Does swapping two atoms in the ordered crystal cost what a nearest-neighbour model predicts?

E213b — the local check, pre-registered before measuring (2026-09-21 07:2x). A high R² on the global order parameter is not the same statement as "the local excitation spectrum is nearest-neighbour Ising". Bragg–Williams cares about the first; the Monte Carlo is driven entirely by the second — the energy cost of one swap. A model whose energy is linear in mean α₁ can still have soft local swaps if the many-body part absorbs them, and that would produce exactly the observed low T_c with no sampler defect at all. So: take v5's Mo–Ta B2 ground state, swap one Mo with one Ta on non-adjacent sites, and measure v5's ΔE directly. The bar, derived not guessed. ΔE_order = 76.5 meV/atom over 4 bonds per atom, with B2 at 4 unlike bonds and random at 2, gives a per-bond like−unlike difference Δ = 38.2 meV. One far-apart unlike swap flips 8 + 8 bonds, so a nearest-neighbour Ising with this Δ costs 612 meV total = 11.33 meV/atom. Prediction: if v5 is NN-pair-like locally, the mean measured swap cost is 11.3 ± 1.5 meV/atom. If it comes in below 7, the local spectrum is soft, the Bragg–Williams yardstick is being fed a global number the sampler never sees, and the fault is the model's many-body partition, not the Monte Carlo — which reverses the conclusion E213's decision rule would otherwise deliver.

E213b CONFIRMED. 40 far-apart Mo↔Ta swaps in B2 cost 9.96 ± 0.02 meV/atom against the derived nearest-neighbour Ising bar of 11.33 — ratio 0.88, inside the pre-registered 11.3 ± 1.5. So the local excitation spectrum the Monte Carlo actually samples is nearest-neighbour-like too, not just the global correlation.

A methods trap, recorded because it nearly produced a published-looking wrong answer. The first version of this script built the cell itself from BCC.ideal_sites and an assumed species order instead of reusing ground_state.py::to_structure. That scrambled every decoration: energies came out −93 to −49 (straddling the random value) with α₁ never below −0.21, and it still returned a confident R² = 0.9685. A scrambled dataset produced a number two hundredths away from the right one. The script now asserts that the deepest saved state reproduces the recorded ground-state energy and refuses to report otherwise. Second trap, same session: a follow-up check reported a swap cost of exactly 0.00 and I briefly took it for a predictor defect — the two sites I swapped held the same species. Both were caught by controls, and neither would have been caught by the test suite.