# E1 — Distillation without grounding collapses, tail-first

**Claim tested:** if a model is trained only on the previous model's output, generation after
generation, does it lose knowledge — and does the *rare* knowledge go first?

**Setup (Layer 1, pure math).** A "population" of `K = 500` items with a fixed true frequency
`p*` shaped like a Zipf curve (a few common items, a long tail of rare ones). Each generation we
draw `n = 100` samples from the current model and refit — **no real data is ever added** (`g = 0`).
Run for 600 generations, averaged over 100 independent repeats.

### Symbols
- **`p*`** — the true frequencies (fixed reality). **`p_t`** — the model's frequencies at generation *t* (drifts).
- **`H`** heterozygosity = diversity (1 = everything equally likely, 0 = one item left). **`H*`** = diversity of the truth.
- **forward-KL** `D(p*‖p_t)` — how far the model has drifted from truth (0 = perfect, grows without bound as the tail is forgotten).
- **support** = how many items still have any probability. **head/tail** = common/rare items.

### The three panels
1. **Geometric decay.** Blue = the simulated diversity `H`; black dashed = the exact textbook law
   `H₀·(1 − 1/n)^t`. They sit on top of each other — the loss of diversity is *exactly* the
   population-genetics drift law, not an approximation. (This is the validation gate: if these two
   curves disagreed, the simulator would be wrong.)
2. **Tail dies first** (log axis). Red = fraction of *rare* (tail) items still alive; green =
   fraction of *common* (head) items still alive. The red curve plunges far faster — rare knowledge
   is lost roughly an order of magnitude sooner than common knowledge.
3. **Collapse.** Purple (left axis, log) = number of distinct items surviving, falling from 500
   toward ~1 (everything collapses onto a single dominant item). Orange (right axis) = forward-KL to
   truth, diverging as the tail vanishes.

### Takeaway
Unchecked model-on-model training is a ratchet: diversity decays on a precise mathematical schedule,
and the rare tail is destroyed first. **Falsifier (not triggered):** if `H` had stayed flat, the
whole thesis would fail. It didn't.