# grounding — the grounding response in real weights (and why the ruler matters)

**Claim tested:** does the E2 result — a small dose of real data (`g* ≈ 0.05`) rescues diversity —
reproduce in a trained GRU? The honest answer reframes the question.

**Setup (Layer 1.5).** Autoregressive GRU, `K = 256` modes, `n = 200`, 30 generations, grounding
swept over 9 values `g ∈ {0, 0.005, …, 0.2}`, **18 repeats** (many repeats are needed because each
lineage's fate is genuinely noisy under `n = 200` drift). The falsifier was pinned in the config
*before* running.

### Symbols
- **`g`** grounding fraction (share of real data), **`g*`** its critical value.
- **forward-KL** distance from truth (the operative neural collapse metric here).
- **tail survival** `tail_truth_mass_alive` — truth-weighted fraction of the rare tail retained. **`H`** diversity.
- **recovery fraction** — how much of the achievable forward-KL reduction a given `g` has bought (0 = dry, 1 = best observed).

### The four panels
1. **Trajectories.** Forward-KL over generations per `g`: grounding suppresses the climb.
2. **Phase boundary.** Stationary forward-KL vs `g` **falls monotonically** (dry ≈ 2.08 → `g = 0.2`
   ≈ 0.75); the effect is statistically significant (paired *t* up to 3.3; 89% of lineages improve at
   `g = 0.2`). **The SIGN is confirmed.**
3. **Recovery curve.** Fraction of the divergence gap closed vs `g`. **Half the gap closes by a
   median-recovery grounding of `g ≈ 0.04`** (CI [0.004, 0.116]) — a striking echo of Layer-1's 0.048
   (black dashed) — **but full recovery needs `g ≈ 0.19`**, far more than the exact histogram: the
   GRU's smoothing both caps the collapse and slows the rescue.
4. **Why forward-KL (the key methodological panel).** Normalised responses of three rulers vs `g`:
   `H/H*` (flat ~0.8) and **tail survival (flat / non-monotone — dry is as high as grounded!)** both
   fail to register the effect, while **forward-KL recovery** responds cleanly. A smoothing model keeps
   *spurious* tail support alive, so counting surviving modes is misleading; only distance-from-truth
   is honest.

### Takeaway (an honest reframing)
Two results: **(1)** the operative neural collapse metric is **forward-KL**, not `H` or tail-survival —
smoothing decouples "modes alive" from "close to truth." **(2)** The *sharp* threshold `g* ≪ 1` is a
property of the exact operator, carried quantitatively by the histogram **bridge** (`g* = 0.047`); the
trained GRU confirms grounding's **direction** and **softens** its sharpness. The pre-registered
95%-of-`H*`/tail falsifier is *not* met — but because those are the wrong rulers for a smoothing
model, not because grounding fails; the blueprint §3.5 directional claim holds robustly.