# bridge — the histogram model reproduces Layer-1 E2 exactly (the HARD GATE)

**Claim tested (a plumbing check, not science):** does the neural experiment harness, when run with a
*trivial* model, reproduce the Layer-1 math exactly? If not, no later neural result could be trusted.

**Setup (Layer 1.5).** Same generational loop as every neural experiment — each generation, draw the
parent's samples, optionally mix in real data, retrain, measure — but the "model" is a **histogram**:
it just counts which modes appeared and resamples them (no neural net, no smoothing). This reduces the
neural setup *exactly* back to Wright–Fisher drift. `K = 200` modes, `n = 200`, 200 generations,
60 repeats, grounding swept `g ∈ {0, …, 0.4}`.

### Symbols
- **mode** = one of the `K` items (Layer-1.5 word for "item"); read off each generated sequence by a zero-error oracle.
- **`g`** grounding fraction, **`g*`** its critical value, **`H`** diversity, **`H_eq`** the exact closed-form equilibrium diversity.

### The four panels
1. **Trajectories.** Diversity `H` per `g`. `g = 0` collapses; `g > 0` plateaus — the E2 picture,
   now produced by the *neural runner*.
2. **Bridge = Layer 1 (the gate).** Dots = the neural histogram runner's stationary `H` vs `g`; the
   black dashed curve = the exact `H_eq` closed form from Layer 1. The dots sit **on** the curve, and
   the recovered critical grounding is **`g* = 0.047` (CI [0.045, 0.052])** — matching Layer-1's
   0.048. This equality is what licenses every trained-model result to be read against the analytic
   core.
3. **Tail survival.** Fraction of the rare tail retained vs `g` (item-count red, truth-mass purple) —
   rises with grounding, deep tail lags, exactly as in E2.
4. **Per-rarity band.** Survival by rarity band vs `g`; deep bands need more grounding
   (`m·p*ᵢ ≳ 1`).

### Takeaway
The harness is faithful: with a memoryless model it reproduces Layer 1 to the decimal. **This is a
gate, not a finding** — passing it is the precondition for the RNN/MLP experiments
(`collapse`, `grounding`, `architectures`, `recombination`), where the *model* is what changes.