Paper-wide rename: the pop-gen construct is "the biological model"; "(exact)" dropped

"Model" now means an AI model everywhere; the Wright-Fisher construct is "the
biological model" throughout (19 occurrences): tier header, section title (now
"The biological model, and where trained learners depart from it"), Table 1
support column ("Exact" -> "Closed form"; "Analytic model" -> "Biological
model"), Results, Discussion, Methods ("Biological-model tier"), and all
figure captions. "Exact" survives only in technical noun phrases (exact-match
verifier, exact oracle, exact equilibrium, exact recovery); the abstract's
"exactly Wright-Fisher" is now "literally Wright-Fisher".

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

paper/pnas/main.md

@@ -25,8 +25,8 @@ systems whose inheritance, diversity, and compatibility must be managed, and tra
 quantitative framework of the evolution of sex. Its starting point, that training on model output is
 genetic drift and model collapse its signature, is by now established from several independent
 directions; I develop the structure that follows from it. In a minimal
-inheritance model that is exactly Wright–Fisher, and measurably Wright–Fisher plus estimator bias in
-trained networks, I derive and test remedies. Grounding acts as immigration: a real-data fraction
+biological model of inheritance that is literally Wright–Fisher, and measurably Wright–Fisher plus
+estimator bias in trained networks, I derive and test remedies. Grounding acts as immigration: a real-data fraction
 far below one retained most equilibrium diversity, with a per-capability observation floor that
 makes the rarest knowledge expensive under unstratified sampling. Refitting a child to the mean of
 its parents' output distributions cancels the multi-parent gain to first order in the rare-item
@@ -93,7 +93,7 @@ problem at a new scale: *continual learning*, the struggle to acquire new abilit
 ones (26, 27), transposed from a single network to a population whose members inherit from one
 another. Population genetics, I will argue, prices these decisions. Table 1 summarises the
 correspondences on which the argument runs. Fig. 1A maps the experimental programme built on
-them: the same abstractions tested at three tiers — an exact population-genetic simulator,
+them: the same abstractions tested at three tiers — a biological model in simulation,
 trained neural networks, and language models — with the sections that follow climbing that ladder.
 Fig. 1B draws the shift of perspective the whole transfer rests on: the model ecosystem read not as
 a society in space, contemporaries exchanging messages, but as a society in time, generations
@@ -101,13 +101,13 @@ coupled by inheritance, recombination, and immigration.
 
 *(FIG:fig1)*
 
-## The minimal model, and where its exactness ends
+## The biological model, and where trained learners depart from it
 
 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, population genetics' canonical model of neutral
+real samples ("grounding", `g = m/(n+m)`), and refit the child*. This construction is the paper's
+*biological model*, deliberately minimal, and its 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
@@ -117,7 +117,8 @@ decay `E[H_t] = H_0(1 − 1/n)^t`; the exact immigration–drift equilibrium; th
 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
+Where a real learner departs from this model matters, and I measured that boundary 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
@@ -131,32 +132,33 @@ of *Muller's ratchet* (28), the mechanism by which lineages that never recombine
 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
+survive somewhere (a consequence-level correspondence; the biological model lacks the ratchet's
 recurrent-mutation driver).
 
 **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 =
+(closed form = derived in the biological model and verified against simulation; 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 |
+| Genetic drift in a finite population | Training on finite samples of model output | Closed form (biological model); signs in trained nets; diagnosis conceded to prior work |
+| Immigration from a fixed source | Grounding with verified real data | Closed-form 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 |
+| Fisher–Muller effect | Merged specialists exceed every parent | Biological model; replicated in LLMs |
+| Outbreeding depression under epistasis | Merging entangled skills harms offspring | Biological model (NK landscapes); hypothesis at LLM scale |
+| Mating systems / population structure | Who merges with whom (breadth of the parent pool) | Biological 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) |
+| Selection on a fitness function | Verifier-anchored selection ("reality that can say no") | Biological 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
+In the biological model, grounding from a fixed real source is *immigration* into a drifting population
 (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
@@ -178,8 +180,8 @@ still retained across complementary parents* (next section). In trained networks
 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. 2B), 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).
+mode while ~10% grounding holds all thirty (the trained model needs roughly twice the biological
+model's fraction, the measured price of the estimator bias).
 
 *(FIG:fig2)*
 
@@ -204,7 +206,7 @@ reduction from averaging can *reduce* extinction relative to a randomly chosen s
 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
+expected retention with K in all regimes in the biological 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* (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,
@@ -219,7 +221,7 @@ adaptation: in an asexual population, beneficial variants arising in different i
 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
+In the multi-locus biological 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. 4). In real language models the signature replicates under seed replication: merges of three
 LoRA (37) specialists beat every parent overall (decisively at 7B: 0.87 vs 0.77), and on the sharper
@@ -282,11 +284,11 @@ separate species (*reproductive isolation*) through Bateson–Dobzhansky–Mulle
 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. 7A): hybrid fitness tracks the parents while
+exposed hybrid. I built the biological model of the process (Fig. 7A): 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
 (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
+the count-to-effect-size link, which the biological 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
@@ -382,7 +384,7 @@ conventions on shared circuitry, which is the thing to detect.
 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 (48) and 5% to 25% by distribution-shift strength in
-continual pretraining (49), sit where the minimal model's operational threshold lies.
+continual pretraining (49), sit where the biological 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)
@@ -426,7 +428,7 @@ null that bounds it; and the controlled predictive test with its controls. I cla
 these measurements and experiments; I do not claim that 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,
+**Limits and open problems.** The demonstrations are deliberately small: closed-form 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
@@ -470,7 +472,7 @@ paper tested.
 
 ## Materials and Methods
 
-**Analytic tier.** Pure NumPy/SciPy Wright–Fisher simulator over `K`-item distributions (knowledge as
+**Biological-model 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;
@@ -480,7 +482,7 @@ equilibrium, closed-form union) to <0.5% and run in CI with 151 further correctn
 
 **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
+exactly to the biological model — 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