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An estimator, and what its tests found

For phonon pairing the first paper, "Room temperature, one atmosphere", uses the known limit $k_BT_c\to0.1827\,\hbar\sqrt{S/M_H}$ at large coupling, where the Hopfield sum $S$ is the sum over atoms of the Hopfield parameter times the ratio of the hydrogen mass $M_H$ to the mass of the atom [1, 2]. $S$ is an upper scale for $T_c$. A transition at 300 K and one atmosphere needs a hydrogen part, $\eta_H$, of 8.7 to 12 eV/Ų with a single mode, and the largest among the 4,619 hydrides of that paper's census is 5.6.

$S$ comes from a linear-response calculation, which the Alexandria release of 11 August 2025 holds for 27,151 compounds [3, 4]. An estimate from the crystal structure alone would rank candidates before that calculation is made. A search for a record also needs the estimate to recognise a compound stronger than any in its training set, which models trained on measured $T_c$ rarely did in published tests [5, 6].

roomtsc takes a crystal structure and returns an estimate of $S$ and, for a hydride, of the scattering strength per proton, $h=\eta_H/\rho_H$, with $\rho_H$ the hydrogen number density. It predicts no $T_c$. A sum rule fixes $S$ at fixed structure and linear coupling, so errors in phonon frequencies do not enter it, and $T_c$ needs in addition the phonons and the division of $S$ among them. We estimate $h$ because $\rho_H$, read from the structure, accounts for two thirds of the variance of $\ln\eta_H$ in the census.

The two estimators are gradient-boosted trees on 72 descriptors of the cell and a message-passing network on the crystal graph. Draft 4 of the first paper fixed their tests. Before either was trained on the full release we deposited the splits, the baseline predictions and the mark, a mean absolute error in $\ln h$ at most half that of the best of three baselines. Three of the four splits hold out whole classes of structure and carry the mark. The one that holds out the classes with the largest $h$ also requires that at least half of the held-out hydrides above the training maximum be estimated above it.

That deposit labelled a class by Wyckoff letters, which depend on the choice of origin, so one structure type could carry two labels and lie on both sides of a split, and a compound stored in the release as two records could have one on each side. We found the leak while writing up the first run. A second deposit, written with those results known, labels structure types by site symmetry and keeps one record of each compound. The second run is the result of this paper, and the first is reported beside it with the measure of the leak (Table 2).

In the second run no structure type and no compound is on both sides of a marked split, and the estimators' errors are 0.40 to 0.54 of the best baseline's (Table 1). With the 13 structure types of largest $h$ held out, the graph network meets the mark at 0.43 and the trees miss it at 0.53. With the structure type of Mg₂IrH₆ held out, both meet it, at 0.40 and 0.47. Over five folds of whole structure types both miss it, at 0.51 and 0.54, with intervals that contain one half. On five folds of compounds, which share structure types with training and carry no mark, the ratios are 0.35 and 0.41. The errors of 0.316 to 0.397 in $\ln h$ are 17 to 22% in the upper scale of $T_c$, and the labels change by a standard deviation of 0.31 in $\ln S$ between smearings of 0.005 and 0.030 Ry.

Both estimators fail the extrapolation requirement, in both runs. In the second, 23 held-out hydrides lie above the training maximum of 33.5 eV Å, and the trees estimate none of them above it and the graph network two. For ranking candidates inside the training range the estimators are better than a constant. For finding a compound stronger than any in the training set they failed the test fixed for it, and the released code says so with every estimate for a hydride.

We release the weights, the code, a page at roomtsc.com/model/ and an interface at roomtsc.com/api/predict. The released model averages the two estimators in $\ln h$, a mean that was not deposited, so its scores are after the fact. As a first use we ran it on the 30,822 hydride compounds of the release that have no label. It places them higher than the labelled ones on average and none at the floor for 300 K, and the estimates rank those records for a direct calculation without settling what it will find. The rest of the design this estimator belongs to is not built, and its tests are listed as specified and not run.

What is estimated

The Hopfield sum of a compound is the first moment of its Eliashberg spectral function $\alpha^2F(\omega)$. With the hydrogen mass $M_H$ as the unit of mass,

where $\lambda$ is the coupling constant, $\langle\omega^2\rangle$ the second moment of the spectrum, and $M_j$ and $\eta_j$ the mass and the Hopfield parameter of atom $j$ of the cell. The last equality is the relation of McMillan and Hopfield for a metal with one kind of atom [7, 8], with one term for each atom as it is written for hydrides [2, 9]. $S$ is in eV/Ų.

Its hydrogen part is the Hopfield parameter of hydrogen, $\eta_H$, the sum of $\eta_j$ over the hydrogen atoms. The first paper divides it by the hydrogen number density,

for $n_H$ hydrogen atoms in a cell of volume $V$, and calls $h$, in eV Å, the scattering strength per proton.

$S$ is the label because of its relation to $T_c$ and because a sum rule fixes it. Allen and Dynes found $k_BT_c\to0.1827\,\hbar\sqrt{\lambda\langle\omega^2\rangle}$ at large coupling [1], and the first paper writes, at any coupling,

The efficiency $\Phi$ is below one in every solution of the Eliashberg equations computed there, and is 0.35 to 0.58 for four published spectra of ambient-pressure hydrides and for five megabar hydrides [2]. $S$ is therefore an upper scale for $T_c$, and the estimator returns no $T_c$.

Once the structure and the linear electron-ion matrix elements are fixed, the phonon frequencies and eigenvectors cancel out of equation (1), because the eigenvectors form a complete set. An error in the phonons at a fixed structure changes $\lambda$ and leaves $S$ unchanged, so estimating $S$ requires no phonon model.

The sum rule does not protect $S$ against errors in the electronic factor or in the structure, and the first paper documents three kinds. The first is the sampling of the Fermi surface. The release stores the spectral function at ten smearing widths from 0.005 to 0.050 Ry, and the labels are at 0.030. Over the 27,120 compounds with $S$ at both widths, $\ln S$ at 0.005 Ry exceeds $\ln S$ at 0.030 Ry by 0.15 in the mean, with a standard deviation of 0.31. The five largest $\eta_H$, 4.7 to 5.6 eV/Ų at 0.030 Ry, are larger by factors of 2.4 to 3.7 at 0.005 Ry. We do not know which width is nearer the converged value. The second is a change of structure under nuclear quantum motion. In one path-integral simulation of Li₂AuH₆ part of the hydrogen pairs into molecules, and the $S$ we form from that study's $\lambda$ and second-moment frequency is a factor of 4.2 below the linear-response value for the crystal, a ratio between two different methods [4, 10]. The third is the treatment of the electron-phonon vertex. In PdH, on one set of anharmonic phonons, $\lambda$ is 0.42 with the bare linear vertex, 0.27 with that vertex averaged over the quantum distribution of the nuclei, and 0.64 with averaged second-order terms added [11]. An estimator trained on the labels can at best reproduce them.

The graph network returns a positive strength $a_j$, in eV Å, for each atom and forms the values for the compound from equations (1) and (2) with $a_j/V$ in place of $\eta_j$,

No $a_j$ has a label of its own, and we have not tested the individual values. The mark of the tests is set on $h$, and the second run also scores the network's $S$, without a mark.

What the estimator is one part of

The estimator was designed as one component of a staged model. The others were a closure that divides $S$ among the phonon modes, a phonon model, an estimator of persistence at one atmosphere, an estimate of the phase stiffness, and a decision layer with a dossier for each shortlisted candidate. None of them is built.

$S$ and the phonon frequencies do not give $\lambda$, because each mode contributes its share of $S$ divided by the square of its frequency. The design fixed those shares with a symmetric 3×3 tensor on each atom, of which $a_j$ would be the trace. Without the closure and the phonons the model gives neither $\lambda$ nor $T_c$. The section on the tests lists what was fixed for these components and not run.

Labels and structure types

The labels come from the phonon and electron-phonon release of the Alexandria database of 11 August 2025 [3, 4], harmonic PBEsol calculations at one atmosphere in 93 files, read as the first paper's appendix describes. Of 83,801 records, 56,505 (67%) have imaginary harmonic modes, and none of those carries a spectral function. With repeats of an identifier removed, 27,151 compounds, each one identifier of the release, have a spectral function, and 6,094 of them contain hydrogen. Each of the 27,151 has a label for $S$, integrated from the spectral function stored at a smearing of 0.030 Ry.

The second label, $h$, exists where the hydrogen vibrations separate from the rest. The release resolves the spectral function by phonon branch, and $\eta_H$ of a hydride with $n_H$ hydrogen atoms is the part of $S$ in its $3n_H$ highest branches. The branch criterion of the first paper asks that each of those branches have at least 0.90 of its mass-weighted eigenvector on hydrogen at every stored wavevector. The 4,619 hydrides that pass it with a positive $\eta_H$ are the labelled hydrides. Their $h$ runs from 0.48 to 85 eV Å, with a geometric mean of 7.2 and a standard deviation of $\ln h$ of 0.76.

What the labels leave out

Imaginary harmonic modes leave 32,882 of the 39,002 hydride records without a label. Strong coupling drives phonons toward instability [12], so those records may hold larger $h$ than any record with one. The branch criterion removes 1,474 more hydrides, among them Li₂AgH₆, PdH₄ and Li₂CuH₆, three of the ten with the largest $S$. The release covers the compositions and structures its authors generated, and half of the labelled hydrides fall in 15 of the structure types defined below.

Structure types and prototypes

The splits that carry the mark (the section on the tests) hold out compounds by a label of the atomic arrangement, built from the space group and the Wyckoff positions that spglib finds at a tolerance of 0.05 Å [13]. A prototype, the label of the first deposit, is the space-group number and, for each element, its count in the reduced formula and the Wyckoff letters it occupies, with the element names removed (model/dataset.py). A structure type, the label of the second, has the site-symmetry symbol of each occupied orbit in place of the letters (model/structure_types.py).

The letters depend on which of several equivalent origins a structure is described from [14]. Cubic A₂MH₆, the structure of Mg₂IrH₆, has M on position $a$ of space group 225, A on $c$ and hydrogen on $e$. Moving the origin by half the body diagonal of the cubic cell puts M on $b$, which has the same site symmetry, so the structure is the prototype 225_1a_2c_6e from one origin and 225_1b_2c_6e from the other. The release has 81 compounds under the first and 15 under the second, nine of them labelled hydrides, and the first deposit held out the 81 and left the 15 in training. A shift of origin does not change a site symmetry, and all 96 have the structure type 225_1[m-3m]_2[-43m]_6[4m.m].

The release also holds some compounds under two identifiers, which the first deposit could put on opposite sides of a split. Five A₂MH₆ hydrides are stored under both prototypes, among them Mg₂RhH₆ with $h$ of 24.84 and 24.89 eV Å. The second deposit treats records with the same reduced formula and structure type as one compound and uses only the first record, on either side of a split. That drops 1,093 records, 67 of them labelled hydrides, and leaves 26,058 compounds and 4,552 labelled hydrides. The rule also drops different structures that share a label. $S$ of a dropped record is within 1% of the kept record's in 243 cases and differs from it by more than a factor of 1.5 in 139.

The 27,151 compounds have 2,301 prototypes and 1,740 structure types. The 4,619 labelled hydrides have 837 prototypes, and the 4,552 that are kept have 653 structure types. The structure type is the finer label for 34 prototypes, each of which it divides. Among the 4,224 kept labelled hydrides whose structure type has a second labelled member, the structure type accounts for 52% of the variance of $\ln h$, so a split that puts one structure type on both sides lets an estimator score by recognising the arrangement. The splits that carry the mark therefore assign whole structure types in the second deposit. In the first they assigned whole prototypes, and Table 2 gives the measure of what that let through.

Two estimators and their baselines

Both estimators take the cell and the atomic positions and are given no quantity from an electronic-structure calculation. They are scored separately on every split, with the settings of the appendix, which are the same in both runs.

Trees on descriptors of the cell

The first estimator is gradient-boosted regression trees, in the histogram implementation of scikit-learn, on 72 descriptors of the cell. Nine describe the cell as a whole, among them the hydrogen density and an $r_s$ from the valence-counting rule of the first paper's appendix. Fifty-six are the mean, minimum, maximum and standard deviation of atomic number, electronegativity, radius, mass, group, row and valence count, over all atoms and over the atoms other than hydrogen. Six are the shortest distance overall and for three kinds of pair, and the mean numbers of hydrogen and of other neighbours within 2.4 Å of a hydrogen atom. The last is the space group number. The trees are fitted to $\ln h$ of the labelled hydrides on the training side, 15% of which the fit sets aside to decide when to stop adding trees. They give no $S$.

A graph network with a strength on each atom

The second estimator is a message-passing network on the crystal graph, which joins every pair of atoms closer than 5 Å. An atom starts as a learned vector for its element plus a linear map of its group, row, electronegativity, radius, mass, valence count and a flag for hydrogen. An edge carries its length expanded in 32 Gaussians, so the network is given distances and no angles. In each of four layers of width 96 an atom receives a gated message from every neighbour, and the normalised sum is added to its state. A head with a softplus output returns a positive strength $a_j$ for each atom, and equation (4) gives $S$ and $h$ of the compound. The network's estimates therefore cannot be negative, and its $\eta_H=\rho_Hh$ cannot exceed its $S$.

The loss adds two Huber terms, on $\ln S$ for every compound of the training side and on $\ln h$ for its labelled hydrides. With the structure types of largest $h$ held out in the second deposit, the network trains on 25,303 compounds and the trees on the 4,022 labelled hydrides among them.

Baselines

The estimators are compared with three predictors of $h$ that have at most one fitted constant. The electron-gas constant assigns every hydride 36.6 eV Å, the first paper's value for one proton in an electron gas. The density-of-states baseline multiplies the total density of states at the Fermi level per unit volume by a constant, the geometric mean of the ratio of $h$ to that density over the labelled hydrides of the training side. Draft 4 specified the hydrogen-projected density of states, which the release does not store. The training mean is the geometric mean of $h$ on the training side, 6.8 eV Å with the structure types of largest $h$ held out and 6.9 to 7.5 over the five grouped folds of the second deposit. The first deposit added it (the section on the tests), and the mark is set against the best of the three.

The second deposit adds one baseline for $S$, the geometric mean of $S$ over every compound of the training side, 0.067 eV/Ų with the structure types of largest $h$ held out and 0.064 to 0.091 over the grouped folds.

Related work

A graph network trained on linear-response calculations predicts $\lambda$ and $T_c$ and was used to screen hydrides at ambient pressure [15], and another predicts $T_c$ of hydrides up to 500 GPa [16]. Two predict the spectral function itself [17, 18]. All of these targets contain the phonon frequencies, which an estimator of $S$ does not have to learn. Networks that predict the Kohn-Sham Hamiltonian have given electron-phonon interactions for individual systems [19, 20].

The tests, and the two deposits

Draft 4 of the first paper, dated 6 October 2026, fixed the tests and stated that none of the model had been trained. The release holds what two of draft 4's four tests of $h$ need, the one that holds out the largest $h$ (the top split) and the one that holds out the structure of Mg₂IrH₆ (the family split). They were run on each of two deposits, beside five folds of whole classes of structure (the grouped split) and five folds of compounds (the random split).

Rules for every split

A split that carries the mark holds out whole classes of structure, and every compound of a held-out class leaves training, with or without a label. The classes are prototypes in the first deposit and structure types in the second (the section on the labels). The reference value of a held-out hydride is its label. The labels and the published survey [3, 4] come from one group's workflow, so each test measures agreement with it at one smearing. The score is the mean absolute error in $\ln h$ over the held-out labelled hydrides.

The first deposit and its amendments

The first deposit was written at 07:16:08 UTC on 7 October 2026 and committed 18 seconds later (08be0a2), by our own clock, and is at roomtsc.com/model/deposit/. We had read every label before it, because the census of the first paper uses the same file.

Two code tests on nine of the 93 files preceded it. In the first, on grouped folds, the estimators were at 0.39 and 0.42 of the better baseline of draft 4, which meets its mark, and at 0.65 and 0.70 of the training mean. The deposit therefore adds the training mean as a third baseline and sets the mark against the best of the three, with those ratios in view. The second test ran the top split for 2 epochs, and its output was discarded.

The first manifest does not record the number of epochs, and the code committed with it defaults to 120. Every full-data run was given 60, with the network size and tree settings of that code, and the logs show one run of each split that carries the mark, so no setting was tuned on a test split.

The manifest also names commit 2b88f05 as the code before the deposit, in error. That commit holds none of the training code, which was first committed with the deposit. We have left the manifest as deposited.

What was wrong with the first splits

A structure type could be held out under one prototype and stay in training under another. Draft 4 required that every record of a compound go to one side, and the code did not look for one compound stored under two identifiers (the section on the labels). We found both faults while writing up the first run, after its top, family and grouped splits had been scored.

model/leak_check.py counts a held-out hydride as exposed if a labelled hydride of its structure type was in training, and as duplicated if another record of the same compound was (Table 2). The family split had no unexposed hydride, because the second prototype of its structure type stayed in training, and its 68 are among the 76 exposed in the top split. Over the grouped folds 31% were exposed.

The second deposit

The second deposit was written at 11:23:10 UTC on the same day and committed 13 seconds later (a58a0ef), and is at roomtsc.com/model/deposit2/. Its manifest names commit 85e8bde, the first with the code for the new labels, splits and runs. The first of the new runs began within a minute of the deposit.

It changes the label of a class and keeps one record of each compound (the section on the labels). Its manifest lists as unchanged the label, the smearing, the three baselines, the mark, the extrapolation requirement and the settings of both estimators, with the 60 epochs. We wrote the second deposit knowing the results of the first run, and 507 of the 530 hydrides its top split holds out were held out in the first. A formula in another structure type still counts as another compound, and 17 of those 530 and 7 of the 72 of the family split have a labelled hydride of their formula in training.

The four splits

The top split ranks structure types by the largest $h$ among their labelled members and holds them out in that order until a tenth of the labelled hydrides are out. That takes 13 structure types with 755 compounds, 530 of them labelled hydrides.

The family split holds out the structure type of Mg₂IrH₆, with 91 compounds and 72 labelled hydrides, where draft 4 counted eighteen in published tables [4, 15]. Seventy-one are A₂MH₆ hexahydrides and one is U₂Ru₆H. All are among the 530 of the top split, so the two tests are not independent. No compound and no structure type is on both sides of either split.

The grouped split shuffles the 1,740 structure types with a fixed seed and deals them into five folds of 438 to 1,406 labelled hydrides. Draft 4 did not define it, and both deposits give it the mark. The random split deals compounds without regard to structure type, and 4,178 of its 4,552 held-out labelled hydrides (92%) have a labelled hydride of their structure type in training. It carries no mark.

The mark and the extrapolation requirement

An estimator meets the mark if its error is at most half that of the best of the three baselines and the 95% interval of the ratio lies below one. The interval comes from 2,000 resamples of the held-out structure types, or of the compounds in the family split, and may extend above one half. Where an estimator misses the mark, draft 4 requires that the baseline be used for that quantity and that the failure be reported with the output.

Of the 530 hydrides held out in the top split, 23 have labels above the largest training label, 33.5 eV Å. That split also requires that at least half of the 23 be estimated above every training label, and at most a tenth of the other 507.

The Hopfield sum, scored without a mark

Draft 4 asked for the top split in $S$ over all compounds, without a mark, and the first run did not score $S$. The second scores the graph network's $S$ on every held-out compound of every split by the mean absolute error in $\ln S$, beside that of the training-side geometric mean of $S$. The result files hold no estimate of $S$ for single compounds.

Tests not run

The other two tests of $h$ train and test across pressure and need calculations at megabar pressure. The release holds none, and published $\eta_H$ gives $h$ for five megabar compounds [2], where draft 4 required at least twenty held out. We have no reference set of calculations with dense sampling of the Fermi surface for the calibration test. The closure is not built, so neither its fit to the branch-resolved spectra nor the end-to-end check on $T_c$ was run. Nothing has been shortlisted for the prospective test. The persistence estimator is not built, and the appendix keeps its retrodiction test (Table 5).

Results

On the three splits that share no structure type with training, the errors of the estimators in the second run are 0.40 to 0.54 of the error of the best baseline (Table 1). The graph network meets the mark on the top split and the trees miss it. Both meet it on the family split, and both miss it over the grouped folds, at ratios of 0.51 and 0.54. On the random folds, which share structure types with training, the ratios are 0.35 and 0.41.

Against the density-of-states baseline, the better on every split of the two that draft 4 fixed, the ratios on the marked splits are 0.27 to 0.34. The rank correlations of estimate with label there are 0.71 to 0.82.

The first run beside the second

Without the 76, the top split of the first run has errors of 0.386 and 0.310 (ratios of 0.53 and 0.42), so the leak changed that split little (Table 2). Without the 5 that had a second record in training, the family split has errors of 0.309 and 0.334 (0.37 and 0.40). The first grouped result stored no per-compound predictions, so its 1,425 cannot be left out. Over the grouped folds the ratio of the trees moved from 0.48 to 0.51, from met to missed, and the other five verdicts are unchanged.

The extrapolation requirement

Of the 23 hydrides of the top split with labels above the training maximum, the trees place none above it. The graph network places 2, Cs₂AuH₂ and Rb₂AuH₆, and 5 of the other 507. Both fail the requirement of at least half, as in the first run, where neither placed any of 24 such hydrides above the maximum.

The trees are low on all 23, by a factor of 2.9 in the geometric mean, and the graph network on 22, by 2.7 over the 23. The five alkali platinum and nickel hexahydrides have labels of 63 to 85 eV Å and estimates of 12 to 22. An ordinary estimate for a structure type outside the training set therefore does not show that its label is ordinary.

The held-out family

Among the eleven labels above 25 eV Å, five of them in Table 3, the largest misses are Zn₂IrH₆, at 8.5 and 7.5 eV Å against 32.4, and Mg₂IrH₆, at 9.3 and 6.7 against 26.5. Seven of the 72 have a labelled polymorph in training, and without them the ratios are 0.42 and 0.47, computed after the fact.

The whole Hopfield sum

The error of the graph network in $\ln S$ is 0.328, against 2.184 for the training mean of $S$, on the 755 compounds of the top split. It is 0.417 against 2.290 on the 91 of the family split, 0.217 against 1.275 over the grouped folds of all 26,058 kept compounds, with a rank correlation of 0.976, and 0.160 against 1.223 over the random folds. These ratios of 0.13 to 0.18 say less than the ratios for $h$. Draft 4 gave $S$ no mark because composition and cell volume, which the estimator is given, fix part of its ordering. The baseline uses neither, and we have not measured how large that part is.

What the errors measure

On the marked splits the errors are 0.316 to 0.397 in $\ln h$, a factor of 1.37 to 1.49 in $h$. The structure fixes the hydrogen density, so the same error is 17 to 22% in the upper scale $136.5\ \text{K}\times\sqrt{\eta_H}$. The labels depend on the smearing by about as much, with a standard deviation of 0.31 in the change of $\ln S$ between 0.005 and 0.030 Ry (the section on what is estimated).

The mean of the two estimators in $\ln h$, which the released model uses, was not deposited. Computed after the fact from the stored predictions, its error is 0.328 on the top split, a ratio of 0.44, 0.322 on the family split (0.39), 0.303 over the grouped folds (0.48) and 0.216 over the random folds (0.35). No verdict attaches to it, and it places none of the 23 hydrides above the training maximum.

The trees have one seed and each split or fold has one graph network, so the variation between training runs is not measured and the intervals do not include it. Three of the 13 structure types of the top split hold 475 of its 530 hydrides. Five of the six intervals on the marked splits contain one half, so only the verdict for the trees on the family split lies outside the resampling error.

What is released

roomtsc 0.1 is the two estimators trained by model/train_final.py with the settings of the tests and nothing held out, as one tree model and three graph networks of different seeds. The weights were written before the second deposit and have not been retrained, so the training set is all 27,151 compounds with their 4,619 labelled hydrides, the 1,093 records that deposit drops among them. The site build publishes the weights, the scripts, the labels, both deposits and the result files of both runs under roomtsc.com/model/ and roomtsc.com/data/. The page roomtsc.com/model/ sends a structure, as CIF or POSCAR text, to roomtsc.com/api/predict, which returns the estimate as JSON.

For any compound the output gives $S$ from the three graph networks, averaged in $\ln S$. For a hydride it adds $h$, the mean of the two estimators in $\ln h$ with the graph part averaged over the three networks. That mean was not deposited, so its scores (the section on the results) are after the fact, and the average of three networks has no score. From $h$ the output forms $\eta_H=\rho_Hh$ and gives its ratio to the 8.7 eV/Ų of the first paper's single-mode requirement for 300 K. It also lists the strength $a_j$ of each hydrogen atom, averaged over the three networks, and no test has scored those values. The field upper_scale_of_Tc_K holds $136.5\ \text{K}\times\sqrt{\eta_H}$, which leaves out the part of $S$ on the other atoms, with 0.35 and 0.58 of it and a note that it is not a calculated transition temperature. The field baselines_eV_A holds the geometric mean of the training labels, 7.23 eV Å, and the electron-gas 36.6, with a note that the estimators failed the extrapolation test.

The 90% range of $h$ is the estimate divided and multiplied by 1.98, the exponential of 0.683, the 90th percentile of the absolute error in $\ln h$ of the mean of the two estimators on the top split of the second run. It was set after that split was scored, from estimators trained with those hydrides held out and one graph network where the release has three. Of the 23 held-out hydrides above the training maximum, 15 lay above the range. The range does not replace the calibration test, which was not run.

The predictor labels the input with its structure type. Where a training compound has the same reduced formula and structure type, the first flag says that the estimate is not out of sample, and the field in_training_set gives that compound's labels. Formula and structure type do not fix the arrangement of the atoms (the section on the labels), so the labels can belong to a different structure.

Every output carries a flag that the labels are harmonic, at one atmosphere and a smearing of 0.030 Ry, and that persistence is not estimated. Every output for a hydride carries another, that the estimators do not predict above the largest training value, 85 eV Å. That wording overstates the second run, in which the graph network placed 2 of the 23 above its training maximum. Six more for a hydride mark a structure type outside the 654 of the 4,619 labelled hydrides, a cell of more than 68 atoms, the largest in training, an element in no labelled hydride, a hydrogen density outside the training range of 0.002 to 0.148 Å⁻³, an estimate above the 99th percentile of the training labels, 30.6 eV Å, and two estimators that differ by more than a factor of 1.5. A cell without hydrogen is not tested for the six. The service refuses partial occupancies, cells of more than 200 atoms and elements absent from the training compounds.

The output is not a transition temperature. The estimate of $h$ carries the error of the tests (Table 1), and the estimators failed the test of finding a compound stronger than the training set. Draft 4 required that a baseline replace an estimator where it fails. The output gives the baselines beside the estimate, leaves the estimate in place, and does not report the marks missed on the top and grouped splits. The released $\eta_H$, which uses the trees, is not bounded by the released $S$. The service returns $h$ for any cell with hydrogen, though labels exist only where the hydrogen branches separate. Every cell it was trained and tested on is a relaxed PBEsol cell of the release, and it does not relax the input, so an estimate for a cell from another source is untested. It does not test whether the structure is dynamically stable or can be made.

A first use: the records without a label

The first paper names its largest gap. Of the hydride records of the release, 84% have imaginary harmonic modes and carry no spectral function, and strong coupling drives phonons toward instability [12], so those records may hold larger $h$ than the labelled ones. We ran the released estimators on every hydride record without a spectral function. There are 32,891, which are 30,917 compounds once the 206 records whose compound is in the training set are set aside and one record is kept for each compound. Four contain only hydrogen, and 91 have a hydrogen density above the largest in training, 0.148 Å⁻³. The statements that follow are for the other 30,822.

Their estimates of $h$ have a geometric mean of 10.7 eV Å. The out-of-sample estimates for the labelled hydrides in the grouped split have 7.4, and their labels 7.2. The estimators therefore place the unlabelled records higher on average, by a factor of about 1.5. The estimate does not follow the depth of the instability: its rank correlation with the lowest harmonic frequency of a record is −0.02.

No estimate of $\eta_H$ reaches the floor of 4.83 eV/Ų below which 300 K is out of reach at any efficiency. Three exceed 3 eV/Ų and 112 exceed 2. Table 4 lists the largest.

Four of the eight have the cubic structure of the labelled RbPtH₆ and RbNiH₆. Three are alkali platinum hexahydrides: a cubic form of KPtH₆, whose labelled record is a rhombohedral distortion of it, and NaPtH₆ and LiPtH₆, which the labelled set does not contain. Their lowest harmonic frequencies are −16, −50 and −185 cm⁻¹, small beside the median of −323 cm⁻¹ among the unlabelled records.

These numbers are estimates and settle nothing about the gap. The estimators were trained on dynamically stable structures only, and for the hexahydrides they have the labelled members of the same family in training, so the large values for KPtH₆, NaPtH₆ and LiPtH₆ repeat what the estimators were shown. They failed the test of estimating above their training range, so the list cannot show that an unlabelled record is stronger than every labelled one. An unstable structure is not a minimum of the energy either. The Hopfield sum is defined whether or not a mode is unstable, and the list gives an order in which to calculate it directly. The three hexahydrides with small instabilities come first, since quantum motion of the nuclei can remove a harmonic instability, as it does in LaH₁₀ [21].

Limits, and what would change the picture

Every score in Table 1 is agreement with harmonic linear-response labels at one smearing, 0.030 Ry, from one group's workflow, and no test shows how far an estimate lies from a converged value. The released estimators are trained on labels up to 85 eV Å, and at a hydrogen density of 0.10 Å⁻³ the first paper's single-mode requirement for 300 K is an $h$ of 87 to 120. Both estimators failed the extrapolation requirement in both runs.

The splits of the first deposit were fixed before training and leaked (Table 2). The second deposit was written with the first results known, so the second run repeats tests whose outcome on the leaking splits we had seen, and we chose what to correct after seeing scores. One of the six verdicts changed between the runs, that of the trees over the grouped folds, so a verdict near one half can depend on how the splits are drawn. We checked the second splits for the two faults we had found, which does not exclude a fault we have not found.

The five hydrides of largest $\eta_H$, whose $h$ both estimators put at 12 to 22 eV Å against labels of 63 to 85, have an $\eta_H$ larger by a factor of 2.4 to 3.7 at a smearing of 0.005 Ry, and at 0.050 Ry their $h$ is 52 to 67 eV Å. Calculations converged in the sampling of the Fermi surface would say how strong these compounds are, and a random sample of such calculations is the reference set that the calibration test lacks.

No estimator was trained or tested on a structure with imaginary harmonic modes, which 84% of the hydride records have. By the sum rule $S$ does not depend on the phonon frequencies, so it is defined for such a structure. Computing it for a random sample of the unstable hydride records would measure the bias of the labelled set, and we have not done so.

The twelve megabar pressure points of the first paper [2] have hydrogen densities of 0.23 to 0.45 Å⁻³, against at most 0.148 among the labelled hydrides, and $h$ of 24 to 45 eV Å, inside the range of the labels. Megabar labels would test the estimators at densities they have not seen.

A transition temperature also needs the efficiency, which is 0.35 to 0.58 for the four ambient-pressure spectra and the five megabar hydrides of the first paper, a factor of 1.7, against the 17 to 22% by which the estimators' errors move the upper scale. The efficiency depends on the phonon frequencies and on how $S$ is divided among the modes, and neither a phonon model nor the closure is built.

Nothing in the model estimates whether a structure can be made or kept at one atmosphere. With the structure type of Mg₂IrH₆ held out, the estimators give Li₂AuH₆ 22.9 and 33.7 eV Å against a label of 33.5, and the $S$ we form from the one path-integral simulation of that compound is a factor of 4.2 below the published linear-response value for the crystal (the section on what is estimated) [4, 10]. An estimate that matches the label describes the static crystal, as the label does.

Appendix: settings, files and a test not yet run

Settings

Both runs of the tests and the released model share the settings of model/train.py. The second manifest records that the trees have at most 600 boosting iterations at a learning rate of 0.04, 24 leaves and early stopping, and that the graph network has width 96, four layers and 32 distance features to 5 Å and is trained for 60 epochs in batches of 96 at a learning rate of 0.002, with one network per split from seed 0, or from the fold index for the folds.

The other settings are in the code, which trains the network with AdamW, a one-cycle schedule and Huber losses on $\ln S$ and $\ln h$ in equal weight. The released networks have seeds 0, 1 and 2. Settings the code does not name are the defaults of scikit-learn 1.9.1 and PyTorch 2.14.1.

The files of the deposits and of the runs

Each deposit is a manifest and two tables. splits.csv has one row for each of the 27,151 compounds, with its prototype or its structure type, its reference values and its side in every split, and the second marks 1,093 rows as dropped. baselines.csv has one row for each held-out labelled hydride of each split and fold in the first deposit, 9,822 in all, and one for each held-out compound in the second, 52,962, with the training mean of $S$. Between the deposits train.py changed only in the choice of device and in storing predictions for the grouped and random splits. Both runs read the labels from a binary file that neither deposit checksums, and the reference values in both splits.csv files equal the label table for every compound.

model/results/ holds the first run, with leak_check.json and the output of the first code test, and model/results2/ the second. The run logs, the output of the code test, the text of draft 4 of the first paper and the commits named in the section on the tests are in the project's git repository. It is not public at the time of writing, and the site build publishes none of them. The first run's top.json and a2mh6.json store predictions by formula without an identifier, which model/leak_check.py recovers from formula and label, and its grouped.json stores scores alone. Each file of the second run stores, for every held-out labelled hydride, its identifier, structure type, label and the five predictions of $h$.

The retrodiction test of a persistence estimator

No persistence estimator is built, so this test has not been run. Table 5 holds its cases as they stood in draft 4 of the first paper, without the column of filter results. Draft 4 also fixed the rule. A phase on the hull of its enumerated competitors at the temperature and pressure of the protocol is predicted to persist. Off the hull it is predicted to persist if path-integral dynamics at that temperature shows no decomposition event and the computed lifetime exceeds the hold time at every prefactor from $10^{10}$ to $10^{13}$ per second, and to be lost if an event occurs or the lifetime falls short at every such prefactor. Any other case is scored as an error.

The mark is twelve of the fifteen counted rows, with no persisting metal predicted lost and no lost hydrogen predicted to stay. The predicted temperature of loss must fall within 25% of the midpoint of 200 to 270 K for ZrH₃ and of 150 K for RhH₂. α-AlH₃ must be predicted lost within its isothermal runs at 333 and at 413 K, and on one ramp the losses must come in the order RhH₂, ZrH₃, AlH₃.

The baseline is the standard pair of filters, an energy above the hull under 100 meV per atom and no imaginary harmonic phonon, which gives one answer for a structure at every temperature and errs on at least four of the fifteen rows. We had read every outcome in the table before the rule was fixed, so a pass would carry little weight.

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