# ORIGINAL CLAIM — E4-H034

Preserved from the Expedition IV corpus. **Not edited.**

## Statement (Expedition IV, Volume A)

> **E4-H034 — The elastic dipole is the error bar**
>
> The elastic dipole (Kanzaki, double-force) tensor is the leading moment of a defect's strain field.
> A machine-learned interatomic potential trained on bulk configurations has a finite receptive field
> and cannot represent the long-ranged part of that field, so its error on a defect should be
> predictable from the dipole's magnitude — and the coefficient is a property of the model, calibrated
> once, so that the elastic dipole tensor of a defect predicts its own error bar.

## As refined in the dossier (D11)

- **(R1)** The elastic dipole `P_ij = −∂E/∂ε_ij` is well defined and computable from a DFT relaxation.
- **(R2)** The strain field of a point dipole falls as `1/r³`, so a finite-cutoff model truncates a
  non-negligible fraction of the elastic energy.
- **(R3)** The error should therefore correlate with `|P|`, with a material- and model-specific
  coefficient.
- **(R4)** If the coefficient is transferable, the tensor gives an *a priori* error bar without running
  a reference calculation.

## Questions this project treats as open (not settled by the original)

1. **Is the relation linear, quadratic, or saturating?** The original says "correlate" and implies a
   coefficient, which is compatible with all three. This project derives a specific exponent (2).
2. **Does `|P|` add anything beyond simpler descriptors?** The original does not test this. It is the
   whole question, because `|P|` correlates with defect size, formation energy and coordination — all
   of which are cheaply available.
3. **Is `P` available at inference time?** Computing `P` requires a DFT relaxation, so using it as a
   predictor presupposes the expensive calculation one is trying to avoid. Original (R4) claims
   transferability; that claim is untested.
4. **Does the correlation survive material-level holdout?** No prior control for material identity.

## The claim, minimally stated

If true: a single tensor, computable once per defect class, predicts the error of any MLIP on that
defect class to within a stated factor, and the prediction is not reproducible by simpler descriptors.

That is testable. It was not tested here for want of data.