Can a fast stability check scan the whole design space, and where does it point?

EXPERIMENTS.md · lines 3131–3207

E59 — The question that decides costs four milliseconds, and twenty thousand compositions say Mo-Nb-Ta-W

The off-lattice term rejected all seven of the survey's qualifiers (E56) and the ordering temperature decided none of them. That is worth noticing, because the ordering temperature is the expensive part: thirty-six seconds of Monte Carlo. The driving force needs none of it - the expansion's random-limit energy is a dot product, the hull's cheapest mixture is a small linear program, and the mixing entropy is a logarithm.

What stood in the way was relaxation. A cluster expansion lives on a perfect lattice and cannot give back the energy an alloy recovers by letting its atoms move off their sites. Measured across the nine audited compositions that is 16 meV/atom for MoNbTaW and 112 for a V-Hf alloy - not small, not constant, and set against driving forces of 100 to 185 that decide the verdict.

It is predictable. Elasticity puts the distortion energy at the square of the atomic size misfit, so only the constant comes from the data:

relaxation (meV/atom) = 1.83 * delta^2,   delta = rms size misfit in per cent

Correlation 0.83 on nine points, leave-one-out residual 18.5 meV/atom, worst miss 38, against a spread of 27.8 if it were ignored. Against the MACE answer it approximates, the resulting screen has mean error +2 and spread 16 meV/atom at 4.4 ms per composition - eight thousand times faster than the Monte Carlo rung, and it puts MoNbTaW at -51 where the full calculation gives -57.

That is a screening accuracy, so the rung says when it cannot answer: a candidate whose driving force is inside forty meV of zero is promoted rather than judged.

The +2 +/- 16 figure above is in-sample and should not be quoted. The constant k was fitted on those same nine compositions and then the screen was scored against them. Five compositions later confirmed at the dearer rung (E60) give the honest number: a bias of -14 meV/atom with all five errors the same sign and a spread of only 6. The screen is biased rather than noisy in this corner, and the bias runs conservative - it under-reports stability, so it does not manufacture false positives - but it is a systematic and the absolute values it prints are not the answer.

These figures were computed with the relaxation model E70 later replaced. Rescreened with the corrected one, on the same twenty thousand compositions:

The old model under-corrected the relaxation everywhere, so everything looked less stable than it is. The composition of the answer does not move - the most stable are Mo/Nb/Ta/W under both - but the target is five times less rare than this entry reported, and the -40 meV qualifying threshold used in E60 and E61 was one that no uniform draw in twenty thousand could reach. The numbers below are the old ones, left as they were measured.

Twenty thousand compositions, drawn uniformly over the simplex, screened in seconds:

Every one of the eight most stable is dominated by Mo, Nb, Ta and W, and not one contains Hf, Ti, V or Zr above 0.05:

-34   Nb.42 Ta.22 Mo.14 W.14      -28   Nb.66 Mo.22 W.06
-30   Mo.39 Ta.39 Nb.12           -28   Ta.39 Mo.28 W.16 Nb.11
-30   Ta.35 W.30 Mo.21            -27   Ta.57 W.23 Mo.12
-25   Mo.35 Ta.34 Ti.11 Nb.08     -25   Mo.50 Ta.35

The group-5/6 corner, rediscovered from the calculation alone with no literature input - the same corner the original survey rejected, and the same one the ageing experiments have been pointing at since 2011.

And equiatomic MoNbTaW, at -51 meV/atom, beats all twenty thousand. Uniform sampling over an eight-element simplex does not find a composition concentrated in four particular elements with the other four near zero. The target is real, it has a gradient, and luck does not reach it - which is what a generator is for, and what makes the comparison in E60 worth running.