# E2 — A tiny dose of real data rescues diversity (the headline)

**Claim tested:** how much *real* data must you mix back in each generation to stop collapse — a
lot, or a little?

**Setup (Layer 1, pure math).** `K = 1000` items, Zipf truth `p*`, `n = 200` inherited samples per
generation, 500 generations, 100 repeats. Each generation we also mix in `m` fresh **real** samples
drawn from `p*`. The knob swept is the **grounding fraction** `g = m/(n+m)` — the share of the
training pool that is real — across `g ∈ {0, 0.005, 0.01, 0.02, 0.05, 0.1, 0.2, 0.4}`.

### Symbols
- **`g`** grounding fraction (share of real data); **`g*`** the *critical* value that restores diversity.
- **`p*`** truth, **`p_t`** model, **`H`** diversity, **`H*`** truth's diversity.
- **grounding = "immigration"** in the genetics analogy: real samples are migrants that re-introduce alleles drift keeps killing.

### The four panels
1. **Trajectories.** Diversity `H` over generations, one line per `g` (dark = dry, bright = more
   grounding). `g = 0` slides toward 0; any `g > 0` levels off on a plateau — the collapse is
   arrested.
2. **The phase boundary (the headline).** Dots = stationary diversity vs `g`; the black dashed curve
   is the *exact* closed-form equilibrium `H_eq`; the red line marks the critical
   **`g* ≈ 0.048` (95% CI [0.047, 0.050])** where `H` reaches 95% of the truth's diversity. Only
   **~5% real data** buys back essentially all the diversity. The `g = 0` point is drawn hollow (it
   is still sliding — its true equilibrium is 0).
3. **Tail coverage.** Fraction of the rare tail retained, by item-count (red) and truth-mass-weighted
   (purple). Both rise with `g` but stay modest at feasible grounding: a little grounding restores
   *diversity* long before it restores the *deep* tail — which motivates E4 (recombination) and E6.
4. **Per-rarity band.** The tail split into rarity bands (bright = shallowest, dark = deepest). Deep
   bands lag: an item survives only once `m·p*ᵢ ≳ 1` (enough real samples per generation to land it
   at least once). The sharp threshold lives here, in discrete item survival — not in the smooth `H`.

### Takeaway
There is a **critical grounding fraction `g* ≪ 1`**: a small, constant trickle of reality
indefinitely holds off collapse. **Falsifier (not triggered):** if diversity had only recovered as
`g → 1`, the practical thesis would die. It recovers at `g ≈ 0.05`.