# E5 — Quality-diversity selection preserves diversity where greedy selection destroys it

**Claim tested:** if each generation you *select* which outputs to keep, does chasing the "best"
outputs (greedy) accelerate collapse — and does rewarding novelty instead prevent it?

**Setup (Layer 1, pure math).** `K = 500` items, Zipf truth, `n = 200`, 400 generations, 100 repeats,
all arms given the same grounding. Three selection modes: **none** (grounding only, no selection),
**greedy** (keep the fittest — highest-`p*` — items), and **quality-diversity (QD)** (a novelty
bonus `w_i ∝ f_i · p_i^{-α}` that up-weights rare items). The novelty exponent `α` is swept over
`{0.5, 1, 2}`.

### Symbols
- **greedy** — select toward the fittest/most-probable items (directional pressure).
- **QD (quality-diversity)** — select for fitness *and* novelty; `α` = strength of the novelty bonus.
- **`H`** diversity; **support** = number of distinct items surviving.

### The three panels
1. **Diversity trajectories.** `H` over generations: red = greedy (crashes toward ~0, i.e. fixation
   on a few items); orange/blue = QD at `α = 1, 2` (holds a high plateau); green = none (reference).
   Greedy selection is a *second* collapse engine on top of drift.
2. **Novelty dose–response.** Stationary `H` vs the novelty exponent `α` for QD (orange dots), with
   greedy (red dashed) and none (green dashed) as reference lines. QD sits above greedy for **every**
   `α`, and rises as the novelty bonus strengthens.
3. **Surviving items per arm.** Stationary support (number of distinct items alive) as bars: greedy is
   lowest; QD arms keep progressively more items alive as `α` grows; none is the reference.

### Takeaway
Optimising only for "what looks best" (greedy) collapses the population onto a handful of winners; a
novelty-rewarding, quality-diversity objective actively **re-introduces and holds the tail**. Key
numbers: greedy `H ≈ 0.01` (near-total fixation) vs QD `H ≈ 0.48–0.88` rising with `α`.
**Falsifier (not triggered):** if QD's stationary `H` had been ≤ greedy's, quality-diversity would be
doing no work.