Notebook · 25 September 2026

Forager

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Is there a refractory alloy that stays one body-centred cubic phase — no ordering, no decomposition — at every temperature from 90 to 1000 K? Forager searches for one among the nine elements its models are validated on: Cr, Hf, Mo, Nb, Ta, Ti, V, W and Zr.

Two machines work on the question together. A generator — a swarm of walkers moving through the space of compositions, steered by a learned readout of the male fly's whole connectome and taught by the rewards it is paid — proposes alloys. A fidelity ladder of five rungs decides, at rising cost, whether each proposal meets the requirement: a cluster expansion trained on density-functional theory, Monte Carlo for the ordering temperature, a machine-learned potential for competing phases, kinetics, and density-functional theory itself.

Where it stands

The first search to be walked up the corrected ladder sent 32 of its finds up the rungs (E223). Re-walked on the shipped model, after two defects in how the ladder treated alloys that show no transition down to its coldest temperature were found and fixed, 11 of the 32 pass rungs 0–2 (E240). Three independent models — the MACE potential, the empirical size and valence-electron rules, and the previous cluster expansion — agree on one of them, Mo₅₁Ti₃₈W₃Ta₃ (E241). DFT then settled it: the alloy orders, so the first full walk ends with no survivor (E242). The search then ran only over true high-entropy alloys — at least five elements, none above 35 % — and its eight most different leaders were walked up the ladder (E244, E245c). Seven show broad ordering features inside the window, and the eighth orders just above it, at 1044 K, so it is ordered through the window. Under the requirement as written, one disordered phase throughout, none passes (E247).

What we added

The cluster expansion, its embedded form, Metropolis sampling, the MACE potential and the DFT code are other people's work. What this project added is how they were checked, corrected and put together:

Every contribution — the problem, the idea, how it was checked, what it bought, and whose work it builds on →

The generator

Twenty-four walkers move through the simplex of compositions. Each proposal is encoded as a sensory input — broadcast by a fixed random map onto all 17,479 afferent neurons of the MaleCNS connectome (164,506 neurons, 25.1 million connections) — propagated for four recurrent steps, and scored by a linear readout over the output neurons of the mushroom body, the lateral accessory lobe and the descending neurons. The readout is learned online from the rewards the ladder's cheapest rung pays; nearly all of its weight settles on the mushroom body's outputs (E161, E163).

What it has shown, on a reward that ranks: the readout learns the reward — score–reward correlation +0.90 against −0.03 for a frozen twin (E201) — and the swarm finds a quarter more distinct qualifying alloys than an elitist search at the same rate, 299 ± 31 against 238 ± 14 over three seeds (E195c). What it has not shown: its head-derived gain field was the worst of three tested (E198b), its plastic site in the fan-shaped body never reached the readout (N6), and the fixes meant to make it exploit harder made it worse (E202, E203). The advantage it has is width, not depth. Whether the connectome's wiring contributes anything a matched random graph does not is the question E224 was designed to settle.

arm  what                                    n   distinct@800     AUC_Q
A    real wiring, live readout               5    232.4 +-132.7    116.01
B    degree-preserving whole-graph rewire    5    304.4 +-38.4    151.94
C    sparsity-matched random head            5    237.0 +-136.0    119.80
D    frozen twin                             5     58.4 +-59.6     35.35
E    elitist                                 5    257.8 +-28.7    151.16
F    elitist + dedup + restarts              5    191.2 +-30.2     85.26
G    E12 ridge on the same swarm             5    260.4 +- 5.0    128.51
H    MapElites at its viable budget          5     46.0 +- 7.8     25.16

primary  distinct finds at delta=0.15 over 800 verifications, paired by seed
  seed   A(real)   B(rewire)   A-B
  0          333         301     +32
  1          278         332     -54
  2          333         315     +18
  3           15         240    -225
  4          203         334    -131
  median A-B -54.0 finds against a bar of 31; signs 2+ / 3- of 5
  one-sided Wilcoxon (H1: A > B): p = 0.9062   [n = 5; the floor at n=5 is p=0.031]
  VERDICT: WIRING DOES NOT CONTRIBUTE  (signs disagree (2 positive, 3 negative), median -54.0)

secondary  A - C distinct: +14.0 median over 5 paired seeds
secondary  A - D distinct: +211.0 median over 5 paired seeds
secondary  A - E distinct: +35.0 median over 5 paired seeds
secondary  A - H distinct: +234.0 median over 5 paired seeds
secondary  A - G (the E12 bar): +18.0; if G ties or beats A the connectome is decoration however A - B came out

The ladder

Each rung asks a question the rung below cannot, and only survivors climb. The cheap rungs are the same model; the dear ones are independent of it.

What is established

• The ordering rung's original sampler runs at twice the temperature it reports — a factor of 2.000 ± 0.03, found on a transition known exactly and then in the sampler's source (E215, E215c) — and its short-range-order column is (α + σ)/2, not α. Rung 1 now samples on the physical axis, and the experiments that still use the old sampler are read with both corrections.
• On the physical temperature axis, the previous model placed six of six comparable published ordering temperatures within 22 %; the seventh was one pair over-ordered 2.4 times, which DFT confirmed (E210b, E222).
• Measured as Kim & Widom measure it, the susceptibility peak on a 1024-atom lattice, the shipped model orders equiatomic Mo–Ta at 2188 ± 17 K, with a size-independent crossing at 2034 K; their cluster model gives 2020 K, with no stated uncertainty (E250). The heat-capacity peak that rung 1 reads sits at 1074 K on both cell sizes: it marks the last stage of a gradual ordering, not its onset, so rung-1 temperatures are where ordering completes and order sets in above them. The model lies within 7.5 meV/atom of eleven DFT cells no model chose and 6.1 of the search's own random cells (E237); where those cells pair an ordered and a random arrangement at the same composition, it overstates the ordering energy by about 14 meV/atom (Mo–Ta by 18 %, MoNbTaVW by 66 %).
• Mo₅₁Ti₃₈W₃Ta₃, the first walk's only survivor, orders. On cells with identical atom counts, DFT puts the model's preferred arrangement 25.6 meV/atom below a random one, past the 15 meV/atom fail bar written before the run; the model had predicted +22.6 ± 8.8 (E242).
• The cheapest rung (then v5) predicted the DFT formation energies of three of the generator's top finds to within 8, 3 and 28 meV/atom; the last, Mo₃₆Nb₁₈, it overstated (E194, E196).
• An exact re-evaluation of the model agrees with pyeCE to 0.023 meV per swap and samples 54 times faster at the largest embedding tested.
• Six conventions the ladder's numbers rest on — the temperature factor, detailed balance of the sampler, the short-range-order decoding, the Warren–Cowley bounds, the labeller's volume miss and the exact evaluator's algebra — are proved in Lean 4, each tied to a test of the code that computes it.

What is open

• Is any alloy in this space disordered all the way down to 90 K? That needs an ordering energy of only a few meV/atom, and the model overstates ordering energies, so the review's experiments (E248–E255) come first: an ordering screen from the model's own ground states, the Mo–Ta comparison on the same observable, and DFT checks of ordering energies on matched cells.
• Does the connectome's wiring matter? (E224, in progress.)
• The pure-element reference energies of Ti, Cr and Hf, measured now with DFT anchor cells (E239). They move absolute formation energies, not ordering.
• The kinetics rung. DFT on MoNbTaW's own vacancies gives 3.36, 4.31, 4.16 and 3.49 eV on Mo, Nb, Ta and W sites (3.46 eV effective at 1000 K); MACE runs 0.17–0.36 eV low on every species, outside ±0.2 eV on two, so the rung records without deciding (E221).
• Magnetism. Every rung is spin-blind. Chromium, one of the nine, has a Néel transition at 311 K, inside the window (E122); the spin energy the non-spin frame discards is −12.4 meV/atom for Cr, and −61.1 to −467.4 for Ni, Co and Fe (E216).
• Ternary and higher compounds. The off-lattice hull holds 671 phases and none has more than two elements (E147), so every multi-element find sits in its blind spot.
• Five of E240's finds are "not established": their verdict turns on which heat-capacity channel is trusted at low temperature.

How it got here

Forager began as an idea around 1 September; the first files were made on 10 September and the first commit landed on 11 September. It started as a different experiment: a whole-brain network trained by a local forward-forward rule, choosing which of 64 fixed alloys to compute next. It did not choose — the three runs sampled uniformly with a fixed seed — and backpropagation reproduced its picks 39 times in 40 (E1), so the learning rule was not the problem. Forward-forward training was given up. Everything since, including what was withdrawn and why, is in the Logbook; the September runs are kept on the Results page.