Is the energy model to blame for ordering temperatures coming out half the published values?

EXPERIMENTS.md · lines 6383–6431

E109 — Why rung 1 halves every ordering temperature: energies or sampling?

E108 measured rung 1 at 0.54 of the published scale on 74 of 75 compositions, with Kim & Widom's replica-exchange Monte Carlo agreeing with the published scale on MoNbTaW where we report half. The fault lies in one of two places and they are separable:

• The energies. If the expansion compresses ordering energies, every transition temperature follows down roughly in proportion. A truncated cluster basis - pairs to 6 A, triplets to 4.5 A - does exactly this, because the neglected longer-range and higher-order terms contribute to ordering and not to the random-limit average.
• The sampling. 128 sites, 90 sweeps, a 200 K temperature grid and a centre-of-mass estimator on the excess heat capacity could each pull a transition low.

The test separates them cleanly and needs no new convention. For a composition, build the random occupation and the B2-ordered occupation of the same 128-site cell at the same ideal lattice constant, unrelaxed, and take the ordering energy E(random) - E(ordered) from the expansion and from MACE directly. Everything cancels in the difference: same lattice, same references, same relaxation state. Whatever disagreement remains is the expansion's.

Predicted:

1. The expansion underestimates the ordering energy against MACE on the same structures.
2. By roughly the factor needed to explain the temperature gap - a ratio near 0.5, since an ordering temperature tracks the ordering energy close to linearly.
3. If instead the two agree to within 20 per cent, the energies are sound and the fault is in the Monte Carlo or the transition estimator, which is a different repair entirely.

Falsified if the expansion and MACE agree, in which case prediction 1 is wrong and the next thing to test is the sampling - sweeps, cell size, grid spacing, and the centre-of-mass estimator against a peak-position estimator.

Result: prediction 1 is FALSIFIED, and in the opposite direction. Five compositions, the same 128-site cell, same ideal lattice, unrelaxed, E(random) - E(ordered) in meV/atom:

The expansion overestimates the ordering energy on every one of the five, by 4 to 65 per cent. Nothing here can halve a transition temperature; if anything these energies should push it up. The prior-art note already said the expansion runs about 30 per cent high because it was fitted to unrelaxed MACE, and this measures the same thing from the other side.

Prediction 3's branch is therefore the live one: the fault is in the sampling or the estimator, not the energies.