Did the larger set of competing phases, not the new energy model, shift old results?

E126 — Separating the expansion change from the hull change

"Re-measure the eight-element record on twelve" was written as though only the element set had changed. E120 showed two things changed together: the expansion (344 parameters, CV 5.68 -> 1068 parameters, CV 9.09) and the hull (136 phases, all prototypes -> 671, the same 136 prototypes plus 535 database phases). A driving force is a distance below the hull, so a straight 8-vs-12 comparison confounds the two.

They separate with a third scoring, because the hulls are nested. Filtering offlattice_hull12.json down to its 136 prototypes reproduces the eight-element hull's contents on the twelve-element indexing:

A   8-element CE   +  8-element hull (136 prototypes)      the old record
B   12-element CE  +  12-hull restricted to 136 prototypes  isolates the CE change
C   12-element CE  +  full 12-element hull (671)            current default

A against B is the expansion alone. B against C is the hull alone.

Predicted:

1. B minus C is one-signed: every composition's driving force is greater (less negative) under the full hull than under the restricted one, with no exceptions. This is close to a theorem rather than a guess - a convex hull is a lower envelope, so adding phases can only lower it, and drive = E_alloy - E_hull can only rise. A single composition moving the other way is a bug in the hull code, not a finding, and that is the most useful thing this experiment can do.
2. The hull change dominates the expansion change. Median |B - C| exceeds median |A - B|, because 535 new competitors is a larger perturbation than re-fitting the same MACE energies on the same lattice with more elements.
3. A minus B is small and two-signed, of order the difference in cross-validation, 10 to 20 meV/atom, scattering both ways rather than trending.

Falsified if A and B differ by more than the hull effect, which would mean the twelve-element expansion is not merely a wider fit of the same physics and the old eight-element numbers cannot be carried forward at all.

E126 result: prediction 1 confirmed exactly, prediction 3 confirmed, prediction 2 FALSIFIED — and the falsification corrects E120.

210 compositions drawn only from the refractory eight, scored three ways:

effect                         median      max      sign
expansion  (B - A)                4.5     48.1      138 up, 72 down
hull       (C - B)                0.0      0.8      194 up, 0 DOWN

Prediction 1 confirmed, and this is the result worth having. Adding phases to a convex hull can only lower the envelope, so drive = E_alloy - E_hull can only rise. Zero of 210 compositions moved the other way. That is a property the hull code must have and now demonstrably does, on a test that would have caught a sign or indexing error immediately.

Prediction 3 confirmed: the expansion change is two-signed, 138 up and 72 down, median 4.5 meV/atom - somewhat smaller than the 10-20 predicted, with a tail to 48.

Prediction 2 falsified. The hull effect is not merely smaller than the expansion effect, it is negligible: median 0.0, maximum 0.8 meV/atom.

The reason is obvious once measured and I should have seen it beforehand. Of the 535 database phases added going from eight elements to twelve, 297 contain Cr, Ni, Cu or Co. A phase containing an element can never lie on the hull at a composition where that element is absent, because hull membership needs non-negative mixing amounts and nothing can cancel the extra element. So for a refractory-only composition those 297 are not competitors at all, and the 238 that are refractory-only barely move the envelope.

This withdraws a claim in E120. That entry stated the hull change "is the one that moves driving forces" and that "any comparison between a pre-E114 number and a current one is confounded by it unless the hull is held fixed". Measured, that is wrong for the refractory compositions the old record consists of. The hull change moves them by at most 0.8 meV/atom. The confounding claim was reasoned, not measured, and is withdrawn.

What stands from E120: the hulls genuinely are nested and the twelve-element one genuinely holds 535 more phases. What was wrong was the inference about magnitude. The correct statement is: the eight-element record can be carried forward for refractory compositions, with an uncertainty from the expansion refit of about 4.5 meV/atom median and 48 worst case, and an uncertainty from the hull of under 1. For compositions containing Cr, Ni, Cu or Co the hull change is of course decisive - those compositions had no hull at all before.

230 passing.