Does a bigger simulation box raise molybdenum–tantalum's ordering temperature?

EXPERIMENTS.md · lines 12540–12568

E200 — is v5's Mo–Ta T_c finite-size? (2026-09-19 00:01; queued)

E188 gave T_c(Mo₀.₅Ta₀.₅, v5) = 500 ± 50 K on 6³ = 432 sites, 100 samples. E191 says v5's Mo–Ta ordering energy is right (ratio 1.05, partial), so the 500 vs 600–1000 K gap must be on the MC side or in how v5 distributes that energy over clusters. Same sweep at 8³ = 1024 sites, 200 samples, 20 steps 1100 → 300 K. Predictions: A. T_c rises by ≥ 100 K (finite-size rounding of a first-order-like B2 transition at 432 sites); B. if it stays within 50 K of 500, the sweep is converged and the deficit is in the model's cluster partition of the ordering energy, not the ensemble — then rung 1's fix is the fit after all, and v6b is tested against it. Chain runs/e200_chain.sh (CPU, after E173c).

A yardstick, derived (2026-09-19 00:02): forager/physics/bragg_williams.py. SymPy: for a B2 order parameter with E(η) = E_rand − ΔE·η² and ideal sublattice entropy, the disordered state loses stability at T_c^BW = 2ΔE/k_B (test pins the closed form); the bcc nearest-neighbour Ising model with the same ΔE orders at 0.79× that (J = ΔE/4, T_c = 6.3508 J/k_B). Numbers: Mo–Ta ΔE(DFT) 76.8 → BW 1782 K, Ising 1415 K; published CE+MC 600–1000 K; v5 MC 500 K. Quinary ΔE(DFT) 21.6 → BW 501 K, Ising 398 K; published 745 K.

Caveat this forces on the quinary reading above. E191's MoNbTaVW "ordered" cell was a guessed arrangement, not a known ground state (Mo–Ta's B2 is). So its 21.6 is a lower bound on the ordering energy — the published 745 K needs ΔE ≳ 32 (BW) to 47 (Ising) meV, i.e. a deeper ordered state exists that our cell did not reach — and v5's 3.6 on that cell cannot be the ordering energy its own MC uses either: E190b's 367 K exceeds the 84 K BW bound of 3.6 meV, so v5's sampler orders into a different, deeper state. What stands: on the same structure v5 is 18 meV too shallow (the under-fit of deep states, measured on the training set). What is withdrawn: "v5 reads the quinary ordering energy at one sixth" — the ratio compares two lower bounds of different tightness. E192's MC-sampled cells (v5's own ordered states, to DFT) are the right comparison for the quinary; E191 is the right one for Mo–Ta.

EXPERIMENTS.md · lines 12986–12991

E200 — Mo–Ta at 8³ = 1024 sites, 200 samples. T_c = 457 ± 147 K vs E188's 6³ 500 ± 50. Prediction B (converged, within 50 K) confirmed; A (rises ≥ 100 K) falsified. The sweep is converged and the drift is down with size, exactly as the finite-size literature says. So the Mo–Ta 500 vs 600–1000 K gap is in the model's cluster partition of a correct ordering energy, not in the ensemble. Note the ± 147: the peak width now agrees with the CV-derived error bar (± 150 K) from the literature, which E188's ± 50 (seed scatter) hid.

EXPERIMENTS.md · lines 14307–14325

E200's peak height, read at last (2026-09-21 17:3x) — the transition is continuous, and pyeCE's variance factor is a fixed 2. Mo–Ta cooling legs, dE/dT estimator:

A first-order-like transition's peak height grows with volume, (8/6)³ = 2.4× here; a continuous 3D-Ising one's is nearly size-independent. Measured 8³/6³ = 1.1–1.4×: referee 2's candidate C ("Cv peak ≠ T_c for a sharper-than-Ising transition") is excluded. The variance channel's excess over the derivative is 1.91 / 1.97 / 1.94 across two cell sizes and three grids — a fixed factor of two, which is the number of sites in pyeCE's PRIM cell; the working hypothesis is that <E E>-<E><E> is the variance per conventional cell, not per site. It changes no peak location; it halves every peak height and prominence this record has quoted from the variance channel, which is why they are now read from dE/dT. Across the three runs and both estimators, T_c(Mo–Ta, v5) = 470–530 K at 40–85 K resolution, the number a second sampler must now reproduce or refute.