# D11 — E4-H034: The elastic dipole is the error bar **Batch:** 2 · **Selected rank:** 11/30 · **Tractability:** 7/10 · **Significance:** 6/10 ## 1. Original hypothesis and refined claim The entry claims the transfer error of a learned potential on a defect property obeys δE ≈ α|P| with α calibrated once, so that the elastic dipole tensor of a defect predicts its own error bar and the ordering of defect classes by error reproduces their ordering by |P|. The mechanism given — a constant error density over the volume where the defect's strain exceeds a threshold — does not survive its own amplitude. That volume is V = (4π/3)|P|/(μ ε_t), which for |P| = 4 × 10⁻¹⁹ J, μ = 50 GPa and ε_t = 10⁻³ is 3.4 × 10⁻²⁶ m³, some 2800 atomic volumes. A constant error density of 1 meV Å⁻³ over it gives some 70 eV, against a plausible 0.05 eV — a factor of order 10³ too large. To match, the error density inside the region the potential is supposed to fail in would have to be ~10⁻⁶ of a typical energy density. The linear law therefore cannot be derived from the stated mechanism; it is at best an empirical regularity with a fitted α. The repaired claim is a *slope* statement. If the local error density scales as e(r) ∝ ε(r)^m, the volume integral gives δE ∝ |P|^{max(m,1)}: slope 1 for a saturated, |P|-independent density (the entry's case), slope 2 for a strain-quadratic density, slope 0 if the error is confined to the core. Which holds is one plot, and the entry's slope of one is the least likely. The ordering claim — rank correlation above 0.7 — survives under all three. ## 2. Refined scientific description A defect is characterised, to leading order in its far field, by the first moment of the forces it exerts on its neighbours. That moment fixes the 1/r³ strain field, the interaction energy with any stress, and the relaxation volume. A learned potential errs most where strains are unlike its training data, so the predictor of a defect's error is the volume over which its strain exceeds the training-set scale — and that volume is set by |P|. What the entry misses is that the error must *fall* with distance inside that volume, since the potential was trained on small strains; a saturating density is the one choice that makes the integral linear, and it is the one that fails on amplitude. ## 3. Verified supporting literature and prior-art assessment [EST] The elastic dipole (double-force, Kanzaki) tensor as the leading moment of a defect's strain field; the 1/r³ decay of its strain; the relation Tr P = 3K V_rel between its trace, the bulk modulus and the relaxation volume. [EST] The Peierls–Nabarro picture in which the Peierls stress depends exponentially on the core width, τ_P ∝ exp(−2πξ/b). [REC] High-throughput density-functional defect databases with formation energies and relaxed geometries. [REC] Foundation potentials with open weights, and reports that their defect and surface errors exceed their bulk errors. [UNV] A published, cross-material error-versus-|P| correlation — I have not verified that one exists, and the entry's case rests on its absence. ## 4. Proposed physical or mathematical model Definition. For a defect in an infinite crystal, let F_i(R) be the force the defect exerts on the atom at R, i.e. minus the gradient of the total energy with respect to that atom's position. Then P_ij = Σ_R R_j F_i(R), Tr P = 3 K V_rel, u_i(r) = P_jk G_ij,k(r), (1) with G the elastic Green function, so |ε(r)| ≈ |P|/(μ r³) and u ∝ 1/r² (the 4π is absorbed into the numerical factor, which is O(1) throughout). Equivalently P_ij = C_ijkl V ε*_kl, the eigenstrain form, which is how it is computed from a relaxed-cell calculation. Error model. The potential's local energy-density error is e(r) = e₀ for ε > ε_t and e₀(ε/ε_t)^m below it, with r_t = (|P|/(μ ε_t))^{1/3}. Then m = 0: δE = 2 e₀ (4π/3) r_t³ = (8π/3) e₀ |P|/(μ ε_t) slope 1 m = 2: δE ≈ e₀ (4π/3) r_min³ slope 2, core-dominated core: δE ≈ e_core slope 0 The m = 0 case is the entry's. The m = 2 case is the natural one for a potential trained on small strains, and it makes the error independent of r_t, so the "volume grows with |P|" argument cancels. ## 5. Variables, assumptions and boundary conditions P_ij [J] elastic dipole tensor, symmetric; |P| its Frobenius norm; K [Pa] bulk modulus; V_rel [m³] relaxation volume; μ [Pa] shear modulus; C_ijkl [Pa] elastic tensor; G_ij [m/N] elastic Green function; ε(r) strain; ε_t the training-set strain scale; r_t [m] the radius where ε = ε_t; e₀ [J m⁻³] saturated error density; m the error-density exponent; r_min [m] the core cutoff, of order one atomic volume to the one-third power; α the calibrated constant; b [m] Burgers vector; ξ [m] core half-width; Ω_P = b²ξ [m³] the Peierls activation volume; τ_P [Pa]. Assumptions: (i) linear elasticity applies beyond r_min; (ii) the error density is a function of strain magnitude alone, not of chemistry; (iii) ε_t is a single number per potential, estimable from the training set's strain distribution; (iv) the defect is isolated, at least 2r_t from another, which for r_t ~ 1 nm means supercells beyond 500 atoms. Domain: |P| between roughly 10⁻¹⁹ and 10⁻¹⁸ J (a monovacancy to a small cluster), strains below the harmonic limit in the far field, and a potential whose training set spans strains no larger than ε_t. ## 6. Quantitative predictions *Slopes.* log δE against log|P| has slope 1 (saturated), 2 (strain-quadratic) or 0 (core). The entry's slope of one is the one whose amplitude fails by ~10³; slope 2 is the prediction of this dossier. A measured slope of 1 with a small α would mean the error density is not strain-driven at all. *Peierls error bar.* With Ω_P = b²ξ = (0.25 nm)²(0.375 nm) = 2.3 × 10⁻²⁹ m³, one electronvolt of defect energy error becomes 1.6 × 10⁻¹⁹/2.3 × 10⁻²⁹ = 7 GPa of Peierls-stress error, i.e. δτ_P[MPa] ≈ 7000 δE[eV]. A 0.05 eV error gives 340 MPa, comparable to the whole Peierls stress of a close-packed metal. And the propagation is exponential: τ_P ∝ exp(−2πξ/b) with 2πξ/b ≈ 9.4 at ξ = 1.5b, so a 10% error in core width is a factor 2.6 in τ_P, and δE = 0.05 eV over b² = 0.13 J m⁻² against a typical γ_us ≈ 0.15 J m⁻² is a 85% error in the unstable stacking fault energy. **The Peierls stress is exponentially ill-conditioned in exactly the quantity the dipole measures**, so an a priori Peierls error bar can rank materials but cannot bound the value. ## 7. Comparison against conventional explanations The conventional account is that a model's error tracks how well its training set covers the local environment — coordination number, chemical species, charge state — and that a mechanical moment is irrelevant. It predicts no correlation with |P| and no slope. Separating observable: an isovalent substitution series. One host, one potential, five solutes from the same group with |P| tuned by size mismatch from 2 to 10 eV at essentially fixed chemistry. The complexity account predicts flat errors, the dipole account a rise with slope 1 or 2. A sharper test is orientation: a volume-compensated pair with equal |P|, one dilatational and one deviatoric, must give equal errors under an |P|-only law and unequal errors if the potential's strain covariance is anisotropic. ## 8. Specific experimental or computational method One method, on stored data. Select a public defect database for one metal and one oxide, restricted to supercells of at least 200 atoms and to defects with a computed relaxation volume. Evaluate two foundation potentials with open weights on the *stored* DFT geometries (single-point energies and forces, no relaxation), and take the error as the formation-energy difference from the reference, with the reference chemical potentials held fixed across the series. Compute P from the same configurations by two routes — the Kanzaki force moment of (1), and the strain-response route P_ij = −∂E/∂ε_ij from six applied strains of ±0.5% — and require them to agree to 10%. Rank-correlate |error| against |P| within each material, fit the slope, and repeat with |P| taken from the potential rather than the reference. Control arm: the same analysis against a *second* reference functional, whose difference should be uncorrelated with |P| if the effect is a model-transfer effect rather than a DFT-functional effect. ## 9. Necessary controls (i) The second functional, which must show no |P| correlation. (ii) A strain-matched null: two defects of very different |P| in cells strained by the same amount, so training-set distance is matched and only the dipole differs. (iii) A bulk reference, whose error must be far below every defect error. (iv) Two independent P routes, which must agree. ## 10. Sources of measurement or numerical error *Dominant systematic: the dipole definition.* Kanzaki-moment and strain-response dipoles differ by the surface term and by the choice of reference cell; in published workflows the two disagree by 10–30% for vacancies. Since the predicted error is linear (or quadratic) in |P|, a 20% dipole error is a 20% (or 40%) error in the predicted bar — comparable to the entire spread across defect classes, which is the effect being claimed. ## 11. Falsification criteria (i) A measured slope near 1 with an α so small that the implied e₀ is below 10⁻⁶ eV Å⁻³: the mechanism is not operative and only a fitted regularity remains. (ii) Error correlating more strongly with formation energy, coordination or chemical complexity than with |P|, by more than 0.15 in rank correlation — the entry's own failure condition. (iii) Two defects with equal |P|, matched cells, different orientation, differing by more than the 1σ spread: |P| alone is insufficient and the tensor must enter. ## 12. Expected scientific contribution An a priori error bar per defect class from a quantity that costs one static calculation, and a test of whether transfer error in learned potentials is organised by mechanics or by chemistry. If mechanics organises it, the same construction applies to surfaces, interfaces and dislocation cores. ## 13. Resource and feasibility assessment Estimates. Compute: single-point evaluation of two potentials on ~10³ stored configurations of a few hundred atoms — hours of GPU time, not days, multiplied by seven for the six-strain dipole route. Software: an ASE workflow (~400 lines), a database client, and the potentials' inference stacks. The binding constraint is the absence of a defect database with consistent supercells and published relaxation volumes; only a few chemistries have one. ## 14. Reproducibility plan Configuration list, potential version hashes and derived |P| per defect released as one table. Bootstrap seed 40 341 001, 1000 resamples, for the rank-correlation interval. Tolerances: the two dipole routes must agree to 10% before any correlation is reported, and the slope's 1σ interval must exclude both 0 and 2 for a definite claim. Replication: a third potential on a different database, same ordering with ρ > 0.6. ## 15. Principal remaining uncertainties The likeliest killer is that the "error" being correlated is mostly the reference's, not the model's: DFT formation energies for the same defect differ between functionals by amounts comparable to the transfer error being measured, and that difference is smooth in defect size and will correlate with |P| for trivial reasons. The second is that |P| correlates with defect size, and defect size correlates with the number of atoms whose local environment is unusual, so the correlation may be with "bigger defect" rather than with the multipole; only the chemistry-matched and strain-matched controls separate these. Third, the Peierls propagation is so strongly amplifying that any error bar derived this way will span a factor of order e, which may be honest but is of little use to a practitioner deciding whether to trust a number.