Was the ordering temperature truly that uncertain, or was our reading of it too noisy?

EXPERIMENTS.md · lines 2700–2764

E52 — The transition was never +/- 173 K; the estimator was throwing the precision away

The order-disorder map was too noisy for any learner: a small neural network reached AUC 0.998 on a smooth synthetic target drawn from the same 150 compositions and 0.565 on the real objective, which says the labels carried nothing, not that the learner was weak. Four things were measured to find out why.

The heat capacity was the wrong quantity. It was being reconstructed as a finite difference of six mean energies; CEThermo.integrate computes the Monte Carlo trace and kept only its mean, discarding the variance that is the heat capacity. Returning the variance changed the physics: genuine transitions went from 33 of 150 to 138 of 150, and peaks clipped at the edge of the scanned range from 69 to 7.

Wang-Landau is not affordable here, measured rather than assumed. The literature's first recommendation. order_disorder_wang_landau had never produced a number - it returns a fill factor of exactly 1.0, because with no energy limits the histogram must cover arrangements that are never visited and so never flattens. Bounded to the window the alloy actually occupies, 60 bins took 16,000 sweeps and 123 s for the fill factor to halve once, against the thirteen more halvings a usable density of states needs; 120 bins never moved at all. Hours per composition against eleven seconds for a temperature ladder.

A finer ladder made it worse. 29 rungs at 50 K scattered by 271 K against 198 K for 8 rungs - and the two runs also differed in sampling, which was a second fault in the comparison. Grid resolution was not the limit.

The limit was taking the largest of a set of noisy numbers. Each rung's heat capacity is a variance estimated from a finite trace and carries its own error, so argmax moves whenever noise outbids the true maximum, and more rungs give noise more chances. Three estimators, same compositions, same five seeds:

The centre of mass uses every rung, so no single noisy one can carry it. It cuts the scatter nearly fivefold at identical cost, and is now what order_disorder reports; the largest rung is still returned alongside as temperature_argmax.

This supersedes the +/- 173 K of E41, which measured an estimator rather than a transition.

Against the literature, and what the estimator was costing. 69 published transition temperatures for equimolar alloys inside these eight elements (Lederer, Toher, Vecchio and Curtarolo, Acta Mater. 159, 364), the same compositions through both estimators:

Reading the same curves differently nearly doubles the agreement, which is the clearest evidence that the estimator was the limitation and not the physics. At +0.638 the map has the ordering of these alloys substantially right, which is the part a search needs.

A scale error remains: ours run about 2.2x low. A constant factor is worth far more than scatter, because it can be characterised and quoted rather than merely suffered. Two candidates, neither yet tested. The expansion is fitted at one fixed lattice constant, 3.2935 A, for every composition, while a real alloy's lattice parameter moves with what is in it. And the published values are themselves mean field, which the literature puts 25-30% high, closing part of the gap from the other side. Note the direction: an expansion fitted to unrelaxed energies was expected to run high, and it runs low, so whatever is happening is not the relaxation bias.