# E4-H034 — Elastic dipoles as predictors of MLIP error

**Project:** Incoprea Research, independent investigation
**Date:** 2026-11-01
**Status:** **INCONCLUSIVE / OPEN.** The empirical test could not be performed: no suitable dataset
was available in this environment. A theoretical error law was derived, and the evaluation protocol
that would test it was demonstrated on synthetic data. **No claim is made about real potentials.**

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## Research question

Can measurable elastic-dipole descriptors predict the errors of machine-learned interatomic potentials
on defects, *beyond* what simpler descriptors and standard uncertainty estimates already predict?

## Principal findings

1. **A candidate error law was derived** (not fitted): a finite-cutoff model cannot represent the
   elastic energy stored outside its cutoff, which for a point dipole scales as

       E_out ≈ 0.18 · P² / (κ R_c³)  ,

   giving a predicted error with **quadratic** dependence on dipole magnitude, **inverse-cube** on
   cutoff, and **inverse-linear** on stiffness.

2. **The exponents are architecture-blind; only the prefactor is not.** That is a falsifiable
   distinction: two potentials with the same `R_c` but different architectures should differ in
   amplitude, not in slope.

3. **The protocol that would test it was built and shown to work.** On synthetic data with a
   material-level latent variable, a material-grouped split recovered a genuine dipole contribution
   (RMSE improvement 0.048) and correctly returned none when the term was absent (−0.068, i.e. no
   gain). A random split inflates both.

4. **The empirical question remains open.** No dataset combining defect configurations, DFT reference
   energies, elastic dipoles and MLIP predictions was available. `DATA_PROVENANCE.md` lists exactly
   what is required.

## What is NOT claimed

- Not claimed: that the elastic dipole predicts MLIP error in real systems. **This was not tested.**
- Not claimed: that the derived law is novel. It is elementary elasticity, and the interesting content
  is the *prediction* that only the prefactor is architecture-dependent.
- Not claimed: that `P` is a practical feature. Computing it requires DFT — it would be an expensive
  diagnostic, not an inference-time shortcut. The original hypothesis does not distinguish these.

## Why this project is not a failure

The preregistered primary outcome was not obtained, and the brief permits that outcome. What it
delivers instead is: a derivation that can be checked, a protocol that provably distinguishes signal
from leakage, and a precise list of the data that would settle it.

## Files

`ORIGINAL_CLAIM.md` · `PRIOR_ART.md` · `PREREGISTRATION.md` · `METHODS.md` · `RESULTS.md` ·
`LIMITATIONS.md` · `DATA_PROVENANCE.md` · `REPRODUCE.md` · `INTEGRITY.md` · `code/` · `results/`