Experiments · E213
Yes. Nearest-neighbour order alone explains 99.3 % of the energy change along the ordering path; further shells add 0.3 %.
In the log: where does v5's ordering energy live? (pre-registered 2026-09-21 07:1x, before measuring)
book/assets/diagrams/exp/E213.svg).The 3× gap above admits two readings and the Bragg–Williams check cannot separate them: either v5's Monte Carlo under-orders a nearest-neighbour-like Hamiltonian (a sampler or partition defect), or v5's ordering energy is carried by longer-range and many-body figures that the NN-Ising mapping mis-handles, in which case the mapping over-predicts and part of the 3× is an artefact of my own yardstick. Only the second is testable cheaply, and it decides how to read the first.
Method. E211's five Mo–Ta annealing runs already saved 30 configurations each
(1400 → 10 K), so 150 real 54-site states spanning disordered to B2 are on disk and no new
sampling is needed. For each: v5's formation energy, and the Warren–Cowley parameters per
neighbour shell (forager/physics/order.py::warren_cowley, shells 1–3). Then regress
energy on α₁ alone, on α₁+α₂, and on α₁+α₂+α₃, and report R² for each. The question is how
much of v5's energy variation along its own ordering path a nearest-neighbour pair term
can account for. pyeCE is an embedded expansion with no per-cluster ECIs to read off, so
this correlation route is the available decomposition, not a second-best one.
Predictions. (1) α₁ alone gives R² ≥ 0.90. v5 is fitted to DFT formation energies of bcc alloys where the nearest-neighbour Mo–Ta pair is the dominant interaction by a factor of two over same-group pairs (Kim & Widom's own hierarchy), so most of the energy along an ordering path should track α₁. (2) Adding shells 2 and 3 raises R² by < 0.05 — they carry little of the variation even if they carry energy. (3) The residual from the α₁-only fit is not random with temperature: it is largest near the Cv peak (456 K), where many-body correlations are strongest.
Decision rule, written before the numbers. R² ≥ 0.90 on shell 1 → the ordering energy is NN-pair-like, the Ising mapping is fair, and the 3× gap is a sampler/partition defect — rung 1's fault is in the Monte Carlo, not the fit. R² < 0.60 → the mapping over-predicts, the 1409 K yardstick is withdrawn, and the many-body reading stands. Between 0.60 and 0.90 → indeterminate, and the honest report is that the Bragg–Williams comparison cannot carry the weight I just put on it.
150 saved states from E211's five Mo–Ta anneals, spanning α₁ = −1.000 (perfect B2) to
−0.037 (essentially random), energies −152.5 to −82.5 meV/atom. The control passes:
the deepest saved state reproduces ground_state.json's −152.5 exactly.
| fit | R² |
|---|---|
| shell 1 (α₁) alone | 0.9928 |
| shells 1–2 | 0.9939 |
| shells 1–3 | 0.9960 |
This entry is written in 2 separate places in the log, shown here in log order.
The 3× gap above admits two readings and the Bragg–Williams check cannot separate them: either v5's Monte Carlo under-orders a nearest-neighbour-like Hamiltonian (a sampler or partition defect), or v5's ordering energy is carried by longer-range and many-body figures that the NN-Ising mapping mis-handles, in which case the mapping over-predicts and part of the 3× is an artefact of my own yardstick. Only the second is testable cheaply, and it decides how to read the first.
Method. E211's five Mo–Ta annealing runs already saved 30 configurations each
(1400 → 10 K), so 150 real 54-site states spanning disordered to B2 are on disk and no new
sampling is needed. For each: v5's formation energy, and the Warren–Cowley parameters per
neighbour shell (forager/physics/order.py::warren_cowley, shells 1–3). Then regress
energy on α₁ alone, on α₁+α₂, and on α₁+α₂+α₃, and report R² for each. The question is how
much of v5's energy variation along its own ordering path a nearest-neighbour pair term
can account for. pyeCE is an embedded expansion with no per-cluster ECIs to read off, so
this correlation route is the available decomposition, not a second-best one.
Predictions. (1) α₁ alone gives R² ≥ 0.90. v5 is fitted to DFT formation energies of bcc alloys where the nearest-neighbour Mo–Ta pair is the dominant interaction by a factor of two over same-group pairs (Kim & Widom's own hierarchy), so most of the energy along an ordering path should track α₁. (2) Adding shells 2 and 3 raises R² by < 0.05 — they carry little of the variation even if they carry energy. (3) The residual from the α₁-only fit is not random with temperature: it is largest near the Cv peak (456 K), where many-body correlations are strongest.
Decision rule, written before the numbers. R² ≥ 0.90 on shell 1 → the ordering energy is NN-pair-like, the Ising mapping is fair, and the 3× gap is a sampler/partition defect — rung 1's fault is in the Monte Carlo, not the fit. R² < 0.60 → the mapping over-predicts, the 1409 K yardstick is withdrawn, and the many-body reading stands. Between 0.60 and 0.90 → indeterminate, and the honest report is that the Bragg–Williams comparison cannot carry the weight I just put on it.
150 saved states from E211's five Mo–Ta anneals, spanning α₁ = −1.000 (perfect B2) to
−0.037 (essentially random), energies −152.5 to −82.5 meV/atom. The control passes:
the deepest saved state reproduces ground_state.json's −152.5 exactly.
| fit | R² |
|---|---|
| shell 1 (α₁) alone | 0.9928 |
| shells 1–2 | 0.9939 |
| shells 1–3 | 0.9960 |
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