as the natural mathematics of the setting, tho
point: population genetics is not, at heart, a theory of decay; it is a theory of the mechanisms that
maintain and build populations despite decay (immigration, recombination, selection, population
structure) and of where those mechanisms reach their limits. This paper develops that fuller structure
for model populations: the arc from drift through its remedies to its limit, reproductive isolation,
carried as one framework from closed forms to trained networks to language models.
for model populations: the arc from drift through its remedies to its limit, reproductive isolation —
the point at which diverged lineages can no longer produce working offspring, biology's boundary
between species — carried as one framework from closed forms to trained networks to language models.
An operator of a model population faces recurring decisions for which there is no principled guidance: how much verified
real data does retraining need before a lineage decays; will combining two particular models compose
@ -98,10 +101,15 @@ Knowledge is modelled as a distribution `p_t` over `K` discrete items (capabilit
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 I exploit as an
engineering gate: the 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.
resampling step *is* the Wright–Fisher process, population genetics' canonical model of neutral
evolution, in which each new generation is a random sample of size `n` from the previous one and
every statistical property of drift follows from that one step. Diversity throughout this paper is
*heterozygosity*, `H = 1 − Σ p_i²`: the probability that two random draws differ (one minus a
collision probability), high when many items share the mass, zero at total collapse. The identity is
exploited as an engineering gate: the 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 I measured it rather than assumed it. Real training adds
approximation, optimisation noise, and inductive bias, and when trained networks are fit against the
@ -113,11 +121,15 @@ estimator bias*, and the drift signs (rare-first loss; the grounding response)
survived that bias in every architecture I 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). Retraining on a single parent is *asexual reproduction*, and sustained loss under it carries the defining consequence
of *Muller's ratchet* (28): once every copy of a rare capability is gone from all parents and sources,
no recombination can rebuild it, so remedies must act before fixation-by-loss (a consequence-level
correspondence; the minimal model lacks the ratchet's recurrent-mutation driver).
of *Muller's ratchet* (28), the mechanism by which lineages that never recombine decay irreversibly —
the reason non-recombining genomes such as the Y chromosome have shed most of their ancestral genes.
Once every copy of a rare capability is gone from all parents and sources, no recombination can
rebuild it: each such loss is a click of the ratchet, and remedies must act while copies still
survive somewhere (a consequence-level correspondence; the minimal model lacks the ratchet's
recurrent-mutation driver).
**Table 1.** The dictionary. Each correspondence is stated with the level of support it currently has
**Table 1.** The dictionary. Each biological term is introduced in the section that develops it; 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.
@ -139,7 +151,10 @@ known limits is SI Appendix, Table S1.
### Grounding is immigration: cheap, with a floor
In the minimal model, grounding from a fixed real source is *immigration* into a drifting population
(29–31), and the equilibrium diversity has a closed form the simulator matches exactly. That equilibrium is
(29–31). Immigration is what conservation managers prescribe when a fragmented reserve loses
diversity, and its striking property there is how little is needed — the field's rule of thumb is
that one migrant per generation holds an isolated population's diversity (32). The same economy
appears here: the equilibrium diversity has a closed form the 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 I define it as such: under the tested population
size and Zipf source distribution, `g ≈ 0.05` retained most (≥95%) of equilibrium diversity
@ -164,7 +179,14 @@ fraction, the measured price of the estimator bias).
### Recombination: a conservation law, its operators, and offspring that exceed every parent
The largest returns from the transfer concern merging. **Proposition (blending inheritance, rare-item
The largest returns from the transfer concern merging. *Blending inheritance* — offspring as the
average of their parents — is the failure mode at the root of population genetics' founding
controversy: the swamping argument pressed in Jenkin's 1867 review of *The Origin of Species*, that
under blending a rare advantageous variant is diluted toward the common type faster than selection
can multiply it (33), an objection dissolved only by Mendel's particulate inheritance, in which
discrete variants pass through generations undiluted. Refitting a child model to the mean of its
parents' output distributions is blending inheritance, and the proposition below is Jenkin's dilution
made exact. **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
@ -178,7 +200,7 @@ 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* (32) (different storage and inference budgets from a
output-mean) and *routing among intact specialists* (34) (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
@ -186,25 +208,36 @@ quantitative form of the relationship is untested). On easy tasks a capable base
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* (33, 34): recombination assembles, in one offspring,
complementary variants that arose in different lineages, producing a genotype fitter than any parent.
The generative payoff is the *Fisher–Muller effect* (35, 36), the classical account of why sex speeds
adaptation: in an asexual population, beneficial variants arising in different individuals can only
compete until all but one lineage is lost, whereas recombination assembles them in one offspring,
producing a *genotype* (an individual's combination of variants, one at each *locus*, or position)
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 (35) specialists beat every parent overall (decisively at 7B: 0.87 vs 0.77), and on the sharper
LoRA (37) 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 (36)
(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
Sex has risks and, for AI, an unfair advantage, both quantified on Kauffman's NK fitness landscapes
(38), the standard model of *epistasis*, biology's term for interaction between genes: the fitness
contribution of a variant depends on which variants occupy the other loci, much as a component's
value in an ML system depends on the components around it. Each of the landscape's `N` sites
interacts with `K` others (the model's eponymous parameters), and raising that interaction count
tunes the landscape from smooth and additive to rugged and many-peaked (Fig. 3). When skills are entangled, blind recombination produces offspring *below* their parents,
worsening with ruggedness, and the optimal recombination rate shrinks as entanglement grows. Biology
knows this failure as *outbreeding depression*, the reason conservation practice warns against
crossing locally adapted populations: in the textbook case, an ibex herd in the Tatra Mountains
restocked with animals from Turkey and Sinai produced fertile hybrids that bore their young in the
coldest month of winter, and the herd died out (39). 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
population *structure* is itself a knob: sweeping the mate-pool breadth from monogamous (repeated
local pairings) to promiscuous (*panmixia*: any model may merge with any other) 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.
@ -223,7 +256,7 @@ 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 I 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 (37)) climbs to near the global optimum while
directed recombination + diversity-preserving selection (40)) 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
@ -236,18 +269,20 @@ language-model scale this composed loop remains unbuilt; it is the paper's large
### 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 (38, 39):
changes harmless on their own background but deleterious in combination. A merged model is exactly the
separate species (*reproductive isolation*) through Bateson–Dobzhansky–Muller incompatibilities (41, 42):
changes harmless on their own genetic background but deleterious in combination — the mechanism behind
the mule's sterility and the inviability of many between-species crosses, in which two genomes that
each work perfectly cannot run in the same cell. A merged model is exactly the
exposed hybrid. I 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
(39). Note that a super-linear count does not by itself entail a sharp performance cliff without
(42). 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 (40);
richer symmetry groups remove more (41), with known failures beyond the shared-data regime (42). I therefore aligned under the composition of
trained networks are famously *coordinate artefacts*, removable by re-aligning hidden units (43);
richer symmetry groups remove more (44), with known failures beyond the shared-data regime (45). I 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. 5 C and D): two networks trained from different
initialisations on the *same* task have a barrier that this alignment removes essentially entirely
@ -270,7 +305,7 @@ catastrophically-forgetting specialists (parents ≈ 0.50, merge ≈ 0.955, a su
rescue). The same double result appears at the language-model tier (Fig. 5 E and F): 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 (cf. the merging literature's expert-duration effect; 43) produces
disjoint specialists 1→12 epochs (cf. the merging literature's expert-duration effect; 46) 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
@ -290,9 +325,10 @@ divergence with zero conflict). Before merging, six predictors are computed: *co
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 (44), 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
alignment at the shared base (47), LoRA-delta cosine and distance, and a cross-task performance
baseline. The pre-registered outcome is the merge penalty against oracle parent potential (the analogue of
*hybrid load*, the fitness a hybrid loses relative to what its parents' genes could jointly supply),
also reported against best- and mean-parent references because the predictor
ordering is sensitive to that choice.
Across this controlled grid, pre-merge functional
@ -350,16 +386,16 @@ conventions on shared circuitry, which is the thing to detect.
**Continual learning at the population scale.** Within a single network, the discipline's remedies
for forgetting are this framework's operators writ small. Rehearsal and replay of stored data (26, 27)
is grounded inheritance within one lineage, and the replay fractions the field settled on empirically,
on the order of 1% for instruction tuning (45) and 5% to 25% by distribution-shift strength in
continual pretraining (46), sit where the minimal model's operational threshold lies.
*Pseudo-rehearsal*, the replay of a network's own generated samples, proposed as a cure in 1995 (47)
and revived as generative replay (48), is precisely the ungrounded null studied here: immigration from
a drifting source, benign for one hop, compounding over generations, with verifier-filtering (29, 49)
converting it back into grounding. Parameter isolation (50), including frozen-base adapters, which forget far
less (51), is engineered decorrelation; complementary-learning-systems consolidation (52–54) is the periodic
adapter-into-base merge; the recent turn to merging as a continual-learning mechanism (55–58) applies
on the order of 1% for instruction tuning (48) and 5% to 25% by distribution-shift strength in
continual pretraining (49), sit where the minimal model's operational threshold lies.
*Pseudo-rehearsal*, the replay of a network's own generated samples, proposed as a cure in 1995 (50)
and revived as generative replay (51), is precisely the ungrounded null studied here: immigration from
a drifting source, benign for one hop, compounding over generations, with verifier-filtering (29, 52)
converting it back into grounding. Parameter isolation (53), including frozen-base adapters, which forget far
less (54), is engineered decorrelation; complementary-learning-systems consolidation (55–57) is the periodic
adapter-into-base merge; the recent turn to merging as a continual-learning mechanism (58–61) applies
recombination within one lineage over time, where this paper applies it across lineages; and the
observation that rare examples and long-tail knowledge are forgotten first (59–61) is tail extinction
observation that rare examples and long-tail knowledge are forgotten first (62–64) is tail extinction
seen one model at a time. The mechanisms differ (forgetting is largely deterministic interference,
collapse is sampling drift) but the victims and the remedies coincide, and to my knowledge no prior
work carries population-genetic formalism into continual learning. Read into that field, the results
@ -367,26 +403,26 @@ offer: (i) an equilibrium theory for the replay ratio, with the sharper predicti
fraction is set by the rarest capability one refuses to lose (the `1 − e^{−m·p}` law) rather than by
average loss, 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 (29, 47–49); (iii) *pre-merge interference
unless verifier-filtered back into grounding (29, 50–52); (iii) *pre-merge interference
prediction with a mechanism*: where the current state of the art fits regressions over candidate
metrics (44), the functional-conflict measure arrives at a convergent signal from principle and comes
metrics (47), 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
splits the field's practice (keep adapters separate vs merge them; 57–61): 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 I note the standing objection that apparent forgetting can be
skewed task-inference over latent capability rather than erasure (62); the irreversibility results here
skewed task-inference over latent capability rather than erasure (65); the irreversibility results here
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 new.** The collapse-as-drift diagnosis is established prior work
(21–25); so are the empirical facts that merges can beat parents, that decorrelated parents merge better, and that
naive averaging loses to interference-aware or routed merges (4, 63, 64), that model populations can
climb (5, 8–10), and that merge success admits ML-native predictors (44, 65), correlational where this framework
naive averaging loses to interference-aware or routed merges (4, 66, 67), that model populations can
climb (5, 8–10), and that merge success admits ML-native predictors (47, 68), correlational where this framework
supplies mechanism; the reading of sex as an algorithm for mixability in the theory of computation
(66) anticipated the transfer before model merging existed. New here is the framework-level
(69) anticipated the transfer before model merging existed. New here 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
@ -411,8 +447,8 @@ offspring preview, directed mating) that no living system permits.
**Creative diversity.** Collapse is not confined to facts and skills. Homogenisation of *style* is
already measurable: models trained on model output lose lexical and syntactic diversity across
generations (67), writing produced with model assistance is individually better but collectively less
diverse than writing produced without it (68, 69), and the house styles of the large assistants are
generations (70), writing produced with model assistance is individually better but collectively less
diverse than writing produced without it (71, 72), and the house styles of the large assistants are
recognisable enough that their tics serve as signatures. In this framework these are the same
phenomenon at a different locus. A voice is a distribution over rare stylistic variants, exactly the
tail that drift erases first and that blending inheritance averages into a common register. The
@ -432,7 +468,7 @@ the tested regimes found, freely recombinable in the absence of conflicting conv
ecosystem evolves as one interbreeding population, and the levers that matter are grounding budgets
priced per rare capability and diversity preserved deliberately. If instead long-horizon
specialisation at scale begins to produce emergent incompatibility, as the expert-training-duration
observations hint (43) and the small-scale null here does not rule out, then lineages will begin to
observations hint (46) and the small-scale null here does not rule out, then lineages will begin to
speciate, and the ecosystem's future is a set of diverging species connected by routing rather than by
merging. Which of the two it will be is measurable now, with the pre-merge conflict instruments this
paper tested.
@ -504,41 +540,44 @@ publication; every figure in this paper regenerates from committed artifacts wit
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