Was a higher-temperature ordering peak hidden above the simulation's old 1400 K ceiling?

E214 — pre-registered before it runs (2026-09-21 07:4x)

Mo–Ta, the same cell and statistics as E207, swept 3000 → 100 K and back instead of 1400 → 100. Predictions. (1) A specific-heat peak appears between 1100 and 1800 K that no 1400 K sweep could show. (2) The 456 K feature survives as a smaller second peak — if it vanishes, it was an artefact of the old sweep's hot end rather than a real low-temperature transition, which is a different finding and a better one. (3) The SRO onset stays where E207 put it, 1100–1250 K, because that observable was never censored. (4) The high peak lands below the published 2020 K, because v5's ΔE_order (76.5) is short of the ≈110 meV the mapping needs for 2020 K. What it settles. If (1) holds, rung 1's factor-of-four deficit was substantially a measurement window and not a model defect, and every T_c this project has quoted from a 1400 K sweep is a lower bound rather than an estimate. If (1) fails — no peak anywhere up to 3000 K — then the Hamiltonian genuinely does not order despite its ordering energy, and the sampler is at fault after all.

E214 cooling leg complete — prediction (1) FALSIFIED; the ceiling was not the answer (2026-09-21 10:4x)

The 3000 → 100 K leg of E214 is written (35 states, 432 sites, 1000–2000 sweeps per state). Read straight off the energy, because that is the derivative the peak comes from:

There is no feature between 1100 and 1800 K. The transition sits at ≈ 480 K, where E207 (1400 K ceiling), E200 (8³) and E188 all put it. Prediction (1) — a peak in 1100–1800 K that the old ceiling had hidden — is falsified on the cooling leg alone; the heating leg cannot rescue it, only bound the hysteresis. Prediction (3) holds trivially. The ceiling reading, which I called the leading explanation four hours ago, is dead.

What survives is harder, not softer. E213 and E213b stand: the Hamiltonian's ordering energy is nearest-neighbour-like on its own annealing path (R² 0.993) and for first-defect swaps (0.88× the Ising bar). A nearest-neighbour Ising with that ordering energy orders at 1409 K. The same Hamiltonian, sampled by pyeCE's Monte Carlo with a 3000 K ceiling, orders at 480 K. That factor of three is now unambiguously internal to model-plus-sampler, and the same factor on MoNbTaW (3.23) says it is systematic.

The two readings left, and the experiment that separates them. Either (a) pyeCE's sampler under-orders — wrong by a fixed factor on any Hamiltonian — or (b) the eCE is not additive over bonds away from the two regimes E213/E213b probed (the annealing path, and single defects in perfect B2), so the Ising mapping is wrong for the partially ordered states that decide T_c. E215, pre-registered here: run pyeCE's mcmc on a Hamiltonian with a known answer — the nearest-neighbour bcc Ising, which pyeCE can express as a single-pair expansion — at the same 6³ cell and statistics, and read its Cv peak. Prediction (a): peak at 1409 ± 100 K × (J/J₀), i.e. the sampler reproduces 6.35 J/k_B, and the eCE's non-additivity is the fault. Prediction (b): peak near 470 K for the same J, i.e. the sampler is wrong by 3× on an exactly solvable model, and every ordering temperature this project has ever reported from ece mcmc is a factor of three low. One of these is true and the run is cheap. This outranks E210 and v7.

Also on the record from this morning's DFT relaunch: MoNbTaVW crashed at start — the input named V.pbe-spn-kjpaw_psl.1.0.0.UPF and the file on disk is V.pbe-**spnl**-… (pslibrary's semicore-l set for vanadium). write_pw had the name hard-coded; E192's inputs came from a different path and were right. It now resolves the file from the directory and refuses anything but exactly one match; test added. The cell is re-queued with Mo–Ta.

E214 COMPLETE — the ceiling was never the problem (2026-09-21 15:0x)

Mo–Ta, 6³ (432 sites), 3000 → 100 K and back, 35+35 states, 1000–2000 sweeps each. Read with the run-order leg split, never curve_from_json.

• (1) FALSIFIED on both legs. No peak exists between 1100 and 1800 K — peak-finding at a 15 % threshold returns one peak per leg and it is the low one. The 1100–1800 K band carries 5–6× less specific heat than the transition does.
• (2) Wrong in form, and the correction is the point. I predicted the 456 K feature would "survive as a smaller second peak" under a high one. There is no high one: the low feature is the only peak and it is the main transition.
• (3) CONFIRMED exactly. SRO onset 1124 K on both legs, inside the predicted 1100–1250 K, and direction-independent to 0 K.
• (4) Vacuous — there is no high peak to compare with the published 2020 K.

T_c(Mo–Ta, v5) = 483 ± 97 K (leg mean; CV-derived bar), against E207's 456 ± 97 at a 1400 K ceiling and E200's 457 at 8³. Doubling the thermostat ceiling moved the answer by 27 K, inside the error bar. Cell size, sampling depth, sweep direction and now ceiling have each been varied independently and none of them moves this number. The leg gap is 85 K (cooling above heating), just outside the 65 K reversibility bar and inside the 97 K error bar; both legs reach the same −152.5 floor and the same 1124 K SRO onset.

The shoulder is the interesting part. At 1124 K — exactly the SRO onset, on both legs — the specific heat carries a broad shoulder at 0.4 k_B with no local maximum. That is what local ordering without a global transition looks like, and it is the same reading E207 reached from the Cv/SRO split, now seen directly in one curve.

Where this leaves rung 1. Every explanation outside the sampler is now closed: the ordering energy agrees with DFT to 5 % (E211 Mo–Ta, both ends measured), the Hamiltonian is nearest-neighbour-like globally and locally (E213, E213b), and the measurement window is not censoring anything (this experiment). A nearest-neighbour Ising with DFT's own ΔE_order orders at 1488 K; this Hamiltonian's own Monte Carlo gives 483 K. E215 — the same sampler on a Hamiltonian whose T_c is known exactly — is now the only open explanation, and it is running.