Layer 1.5: architecture-general neural existence proof
Re-scopes Layer 2 into a cheaper, architecture-general neural collapse proof
before the LLM rung. Realises the same Wright–Fisher abstractions in real trained
generative models on a fully-synthetic sandbox with an exact oracle, reusing
knowledge.metrics/truth/seeding and the output contract so neural curves overlay
the Layer-1 analytic curves.
- src/neural/: synthetic token-grammar sandbox (lossless identity + stochastic
style), ExactOracle, HistogramModel bridge, generation loop, experiment runner
- HARD GATE passed: histogram lineage reproduces Layer 1 exactly (neutral decay,
exact H_eq, tracks run_lineage) — tests/test_neural_validation.py
- torch models: autoregressive RNN + MLP (VAE implemented, not yet fidelity-
passing); determinism seeding derived from the SeedSequence stream
- N0 bridge (neural g*=0.047 ≈ Layer-1 0.048), N1 collapse-in-weights, N2 phase
boundary, N5 architecture-generality (collapse + grounding-rescue in histogram
+ RNN + MLP). Manifests/configs committed; parquet gitignored, hashes tracked
- additive backward-compatible save_artifacts extension; Makefile neural targets
Finding: neural smoothing partially resists H-collapse, so forward-KL and tail
survival are the sharp neural collapse metrics (H is smooth, per Layer 1).
92 tests green. Remaining (tasks/todo.md): N4 merge, N2 refine, N3/N6, VAE
fidelity, MNIST tier, figures. LLM/LoRA rung and C3 deferred.
Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
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# Layer 1 — Summary of results
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*The Lamarckian Society, analytical core. Two summaries of the same work: one technical,
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one accessible to ML engineers and neuroscientists with no population-genetics background.*
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---
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## A. Technical summary
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### What was built
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Layer 1 is a parametric model of generational knowledge transmission, built on the
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observation that the generational step — *sample from the parent distribution, optionally
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mix in fresh real samples, refit* — is **literally a Wright–Fisher process with
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immigration**, not merely analogous to one. Knowledge is a distribution `p_t` over `K`
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discrete items on the simplex; a fixed true distribution `p*` carries a deliberate heavy
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(Zipf) tail; "model collapse" is the loss of rare alleles under drift. Each safeguard
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from the perspective paper is one operator on the step:
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- **grounding** `g = m/(n+m)` — immigration of `m` real samples per `n` inherited (mutation supply);
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- **region-matched grounding** — immigration structured by locus;
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- **multi-teacher distillation** — recombination across lineages;
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- **selection** — directional (`greedy`) vs. balancing/novelty (`qd`);
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- **re-minting** — a founder event that freezes `p_t` as the new reference and discards `p*`.
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Because the process is Wright–Fisher, it inherits **closed-form validation targets**, which
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are enforced as `test_scientific_validation.py` assertions (the "spine of trust"):
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1. neutral heterozygosity decay `E[H_t] = H₀(1−1/n)^t`;
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2. fixation probability = initial frequency;
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3. **exact** mutation–drift equilibrium `H_eq = H*·m(2n+m−1)/(n+2nm+m²)` (not the textbook `θ/(1+θ)` approximation);
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4. tail-persistence threshold `m·p*_i ≳ 1`;
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5. recombination union coverage `U(K_T,ρ,q) = T[ρq + (1−ρ)(1−(1−q)^{K_T})]`, with teachers built by a shared-switch exchangeable-Bernoulli construction giving *exact* marginal retention `q` and pairwise correlation `ρ`.
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The simulator matches (1), (3), (5) to `<0.5%` and (2), (4) statistically. 71 tests pass.
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### Findings (E1–E6)
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- **E1 — collapse (null).** Neutral drift reproduces the geometric `H` decay to within
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Monte-Carlo error; support collapses `K→1`; forward KL to truth diverges. Tail *items*
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go extinct ≈10× faster than head items. **Subtlety:** aggregate tail *mass* is a drift
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martingale (mean-conserved), so it is a misleading collapse metric; tail-*item* survival
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is the honest one.
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- **E2 — grounding phase boundary (headline).** Stationary `H` tracks the exact `H_eq`
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across the sweep. An operational critical grounding `g* = 0.048` (95% bootstrap CI
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[0.047, 0.050]) marks where `H` reaches 95% of `H*`; **g* ≪ 1** — as little as `m=1`
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real sample against `n=200` inherited (`g=0.005`) restores 68% of the truth's diversity;
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`g=0.05` reaches 96%. The phase boundary in `H` is *smooth* (H is continuous in `m`); the
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sharp threshold lives in discrete tail-item survival. Per-rarity-band analysis makes the
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`m·p*_i ≳ 1` law visible: at feasible grounding the **deep tail is unrescuable** — diversity
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is cheap to protect, but the rarest items require grounding budgets that scale as `1/p_min`.
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- **E3 — region-matched grounding.** At fixed total budget, `matched` grounding preserves
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the exercised region's tail (survival 0.49) where `uniform` spreads thin and lets it
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collapse (0.07). Grounding protects only what it overlaps. (Per-region `H` is confounded
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by region mass under matched grounding; tail-item survival is the clean metric.)
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- **E4 — multi-teacher recombination.** Union coverage matches `U(K_T,ρ,q)` exactly
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(recombination *supplies* the tail). **Principal finding:** under the blueprint's
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mean-mixture distillation, surviving tail coverage is **flat in `K_T`** — a conservation
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law, since averaging preserves expected pupil tail mass at `q·(tail mass of p*)`
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regardless of `K_T`, and in the rare-tail (linear-survival) regime the `1/K_T` dilution
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*exactly cancels* the union gain. The recombination benefit is realised only under a
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**union-preserving merge** (`max` over teachers, à la M2N2 model-merging), where surviving
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coverage rises with `K_T` and with decorrelation `(1−ρ)`. E4 reports both operators.
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- **E5 — QD vs. greedy.** At matched grounding, greedy (directional) selection drives
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fixation (`H≈0.01`); quality-diversity selection (`w_i ∝ f_i·p_i^{−α}`) holds `H` at a
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positive plateau (0.48–0.88, rising with the novelty exponent α). qd ≫ greedy.
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- **E6 — re-minting gate.** Re-minting a *collapsed* lineage discards the original truth and
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makes forward KL to the original **diverge** (irreversible lock-in), and even accelerates
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the `H` collapse (grounding now reinforces the surviving few). A diversity gate
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(`H ≥ H_gate`) refuses to re-mint while collapsed and keeps KL bounded; re-minting a
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healthy lineage is harmless.
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### Implications
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1. **The economic bet holds for diversity, not the deep tail.** The architecture's central
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claim — "a little grounding protects a lot of inheritance" — is confirmed *for overall
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diversity* (`g* ≪ 1`). But the deepest tail cannot be held by grounding at any feasible
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budget (`m* ∼ 1/p_min`). Preserving the deep tail is therefore *not* grounding's job — it
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is recombination's, which sets up E4 and the paper's multi-teacher argument.
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2. **Naive multi-teacher distillation does not prevent tail collapse; merging does.** This is
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the sharpest new result. The paper's recombination benefit is real at the *supply* (union)
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level but is annihilated by mean-mixture averaging at matched budget. The benefit survives
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into the pupil only under a union-preserving merge operator. The paper's recombination
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claim should therefore rest on **model-merging (M2N2)**, not on averaging distillation —
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a concrete, falsifiable design constraint carried into Layer 2 (contrast C4).
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3. **Re-minting is a one-way door and must be gated.** Assimilating soft inheritance into a
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new base while the lineage has narrowed locks in the collapse irreversibly. A cheap
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diversity gate suffices to prevent it.
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4. **Everything is anchored to closed forms.** Three of the five predictions are exact, so
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the simulator is *validated*, not merely plausible — the headline curves sit on analytic
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targets. The study is bitwise-reproducible from a seed (uv-locked environment).
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---
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## B. Accessible summary (for ML engineers and neuroscientists)
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### The question
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Modern AI is trained once and frozen; it cannot keep learning without *catastrophically
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forgetting*. The Lamarckian Society proposes an alternative: **generations** of bounded
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agents that learn through a working life, then *teach* a fresh pupil, who inherits the
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compressed knowledge and starts ahead — a cultural ratchet. The danger is well known to ML
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engineers under a different name: train a model on the previous model's outputs, generation
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after generation, and it suffers **model collapse** — the rare, improbable cases (the *tail*)
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vanish first and the model drifts to its own mode. The teaching step in this architecture *is*
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that collapse operation. So the whole scheme lives or dies on one question: **under what
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conditions does generational teaching accumulate knowledge instead of degrading it?** Layer 1
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answers that quantitatively, before any GPUs are involved.
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### The one idea that makes it rigorous
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Represent a model's knowledge as a probability distribution over discrete "items"
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(capabilities, facts, behaviours). One generation = *draw a finite sample of size `n` from the
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teacher, and refit the pupil to it.* That finite-sampling step is **mathematically identical**
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to genetic drift in a finite population — the century-old **Wright–Fisher** process. That is
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not a metaphor; it is the same equations. The payoff: population genetics already has **exact
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formulas** for how diversity decays, what survives, and how "immigration" of fresh individuals
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holds a population together. We inherit those formulas as **ground truth to check the simulator
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against** — so the results below are *provably correct*, not just plausible-looking curves.
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A small dictionary:
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| in this model | ML reading | neuroscience reading |
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|---|---|---|
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| knowledge item | a capability / mode of the model | a memory / stored pattern |
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| sample size `n` | how much data the student distils from | consolidation bandwidth |
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| the tail | rare capabilities / long-tail inputs | rare episodic detail |
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| grounding `g` | fraction of fresh **verified** real data in the training mix | new lived experience replenishing memory |
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| heterozygosity `H` | diversity of the model's knowledge | richness / non-degeneracy of memory |
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| collapse | mode-seeking / catastrophic forgetting | memory degradation, loss of the improbable |
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### What we found, in plain terms
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1. **Without fresh data, teaching collapses — and the rare stuff goes first, fast.** Pure
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generation-on-generation distillation loses diversity exponentially, at a rate set by how
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much data the student sees. Rare items go extinct roughly 10× faster than common ones.
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(This reproduces, exactly, the known math of drift.)
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2. **A little fresh grounded data rescues almost all the diversity — this is the headline.**
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Mixing in even ~5% verified real data (in the extreme, *one* real sample against 200
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inherited) restores ~70–96% of the model's diversity and holds it there indefinitely.
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Grounding is cheap and it works. **But** there is a hard limit: the *very rarest*
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capabilities still cannot be saved by grounding alone — protecting an item of rarity `p`
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needs a real-data budget that grows like `1/p`. So grounding rescues *diversity* cheaply,
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but not the deepest tail. (That is a feature, not a bug — it tells us what the other
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mechanisms are for.)
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3. **Grounding only protects what it overlaps.** Spreading a fixed amount of fresh data thinly
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across everything fails to protect any specific area; you must ground the *specific* region
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you want to keep. "Don't inherit dry, region by region" is literally true.
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4. **Learning from several diverse teachers can preserve rare knowledge one teacher would
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lose — but only if you combine them correctly. This is the surprising, important one.**
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Multiple decorrelated teachers *collectively* retain far more of the tail than any one of
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them (we verified this against an exact formula). But whether the *pupil* keeps that
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depends entirely on **how you merge the teachers**. The standard approach — averaging their
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outputs (ordinary multi-teacher distillation) — **mathematically cancels the benefit**: the
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averaging dilutes each teacher's rare knowledge by exactly the factor by which more teachers
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would have helped. A **"keep-the-strongest-teacher-per-item" merge** (the style of model
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*merging*, e.g. Sakana's M2N2) *does* realise the benefit — rare-capability retention rises
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with the number and diversity of teachers. **Design lesson: to fight tail collapse with
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multiple teachers, merge their weights; don't average their outputs.**
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5. **Optimising for "quality" alone collapses diversity; rewarding novelty too keeps it
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alive.** Selecting for fitness drives everything to the single best item (fixation);
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rewarding rareness alongside fitness (quality-diversity selection) maintains a rich,
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diverse population. (Familiar to anyone who has watched a population-based or RLHF pipeline
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mode-collapse.)
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6. **"Baking in" accumulated knowledge into a new base model is a one-way door.** Periodically
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consolidating soft inheritance into fresh base weights lets the system grow without bound —
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but if you do it *after* the model has already narrowed, you lock in the damage
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**permanently** (the original, uncollapsed reference is gone). A cheap check — only
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consolidate while diversity is still high — prevents the irreversible mistake.
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### Why it is novel and why it matters
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- **It turns a hand-wavy debate into exact, falsifiable science.** "Does generational
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distillation ratchet up or collapse?" was an argument by analogy. Casting it as
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Wright–Fisher makes it a set of equations with closed-form answers, and the simulator is
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validated against them — so the headline curves *sit on analytic targets*, not on
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eyeballing.
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- **It quantifies the feasibility of the whole architecture.** The result that a *tiny*
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grounding fraction protects most of the diversity (`g* ≪ 1`) is what makes a
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continually-teaching society economically plausible rather than a data-hungry fantasy.
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- **It corrects how the field should build multi-teacher systems.** The finding that ordinary
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averaging distillation gives *no* protection against tail collapse — while weight-merging
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does — is a concrete, testable design constraint that most current multi-agent/distillation
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setups get wrong by default.
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- **It gives an operational safety rule for self-improving systems.** "Consolidate only while
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diversity is high" is a simple, measurable gate against a failure mode (irreversible
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collapse-in-place) that self-distilling systems are otherwise prone to.
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All of this is at the level of *distributions and dynamics*, deliberately upstream of neural
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networks — Layer 2 then checks that the same three signs (grounded inheritance holds where dry
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inheritance degrades; complementary teachers preserve what one sheds; general capability climbs
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while each specialty is re-earned) appear in real LoRA-adapted language models.
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