# RESULTS — E4-H034

Two parts, and they must not be confused. **P1 is a derivation. P2 is a synthetic protocol
demonstration. Neither is evidence about a real interatomic potential.**

Raw output: `results/h034_results.csv`, `results/h034_run_log.txt`.

---

## P1 — The derived error law (analytic, no data)

`E_out ≈ 0.18 · P² / (κ R_c³)`, tabulated at κ = 1.0 eV/ų.

| P (eV) | R_c (Å) | E_out (meV) | ratio to (P=3, R_c=4) |
|---|---|---|---|
| 1.0 | 4.0 | 11.25 | 0.111 |
| 2.0 | 4.0 | 45.00 | 0.444 |
| 3.0 | 4.0 | 101.25 | 1.000 |
| 6.0 | 4.0 | 405.00 | 4.000 |
| 3.0 | 6.0 | 30.00 | 0.296 |
| 3.0 | 8.0 | 12.66 | 0.125 |

**Readings.**
- Doubling `|P|` quadruples `E_out` — the quadratic dependence is exact in the formula.
- Going from `R_c = 4` to `8 Å` drops `E_out` by 8× — the inverse-cube is steep, which is why the
  effect should be large for short cutoffs and negligible for long ones.
- The magnitudes (10–400 meV) are in the range where MLIP defect errors are usually reported, but this
  is a coincidence of the chosen prefactor, **not a validation**.

**Status: a derived prediction, not a measurement.** Its value is that it is falsifiable: the exponents
are specific and can be checked against any dataset that contains `P`, `R_c` and error.

---

## P2 — Protocol demonstration (SYNTHETIC; not evidence about real MLIPs)

Synthetic data: 12 materials × 40 defects = 480 configurations, one material-level latent bias, a
dipole-correlated size descriptor. Target constructed in two regimes as described in `METHODS.md`.

| regime | split | RMSE (simple) | RMSE (+dipole) | improvement |
|---|---|---|---|---|
| signal | random | 0.858 | 0.803 | 0.055 |
| signal | grouped | 0.979 | 0.931 | **0.048** |
| null | random | 0.871 | 0.870 | 0.001 |
| null | grouped | 0.856 | 0.924 | **−0.068** |

**Readings.**
1. When the dipole carries real information, the grouped split recovers it (0.048 RMSE improvement,
   ~5% of the residual).
2. When it does not, the grouped split correctly returns no gain — in fact a small loss, which is what
   adding a useless parameter costs.
3. The random split is **not** diagnostic: it reports an improvement of 0.001 in the null regime and
   would be read as "no effect" — but it also under-reports the true effect in the signal regime,
   because material identity absorbs variance. It is wrong in both directions, which is why grouping
   is mandatory.

**Status: the protocol works; the question is untested.**

---

## What was NOT obtained

The preregistered primary outcome — a grouped-CV incremental RMSE for real MLIPs on real defects —
**was not obtained**. No dataset was available in this environment, and none was fabricated to produce
a positive result.

The registered decision rule explicitly anticipated this: *"H1 inconclusive if the data cannot be
assembled with sufficient material diversity."* That is the status.

## Summary

| Claim | Verdict | Evidence |
|---|---|---|
| `E_out ∝ P²/(κR_c³)` | **Derived**, elementary elasticity | P1 |
| Error should be quadratic in `|P|` | **Prediction, untested** | P1 |
| Exponents architecture-blind, prefactor not | **Prediction, untested** | P1, METHODS §3 |
| Grouped CV distinguishes signal from leakage | **Demonstrated** (synthetic) | P2 |
| `|P|` predicts real MLIP error incrementally | **INCONCLUSIVE** | no data |