No real dataset was available. The central predictive question remains unanswered; the synthetic demonstration is not evidence about real interatomic potentials.
Read the full supplied report and supporting documents
RESULTS.md
Original supplied document, preserved verbatim. Its verdict language is source-reported.
Download original# 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 |
LIMITATIONS.md
Original supplied document, preserved verbatim. Its verdict language is source-reported.
Download original# LIMITATIONS — E4-H034
## 1. The central question was not tested
No dataset was available, so the primary outcome — whether `|P|` adds information beyond simple
descriptors for real MLIPs — is **unanswered**. Everything in `RESULTS.md` is either a derivation or a
synthetic demonstration.
## 2. The prefactor is stipulated, not derived
`E_out ≈ 0.18 P²/(κR_c³)`: the `1/3` comes from integrating `r⁻⁴` from `R_c`, and the remaining factor
is an order-unity angular average that I asserted rather than computed. The *exponents* are robust;
the absolute magnitude could be off by a factor of a few. Anyone using the tabulated meV values should
treat them as order-of-magnitude.
## 3. `P` is not an inference-time feature
Computing the elastic dipole requires a DFT relaxation. So the hypothesis is either:
- **a diagnostic** ("after you have done DFT, predict the MLIP error cheaply"), which is useful but
does not avoid the expensive step; or
- **a transferable coefficient** ("compute `P` once per defect class, apply forever"), which is the
original's actual claim and is **untested**.
The original does not distinguish these, and the distinction decides whether the idea is practical.
## 4. Collinearity may make the test uninformative
`|P|` scales with defect size and formation energy, which are cheap descriptors. If the variance
inflation factor exceeds ~10, the incremental test cannot separate the models and the honest output is
"uninformative", not "null". This was pre-registered and was not reached.
## 5. The derivation assumes a lot
Point dipole, isotropic infinite medium, elastic energy as the only missing contribution, and a single
scalar `κ`. Real defects are anisotropic, surfaces and interfaces break the `1/r³` form, and
**electronic** contributions (charge transfer, magnetism, band-gap error) may dominate the error
entirely — in which case the elastic argument concerns a small correction.
## 6. The synthetic demonstration is easy by construction
I chose a generator where the dipole term was 40× the derived `E_out` and uncorrelated with the
nuisance beyond the size descriptor. Real data will be messier, and the protocol's power on realistic
collinearity is unknown. **The demonstration shows the protocol is not broken; it does not show it will
work.**
## 7. Prior art unverified
The *npj Computational Materials* paper on UQ for misspecified MLIPs may already answer the question.
It was identified by title only.
## 8. No independent implementation
The grouped-CV and OLS are standard and short enough to audit by eye, but they were not re-implemented
by a second route. A bug in the fold assignment would bias the result; the demonstration's *correct*
behaviour in both regimes is weak evidence against that, not proof.
## 9. What would overturn everything
A demonstration that MLIP defect error is dominated by electronic/chemical factors uncorrelated with
elastic dipole magnitude. That is plausible, and would make the hypothesis a second-order effect.
METHODS.md
Original supplied document, preserved verbatim. Its verdict language is source-reported.
Download original# METHODS — E4-H034
## 1. The elastic dipole
For a defect in a periodic cell, the elastic dipole (Kanzaki / double-force) tensor is
P_ij = −∂E/∂ε_ij ,
the strain derivative of the total energy at fixed cell shape. It is the leading moment of the defect's
force field and the coefficient of the `1/r³` term in its strain field.
## 2. Strain field and stored energy outside a cutoff
In an infinite isotropic medium, a point dipole produces
ε_ij(r) ≈ P_ij / (κ r³) (κ = bulk modulus scale)
up to angular factors. The elastic energy density is `∼ κ ε²`, so
E_out(R_c) = ∫_{r>R_c} κ ε² d³r ∼ κ ∫_{R_c}^∞ (P²/κ² r⁶) r² dr
= (P²/κ) ∫_{R_c}^∞ r⁻⁴ dr
= P² / (3 κ R_c³) .
**This is the derived law**: `E_out ∝ P² / (κ R_c³)`. Written with a numerical prefactor chosen for
the tabulation in `RESULTS.md`:
E_out ≈ 0.18 · P² / (κ R_c³) [eV, with P in eV, κ in eV/ų, R_c in Å]
The prefactor 0.18 is `1/3` times the angular average of the dipole strain pattern and is **not
independently derived here** — it is a stated constant, and the tabulated values should be read as
order-of-magnitude, with the *exponents* being the content.
## 3. Why this becomes an MLIP error law
An MLIP with cutoff `R_c` cannot represent any energy contribution arising beyond `R_c`. If the error
on a defect is dominated by the missing `E_out`, then
error ≈ c · E_out ≈ c' · P² / (κ R_c³) ,
where `c` absorbs how much of the missing energy propagates into the predicted quantity (energy,
forces, relaxation geometry). **`c` is a property of the model and the target quantity; the exponents
are not.**
## 4. Consequences, stated as predictions
| Prediction | Test |
|---|---|
| Quadratic in `|P|` | log–log slope 2 at fixed material |
| Inverse-cube in `R_c` | same architecture family, varying cutoff |
| Inverse-linear in `κ` | across materials, after normalisation |
| Same exponents, different prefactor, across architectures | compare fitted slopes, not intercepts |
| Saturates when `R_c` exceeds the defect-separation scale | periodic-cell size series |
## 5. Protocol (the part that was actually executed)
A synthetic generator produces defects with:
- a material identity carrying stiffness `κ`, cutoff `R_c`, and a material-level nuisance bias;
- a dipole magnitude `P`, a correlated size descriptor, and a chemistry proxy;
- an error target that is either (**signal**) `simple + 40·E_out + noise` or (**null**)
`simple + noise`.
Two splits are compared:
- **random** 70/30 — the standard mistake;
- **grouped**, leaving whole materials out, 6 folds.
Four nested models (M0–M3 in `PREREGISTRATION.md`) are fit by ordinary least squares; RMSE is
averaged over folds.
**What this demonstrates:** that the protocol distinguishes "the dipole adds information" from "the
dipole correlates with the material", and that a random split cannot. **What it does not demonstrate:**
anything about real potentials.
## 6. What would be done with real data
1. Join defect configurations → DFT energies/forces → computed `P` → MLIP predictions, on identical
cells and charge states.
2. Fit `log error` on `log |P|` with material fixed effects; report the slope with a CI.
3. Compare M1 vs M3 under material-grouped CV; report the incremental RMSE and its CI.
4. Normalise by `κ` and repeat; report VIF between `|P|` and volume.
5. Report high-error detection ROC.
Steps 2–5 are implemented in `code/h034_theory_and_protocol.py` and would run unchanged on real data
with the same column names.
ORIGINAL_CLAIM.md
Original supplied document, preserved verbatim. Its verdict language is source-reported.
Download original# 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.
PREREGISTRATION.md
Original supplied document, preserved verbatim. Its verdict language is source-reported.
Download original# PREREGISTRATION — E4-H034
Written before any calculation (2026-11-01). **Preserved unedited**, including the outcome that
was not achieved.
## Primary hypothesis
H1: Among defects of a given material, the elastic-dipole magnitude `|P|` predicts MLIP energy error
with incremental explanatory power beyond simple structural and chemical descriptors, under
material-held-out validation.
## Competing models (nested)
| Model | Predictors |
|---|---|
| **M0** | material mean (baseline) |
| **M1** | simple descriptors: defect size / volume, coordination, chemistry proxy |
| **M2** | elastic-dipole magnitude and tensor invariants |
| **M3** | M1 + M2 |
| **M4** | M1 + an ensemble-disagreement uncertainty measure, where available |
## Primary outcome
**Incremental predictive value of M2 over M1**, measured as the reduction in held-out RMSE under a
**material-grouped** split, with a confidence interval from repeated grouped folds.
The comparison that matters is **M3 vs M1**, not M2 vs M0. A correlation between `|P|` and error is
not sufficient, because `|P|` correlates with size and formation energy.
## Secondary outcomes
- Whether the error law's exponent is 2 (as derived) rather than 1 or 3, estimated by regression of
log error on log `|P|` with material fixed effects.
- Whether the prefactor is constant across architectures at fixed `R_c` (C4).
- High-error detection: ROC of "is this configuration in the worst decile?" using M3 vs M1.
## Controls
- **Material-grouped split** (leave-materials-out), to prevent material identity from acting as a
hidden predictor.
- **Configuration-level deduplication**: configurations from the same relaxation trajectory or
rattled snapshots of one structure must not straddle train/test.
- **Stiffness normalisation**: repeat the analysis after dividing error by `κ`, to test whether the
effect is just a material-stiffness proxy.
- **Size collinearity check**: report variance inflation between `|P|` and defect volume.
## Decision rules (fixed in advance)
- **H1 supported** if M3 beats M1 by a held-out RMSE reduction whose 95% interval excludes zero, and
the improvement survives stiffness normalisation.
- **H1 rejected** if the improvement vanishes under grouping or under stiffness normalisation.
- **H1 inconclusive** if the data cannot be assembled with sufficient material diversity (fewer than
~8 materials with ≥20 defects each) — this is the anticipated outcome.
## Falsification conditions
- A material in which error is flat in `|P|` across a decade, with enough configurations to exclude
noise, falsifies universality (though not the law for a specific class).
- If `|P|` is nearly collinear with a simple descriptor (VIF > 10), the incremental test is
uninformative and must be reported as such rather than as a null result.
## Budget
Pilot: assemble at most one dataset, run one grouped comparison, stop. No model training, no
potential fitting — the project consumes existing predictions only.
## Anticipated failure mode (recorded in advance)
Suitable public data may not exist in a form that joins (a) defect configurations, (b) DFT references,
(c) elastic dipoles, and (d) MLIP predictions on the same cells. If so, the correct output is a data
specification, not a result.
DATA_PROVENANCE.md
Original supplied document, preserved verbatim. Its verdict language is source-reported.
Download original# DATA PROVENANCE — E4-H034
## Datasets used
**None real.** The P2 demonstration uses a synthetic generator inside
`code/h034_theory_and_protocol.py`, seeded (`np.random.default_rng(20261101)`) and fully specified in
`METHODS.md §5`. It exists to demonstrate that the *protocol* can distinguish signal from leakage.
**It is not a dataset and must never be cited as evidence about MLIPs.**
The distinction is enforced throughout: every P2 number in `RESULTS.md` is labelled synthetic.
## Data feasibility assessment (the Stage A1 requirement)
The hypothesis needs, **on the same configurations**:
| Requirement | Why needed | Availability in this environment |
|---|---|---|
| Defect configurations with material identity | grouping, descriptors | Not verified |
| DFT reference energies/forces | error target | Not verified |
| Elastic dipole tensors, or stress/strain sufficient to compute them | the predictor | Not verified |
| MLIP predictions on identical cells | error target | Not verified |
| Training-domain information | M4 comparison | Not verified |
| Consistent charge states, cells, relaxation protocols | comparability | Not verified |
**No dataset was downloaded, opened, or verified.** Web search was used only to identify *types* of
source (see `PRIOR_ART.md`), not to obtain data. No file in this project derives from a real dataset.
## Why the join is hard (independent of access)
Even with access, the four required components are usually produced by different groups and are not
guaranteed to agree on:
- **cell size and shape** — elastic dipoles and defect energies are cell-dependent, and a dipole
computed in a 64-atom cell is not the same object as one computed in a 256-atom cell;
- **charge state and Fermi-level reference** — a charged defect's energy depends on the chemical
potential, which is frequently not reported alongside the prediction;
- **relaxation protocol** — fixed-volume vs fixed-pressure relaxation changes both `P` and the error;
- **functional and pseudopotential** — the "DFT reference" is not a single thing.
A study that ignores these would produce a correlation that is partly an artefact of provenance.
## What a valid dataset would look like
A single table with one row per configuration and columns:
`material, config_id, structure_hash, n_atoms, cell, charge_state, functional, relax_protocol,
E_dft, E_mlip, force_rmse, P_tensor (9 components), defect_class, train_flag`
**Minimum size** (from the preregistration): ≥8 materials, ≥20 defects each, with at least two
different `R_c` or architectures for the architecture-blindness test.
## Permissions
Not applicable — no third-party data were accessed, so no licence or permission applies.
PRIOR_ART.md
Original supplied document, preserved verbatim. Its verdict language is source-reported.
Download original# PRIOR ART — E4-H034
**Search performed:** 2026-11-01, two web queries on elastic-dipole descriptors, MLIP error
prediction, and uncertainty quantification for interatomic potentials. **Not a systematic review.**
Full texts were not retrieved; the entries below record what a search surfaced.
## Claims under investigation
| # | Claim | Prior-art status |
|---|---|---|
| C1 | The elastic dipole tensor is the leading moment of a defect strain field | **Well established** (Kanzaki force / dipole tensor literature). Not novel. |
| C2 | A finite-cutoff MLIP cannot represent the long-range part of that field | **Plausible and largely established** in spirit; the `R_c`-dependence of elastic-energy error is elementary. Not novel by itself. |
| C3 | Error scales as `P²/(κR_c³)` | **Derived here from elementary elasticity.** Likely known in some form; I did not find it stated as an MLIP error law. |
| C4 | Only the prefactor is architecture-dependent, exponents are not | **Candidate original element.** Unverified. |
| C5 | `|P|` adds predictive information beyond simple descriptors | **Untested here; unknown in the literature I could see.** This is the load-bearing claim. |
| C6 | The coefficient is transferable across materials | **Untested.** |
## Closest identified prior work
1. **"Uncertainty quantification for misspecified machine learned interatomic potentials"**
(*npj Computational Materials*, 2025; also OSTI 2588080). **Identified by search; full text not
retrieved.** The title indicates that UQ under *misspecification* — the regime where the model is
structurally wrong, not merely noisy — is an active area. This is the closest conceptual neighbour:
if that work already supplies an error predictor that beats simple descriptors, the marginal value
of the elastic dipole falls sharply.
2. **"Atomistic Machine Learning with Cartesian Natural Tensors"** (arXiv 2510.04015, surfaced in
search). Suggests that tensorial (including dipole-like) descriptors are being developed for MLIPs.
If such tensors are already standard inputs, `P` may be subsumed rather than novel.
3. **Radiation-damage MLIP papers for tungsten** (surfaced in search, e.g. a deep-learning potential
for bulk tungsten) — the natural data source for defect-error studies. **Not retrieved**, so
whether any of them report per-defect errors against DFT is unknown.
## What this means for novelty
- **C1, C2 are not novel** and should not be presented as such.
- **C3 is a derivation** that is probably reproducible by any competent elastician; its value is as a
*falsifiable quantitative prediction*, not as a discovery.
- **C4 and C5 are the only candidate original contributions**, and both are unverified.
- **The strongest honest statement** is: *the original hypothesis contains a testable claim (C5) that
appears not to have been tested, and this project builds the test rather than performing it.*
## Honest novelty rating
| Question | Assessment |
|---|---|
| Is the dipole tensor as a defect descriptor new? | **No** |
| Is an elasticity-based error law new? | **Unlikely** |
| Is "only the prefactor is architecture dependent" new? | **Unknown** |
| Is the incremental-information test new? | **Unknown**, but it is the right test |
**Overall: a testable hypothesis with an unverified novelty position and no empirical test performed.**
REPRODUCE.md
Original supplied document, preserved verbatim. Its verdict language is source-reported.
Download original# REPRODUCE — E4-H034
## Environment
- Python 3.12
- numpy (only dependency)
- Interpreter used: `/home/toby/Desktop/300 study 1/04_SUPPORTING/.venv/bin/python`
## Run
```bash
cd Incoprea_E4_H034_2026-11-01/code
python3 h034_theory_and_protocol.py
```
Runtime: under 2 seconds.
## Outputs
| File | Contents |
|---|---|
| `results/h034_results.csv` | 16 rows: P1 (12) and P2 (4) |
| `results/h034_run_log.txt` | full console output of the reported run |
## Determinism
Fully deterministic: `np.random.default_rng(20261101)` seeds both the synthetic generator and the
random split.
## Expected values (spot checks)
- P1 at `P = 3, R_c = 4`: `E_out = 101.25 meV`, ratio 1.000.
- P1 at `P = 3, R_c = 8`: `E_out = 12.66 meV`, ratio 0.125 (exactly `(4/8)³`).
- P2 signal/grouped: RMSE 0.979 → 0.931, improvement **0.048**.
- P2 null/grouped: RMSE 0.856 → 0.924, improvement **−0.068**.
If P2's null/grouped improvement comes out strongly positive, the grouped split is leaking and the
demonstration is invalid.
## What cannot be reproduced
**The empirical test.** There is no data to re-run it on. This file documents how to reproduce the
derivation and the protocol demonstration only.
## Running the protocol on real data
`p2_protocol()` in the script takes a list of per-configuration dicts. To apply it to a real dataset,
supply dicts with the keys `material, size, coord, e_el` and a target vector `y`; the four-model
comparison and the grouped CV run unchanged. `e_el` would be computed from the true
E_out = P²/(3 κ R_c³)
rather than from the synthetic generator.
INTEGRITY.md
Original supplied document, preserved verbatim. Its verdict language is source-reported.
Download original# INTEGRITY — E4-H034
## File inventory
| File | Status | Notes |
|---|---|---|
| `README.md` | final | status INCONCLUSIVE / OPEN |
| `ORIGINAL_CLAIM.md` | preserved | original hypothesis, unedited |
| `PRIOR_ART.md` | final | marks all novelty unverified |
| `PREREGISTRATION.md` | preserved | written before the run, unedited |
| `METHODS.md` | final | derivation + protocol |
| `RESULTS.md` | final | separates P1 (derivation) from P2 (synthetic) |
| `LIMITATIONS.md` | final | nine limitations, including the untested core |
| `DATA_PROVENANCE.md` | final | states no data were used |
| `REPRODUCE.md` | final | |
| `INTEGRITY.md` | this file | |
| `code/h034_theory_and_protocol.py` | final | the only script |
| `results/h034_results.csv` | generated | 16 rows |
| `results/h034_run_log.txt` | generated | console output of the reported run |
## No discarded versions with hidden results
Two bugs were found and fixed *before* any number was recorded:
1. `group_cv` built the design matrix with `column_stack` over rows, transposing it — raised
`LinAlgError: Incompatible dimensions` rather than returning a wrong number.
2. The synthetic signal term was 2.5× `E_out`, too small relative to the material nuisance to be
detectable in 480 samples; increased to 40× so the demonstration exercises the protocol at a signal
level it can see.
Both fixes are recorded here because (2) in particular changed the numbers: the demonstration was
**strengthened** to make the protocol's power visible. A reader should know that the signal strength
was chosen by the investigator, and that a smaller signal would have produced a weaker or null
demonstration. **This does not affect the scientific claim, which is about real MLIPs and was not
tested at all**, but the choice is disclosed rather than hidden.
## Checks performed
- Script exits 0; CSV parses; 16 rows.
- Every number in `RESULTS.md` traces to the CSV or the run log.
- P1 ratio 0.125 at `R_c = 8` verified by hand: `(4/8)³ = 0.125`.
- P2 grouped split verified to hold out whole materials (fold assignment on `material` id).
## Checks NOT performed
- No independent re-implementation of the grouped CV.
- No real data, so no external validation of any kind.
- No statistical test on the P2 improvements (480 samples, one seed). The numbers are illustrative of
the protocol's direction, not estimates with error bars.
## Claims made and not made
| Statement | Allowed? |
|---|---|
| "`E_out ∝ P²/(κR_c³)` follows from point-dipole elasticity" | **Yes** — derivable, and derived |
| "Error should scale quadratically in `|P|`" | **Yes** — as a prediction |
| "The grouped-CV protocol distinguishes signal from leakage" | **Yes** — synthetic demonstration |
| "Elastic dipoles predict MLIP error" | **No** — untested |
| "This is novel" | **No** — unverified |
| "Validated on defect data" | **No** — no data |
README.md
Original supplied document, preserved verbatim. Its verdict language is source-reported.
Download original# 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.**
---
## 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/`