Giorgio Gilestro
ab3dc10587
- paper/pnas -> paper/manuscript (venue-neutral)
- configs/layer1 -> configs/inheritance, src/knowledge -> src/inheritance
(imported as `inheritance`), make layer1 -> make inheritance; layer2 alias dropped
- inheritance and trained-network bundles named after the manuscript figure
they feed (fig2_grounding_sweep, figS3_rebaselining, ...), or descriptively
where they feed none; configs keep their `experiment:` value so parquet
hashes are unchanged, only output.dir moves
- figure scripts, SI figure sources, notebooks, REPRODUCING.md, README and the
SI Methods/tables updated; make clean no longer deletes tracked manifests;
reproduce.sh hashes the s{seed}/ layouts too
Co-Authored-By: Claude Fable 5.1 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01Y64o8FKP7rCuXzC48pxpMm
35 lines
2.4 KiB
Markdown
35 lines
2.4 KiB
Markdown
# E2 — A tiny dose of real data rescues diversity (the headline)
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**Claim tested:** how much *real* data must you mix back in each generation to stop collapse — a
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lot, or a little?
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**Setup (Layer 1, pure math).** `K = 1000` items, Zipf truth `p*`, `n = 200` inherited samples per
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generation, 500 generations, 100 repeats. Each generation we also mix in `m` fresh **real** samples
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drawn from `p*`. The knob swept is the **grounding fraction** `g = m/(n+m)` — the share of the
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training pool that is real — across `g ∈ {0, 0.005, 0.01, 0.02, 0.05, 0.1, 0.2, 0.4}`.
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### Symbols
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- **`g`** grounding fraction (share of real data); **`g*`** the *critical* value that restores diversity.
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- **`p*`** truth, **`p_t`** model, **`H`** diversity, **`H*`** truth's diversity.
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- **grounding = "immigration"** in the genetics analogy: real samples are migrants that re-introduce alleles drift keeps killing.
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### The four panels
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1. **Trajectories.** Diversity `H` over generations, one line per `g` (dark = dry, bright = more
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grounding). `g = 0` slides toward 0; any `g > 0` levels off on a plateau — the collapse is
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arrested.
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2. **The phase boundary (the headline).** Dots = stationary diversity vs `g`; the black dashed curve
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is the *exact* closed-form equilibrium `H_eq`; the red line marks the critical
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**`g* ≈ 0.048` (95% CI [0.047, 0.050])** where `H` reaches 95% of the truth's diversity. Only
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**~5% real data** buys back essentially all the diversity. The `g = 0` point is drawn hollow (it
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is still sliding — its true equilibrium is 0).
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3. **Tail coverage.** Fraction of the rare tail retained, by item-count (red) and truth-mass-weighted
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(purple). Both rise with `g` but stay modest at feasible grounding: a little grounding restores
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*diversity* long before it restores the *deep* tail — which motivates E4 (recombination) and E6.
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4. **Per-rarity band.** The tail split into rarity bands (bright = shallowest, dark = deepest). Deep
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bands lag: an item survives only once `m·p*ᵢ ≳ 1` (enough real samples per generation to land it
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at least once). The sharp threshold lives here, in discrete item survival — not in the smooth `H`.
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### Takeaway
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There is a **critical grounding fraction `g* ≪ 1`**: a small, constant trickle of reality
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indefinitely holds off collapse. **Falsifier (not triggered):** if diversity had only recovered as
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`g → 1`, the practical thesis would die. It recovers at `g ≈ 0.05`.
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