# E4 — Recombination supplies the rare tail; only a union-preserving *merge* realises it

**Claim tested:** if several specialist models each remember a different slice of the rare tail, can
combining them reconstruct the whole tail? And does *how* you combine them matter?

**Setup (Layer 1, pure math).** `K = 500` items. We build `K_T` teacher models that each retain the
rare tail with probability `q`, and we control how **correlated** their retained tails are with a
single knob `ρ` (rho): `ρ = 0` = fully complementary teachers, `ρ = 1` = identical teachers. Swept:
`K_T ∈ {1,2,3,5}`, `ρ ∈ {0, 0.25, 0.5, 0.75, 1}`, grounding `g ∈ {0, 0.02, 0.05}`, 200 repeats.

### Symbols
- **`K_T`** number of teacher models; **`ρ`** how correlated their retained tails are (0 = diverse, 1 = clones).
- **union coverage** — fraction of the tail covered by *at least one* teacher (the raw supply).
- **surviving coverage** — fraction that actually survives in the pupil after it retrains on the combination.
- **mean-distill** — pupil trained on the pooled/averaged teacher outputs. **max-merge** — keep, per item, the strongest teacher (a union-preserving merge, à la M2N2).

### The three panels
1. **Supply.** Union tail coverage vs `ρ`, one curve per `K_T`; solid lines are the exact closed form
   `U(K_T, ρ, q)`. More teachers and more diversity (lower `ρ`) supply more of the tail — and the
   simulation matches the formula exactly.
2. **Realisation.** Surviving coverage vs `ρ`. **Solid = max-merge rises** with more/diverse teachers;
   **dashed = mean-distill stays flat.** Averaging dilutes each teacher's rare items back below the
   survival threshold — the gain is supplied but not realised.
3. **The benefit needs the right operator (`ρ = 0`).** Surviving coverage vs `K_T` under both
   operators. Max-merge climbs with teacher count; mean-distill is flat — a **conservation law**:
   averaging's `1/K_T` dilution exactly cancels the union gain.

### Takeaway — "merge, don't average"
Complementary specialists *contain* enough to rebuild the tail, but **naive multi-teacher distillation
(averaging) throws it away**; you must combine them with a union-preserving merge. This is load-bearing
for the paper's recombination claim and is re-tested in real neural weights in `results/recombination/`.
**Falsifier (not triggered):** if mean-distill had also risen with `K_T`, or max never beat mean, the
recombination story would collapse into "just average your models."