At the centre of its training data, does the cheap model agree with DFT?

E176 — the second rung-4 verdict: a refractory-centred cell

E172 fell at the thin edge of the data (Hf/Zr-rich). The obvious second point is the data's centre: Mo₁₄Nb₁₄Ta₁₃W₁₃, a random decoration (seed 0) on 54 ideal bcc sites, Vegard a = 3.2554 Å (L = 9.7663 Å), the same settings as E172 (50/400 Ry, MV 0.02 Ry, 3×3×3, local-TF β 0.1, mpirun -np 8 -nk 2), single SCF at Vegard — E172's prediction 2 showed the volume correction is 0.02 meV/atom for a cell of this kind, so no volume scan. References: Mo, Nb, Ta, W from refs_v2/ (same settings; k-spacing 0.212 Å⁻¹ vs this cell's 0.214 — a 1% difference that the two-atom cells at k = 10 do not resolve).

The cheap rung's number, before DFT: eCE E_lattice = −21.9 meV/atom (mapping cost 4e-20). Favourable, as MoNbTaW is known to be.

Predictions.

1. E_form(DFT) is within 2 × RMSE = 30 meV of −21.9, i.e. in [−52, +8], and negative. This is the region RHEA samples densely and where the eCE's holdout error was measured; if the eCE misses here by as much as it missed E172 (46), the error is not "thin data at the edge" but something systematic between the eCE's labels and QE-vs-QE formation energies — and E175's per-source offset becomes the first suspect, not the last.
2. Converges in < 40 iterations under local-TF β 0.1 (E172 took 25 with eight species; four species should be no harder).
3. Together with E172 this gives the first two points of the flywheel's calibration: if E172 is +46 off and E176 within ±10, the eCE's error is composition-dependent and the refit should be weighted toward the edge; if both are off by a similar signed amount, a code offset (per-source term) explains both and E175 should absorb it.

E176 result — prediction 1 confirmed at the edge of its window, and the two cells together say something neither prediction 3 case named. JOB DONE in 21 iterations (prediction 2 confirmed). E/atom −7334.68128 eV at Vegard; references Mo/Nb/Ta/W from refs_v2/: E_form(DFT) = −49.1 meV/atom vs eCE −21.9, difference −27.2 — inside [−52, +8] by 3 meV, negative as predicted. Across the two cells:

cell                          eCE     DFT     DFT/eCE
E172  Hf10Mo15Nb2Ta5Ti2V9W1Zr10   +51.2   +97.2    1.90
E176  Mo14Nb14Ta13W13             −21.9   −49.1    2.24

Not an offset (the misses have opposite signs) — a scale: DFT is about twice the eCE in both directions. Prediction 3 offered "edge effect" or "code offset"; it is neither, and I record that its two cases were not exhaustive. The eCE's own holdout says the same thing once asked the right question: on the 159 held-out compositions predicted = 0.586 × true − 9.2 (r = 0.76), against 0.977 × true + 0.8 (r = 0.98) in-sample. The model compresses formation energies of compositions it has not seen by ~40%; the MAE (11.4) hid it because most held-out energies are small. Two QE cells at 1.9–2.2× are that compression seen from outside, plus whatever QE-vs-VASP adds — which the ACWF numbers say is small.

What this changes. The rung-0 driving force for a new composition — the quantity the search rewards — is understated by roughly 0.6×; finds are being under-paid and thresholds (p ≥ 0.832 on a −40 meV drive) are being applied to a shrunken scale. The flywheel's target is therefore the holdout slope, not the holdout MAE: E175's refit is re-scoped to measure slope on the composition-wise holdout before and after adding the QE cells with the per-source term, with the prediction that two cells move the slope by < 0.02 (two points among 4,300 cannot) — which makes the honest statement that the flywheel needs tens of rung-4 cells at new compositions, ~1 h each, before it can change the cheap rung's scale, and that the cells should be chosen at the search's own finds, where the understatement matters. Until then, a provisional correction — dividing the eCE's E_form by 0.586 for compositions outside the training support — is a hypothesis to test on the next cell, not a fix to apply.