MachineSex/results/E2

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.