Are five-element refractory alloys shut out by physics, or only by how they were sampled?

arm        AUC_Q      distinct   best meV/atom
cem        18.01         39.67            -82
archive     0.00          0.00            +35

-79.4   Mo0.522 W0.151 Ta0.135 Nb0.112 Ti0.080
-79.2   W0.412 Mo0.238 Ta0.210 Nb0.077 Ti0.063
-77.4   Mo0.589 Ta0.134 Ti0.098 Nb0.092 W0.079

EXPERIMENTS.md · lines 8159–8214

E134 — The project has never required an alloy to be a high-entropy alloy

The stated goal is a refractory high-entropy alloy - five elements or more, near equiatomic. No such constraint exists anywhere in the code. grep over forager/ and scripts/ finds no element-count requirement in any objective, reward, filter or promotion rule. The ladder has been ranking a 12-simplex by driving force with no notion of alloy class, and driving force structurally favours binaries: mixing enthalpy dilutes as elements are added, while a binary concentrates the single strongest pair bond.

Measured on the corrected 1705-point grid:

elements   in grid   drive < -40   passes every correction
       2       198             7                         2
       3       220             2                         0
       4       495             1                         0
       5       792             0                         0

Zero of 792 five-element compositions clear the qualifying bar. Not one. And the bar is not marginally out of reach:

best binary      Mo0.50 Ta0.50            -86.6 meV/atom
best 5-element   Mo Nb Ta Ti W (equi)     -30.3 meV/atom     gap 56

The best five-element composition in the whole grid is MoNbTaTiW, which is a real and well-studied refractory high-entropy alloy. It misses the -40 bar by 10 meV/atom and it orders at 121 K - inside the 90-1000 K window - so it fails the phase requirement independently. MoNbTaVW is the same story at 367 K.

Two separate problems, and only one is a bug.

• The bug: nothing enforces or even reports alloy class, so a high-entropy alloy can never surface in a ranking dominated by binaries. Every "best find" this project has ever reported was a binary or near-binary for this reason, and the operator has had to point it out more than once.
• The physics, which is not a bug: at equiatomic, where E99 says the ordering gate is most reliable, the refractory five-element alloys order at 120 to 370 K. That is inside the window. If the reading is right they genuinely fail the requirement, and the answer to "find a RHEA that holds one phase from 90 K to 1000 K" may be that none of these does.

The qualifying bar is the other half of the bug. -40 meV/atom was set against what binaries achieve. Within the five-element class the best available is -30.3, so the bar excludes the entire class by construction. A threshold has to be relative to what is achievable in-class or it is not a threshold, it is a class filter wearing a number.

Predicted, before constraining the search to the high-entropy class:

1. A constrained search finds five-element compositions better than -30.3, because the grid sampled five-element space only at exact equiatomic - 792 points of a continuous region - and E133 showed the grid misses whole families off its vertices.
2. None of them clears -40, because the 56 meV/atom gap to the binaries is a property of dilution rather than of sampling.
3. Some order outside the window, since the grid's equiatomic five-element ordering temperatures span 121 to 367 K and non-equiatomic compositions will spread wider.

Falsified if a five-element composition reaches -40, which would mean the gap is a sampling artefact and the class was excluded by my enumeration rather than by physics.

EXPERIMENTS.md · lines 8293–8294

E134/E136 results: prediction 2 of E134 FALSIFIED — the class was excluded by my enumeration, not by physics. And the fly, finally run, is 4 meV/atom off the best.

EXPERIMENTS.md · lines 8296–8313

E134, constrained to five elements or more:

arm        AUC_Q      distinct   best meV/atom
cem        18.01         39.67            -82
archive     0.00          0.00            +35

Prediction 2 is falsified on exactly the stated condition — "falsified if a five-element composition reaches -40, which would mean the gap is a sampling artefact and the class was excluded by my enumeration rather than by physics." It reaches -82. The 56 meV/atom gap I reported between binaries and high-entropy alloys was an artefact of sampling five-element space only at exact equiatomic. E134's claim that the class fails on dilution is withdrawn.

The in-class finds are non-equiatomic refractory quinaries, which is precisely what 792 equiatomic grid points could not represent:

-79.4   Mo0.522 W0.151 Ta0.135 Nb0.112 Ti0.080
-79.2   W0.412 Mo0.238 Ta0.210 Nb0.077 Ti0.063
-77.4   Mo0.589 Ta0.134 Ti0.098 Nb0.092 W0.079