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Let CSP Be Your ANCHOR: Adaptive Crystal Search over Frozen Structure Priors

Emma Lei Hovmand, Jonas Elsborg, Melih Kandemir, Arghya Bhowmik

arXiv:2609.33407v2cond-mat.mtrl-scics.LGphysics.comp-ph

TL;DR

DNG models jointly choose compositions and generate structures, making it difficult to separate where discovery rewards should act. The paper introduces ANCHOR, which trains a composition policy around a frozen CSP model and uses history-aware novelty. Under a shared evaluator, composition-policy substitution improves discovery metrics, while reward fine-tuning through the denoising chain does not raise on-hull fractions.

  • Problem

    DNG jointly learns composition choice and structure generation, so crystal-level rewards can alter both even though discovery chiefly requires deciding what chemistry to investigate.

    Source: §1 · pp. 1–2

  • Method

    ANCHOR trains a GRPO composition policy with discovery rewards around a frozen CSP model, using continuous adaptive novelty against known structures and search history.

    Source: §1 · p. 2, §3 · p. 5

  • Results

    Replacing DNG compositions with ANCHOR’s policy on the same backbone raises MSUN more than 3-fold and SUN more than 10-fold; denoising-chain fine-tuning does not raise on-hull fractions.

    Source: §1 · pp. 2–3

  • Takeaways & Limitations

    Discovery rewards, including history-dependent novelty, belong in the composition policy, while likelihood training on structures found during search can improve the structural prior.

    Source: §1 · p. 2, §5 · p. 9

    On-hull rates depend on the potential: ANCHOR’s stability estimates rely on MatterSim, and evaluation under a different potential, relaxation, and hull can reverse the ranking.

    Source: §4 · p. 9, §5 · p. 10

Abstract

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De novo crystal generation (DNG) models decide where to search in composition space and how to generate structures with one set of weights. We argue that discovery is better served by separating the two. A crystal structure prediction (CSP) model is a physical prior that should be improved by likelihood training, while rewards, including novelty measured against the search's own history, should act on a search over compositions. We introduce ANCHOR, a GRPO composition policy trained with multi-objective rewards around a frozen CSP model, and continuous adaptive novelty (CAN), a graded novelty score against known structures and a growing discovery history. Using the frozen CSP model as a fixed ruler under one evaluator, we test where adaptation should act. Replacing DNG compositions with ANCHOR's policy on the same CSP backbone raises MSUN from 11.4% to 47.6% and SUN from 1.1% to 22.1% at 99.9% formula uniqueness. Fine-tuning DNG models directly on the same rewards instead moves their composition marginal without raising their on-hull fraction. We show that KL-regularized fine-tuning of a DNG model can only reweight chemistry the pretrained model already supports by a bounded factor, while unregularized DNG fine-tunes move toward known or less stable chemistry. Even a stability-only reward routed into ANCHOR's CSP backbone roughly halves SUN relative to the frozen backbone, whereas likelihood training on structures found during search can improve a CSP backbone. Under MatterGen's evaluation pipeline, ANCHOR raises state-of-the-art MSUN from 29.2% to 41.3%, transfers without retraining to two further CSP backbones, and reaches 47.1% after distillation into Crystalite-CSP. As with any model optimised against a potential, its on-hull rate depends on that potential.

1 Introduction & Motivation

The paper separates composition search from structure generation: discovery rewards update a composition policy while a frozen CSP model supplies structures. ANCHOR and continuous adaptive novelty (CAN) implement this separation and improve discovery metrics in the reported comparisons.

Motivation

  • DNG models jointly adapt composition and structure, although discovery mainly requires adapting which compositions are searched.A fixed novelty reference rewards rediscovery, while history-dependent novelty makes structural adaptation respond to campaign history rather than physics.Source: §1 · pp. 1–2

Composition search over a structure prior

  • ANCHOR replaces DNG’s likelihood-trained composition marginal with a GRPO policy querying a frozen CSP model, so rewards change where to search rather than how structures are generated.The CSP model is treated as a physical prior, while discovery signals, including history-dependent novelty, act on the composition policy.Source: §1 · p. 2

Continuous adaptive novelty

  • CAN’s graded and adaptive components together raise SUN from 8.9% with a binary novelty flag to 22.1%.CAN scores against known structures and the search’s own history, so a found crystal stops earning reward.Source: §1 · p. 2

Separation of composition and structure effects

  • Replacing DNG compositions with ANCHOR’s policy on the same CSP backbone raises MSUN more than 3-fold and SUN more than 10-fold under one evaluator.Fine-tuning through the denoising chain does not raise the on-hull fraction, while likelihood training on discovered structures raises Crystalite-CSP SUN from 20.4% to 26.0% on the same compositions.Source: §1 · pp. 2–3

State-of-the-art MatterGen results and transferability

  • 41.3% MSUN versus 29.2% for the previous best under MatterGen’s pipeline; ANCHOR reaches 47.1% after likelihood distillation into Crystalite-CSP.It also transfers without retraining to two further CSP backbones.Source: §1 · p. 3

2 Background

Crystal generation can be framed as choosing a composition and generating a structure conditional on it, but joint generators learn both together. Prior reward-guided approaches adapt the joint generator, leaving composition and structural effects entangled.

Crystal generation and evaluation

  • DNG generates composition and structure jointly, whereas CSP generates structures for a fixed composition; benchmarks commonly summarize performance with SUN or MSUN.A crystal structure specifies a lattice, fractional coordinates, and atom species consistent with its composition.Source: §2 · p. 3

Novelty and continued discovery

  • Conventional novelty uses a fixed reference collection, while uniqueness measures repetition among generated samples; continuous structural metrics remain fixed-reference measures.They therefore do not remember structures found during an ongoing search.Source: §2 · p. 3

Reward-guided crystal generation

  • Reward-guided methods can shift generated compositions away from the likelihood-trained marginal, but adapt the joint crystal generator rather than isolating composition choice.For terminal rewards on decoded crystals, assigning reward across prior denoising decisions creates a temporal credit-assignment problem.Source: §2 · p. 3
  • Strong KL regularization can restrict reward-driven drift toward a creativity objective, but joint optimization still obscures whether adaptation belongs in composition or structure.Reward-driven diffusion fine-tuning can reduce diversity and sample quality, and novelty objectives can favor regions with lower pretrained probability.Source: §2 · p. 3

3 Methods

ANCHOR factorizes search into a composition policy and a frozen structural conditional, directing discovery rewards to composition choice while retaining the CSP model as the structure generator. Its rollout rewards combine stability, novelty, and diversity, with adaptive novelty tracking both known structures and search history.

Controls and reward fine-tuning

  • A frozen CSP model acts as a fixed ruler: holding compositions fixed isolates structural generation, while holding the CSP fixed isolates composition allocation.KL-regularized reward fine-tuning can reweight chemistry already proposed by the pretrained model by at most e^∆R/β; removing KL also removes the conditional constraint.Source: §3 · p. 4

Separating discovery from structure generation

  • ANCHOR replaces the pretrained composition marginal with an adaptive policy and freezes the structural conditional, so reward updates cannot tilt the conditional model.The frozen CSP can still sample different structures at a composition and can be improved between searches through likelihood training on found structures.Source: §3 · pp. 4–5

Discovery as composition-level RL

  • Each rollout selects a composition specifying element count and identities, cell size, and stoichiometry; the frozen CSP samples a structure that is relaxed before scoring.GRPO standardizes each reward within a group of sampled actions.Source: §3 · p. 5

Continuous Adaptive Novelty

  • CAN compares an RDF embedding against same-formula structures in both a fixed reference and growing search history, taking the lower novelty score.After scoring, the structure joins the history, so regenerating a previously found structure drives its novelty to zero.Source: §3 · p. 5

Reward and multi-objective GRPO

  • Valid candidates receive stability, CAN, and structural and compositional diversity scores; a count bonus separately penalizes revisiting a reduced formula.The reward scalarization and element subgroup are sampled per rollout and held fixed across its action group.Source: §3 · p. 5

Exploration-mechanism ablation

  • The exploration ablation evaluates the full method, mechanisms removed, mechanisms alone, and a binary-novelty variant across matched-wall-clock runs.The caption specifies percentage columns except MP-20 counts, with means and standard deviations over three seeds.Source: Table 1 · p. 6

4 Experiments

Experiments use controlled evaluations to separate composition-search effects from structural-model effects. ANCHOR improves discovery under MatterGen, while reward fine-tuning and structural adaptation reveal limits tied to composition and the evaluation potential.

Setup and evaluation

  • 22.1% SUN for the full method versus 16.3% for adaptive novelty alone; replacing history with MP-20 lowers SUN to 14.7%, while binary novelty lowers it to 8.9%.The full method’s MSUN is 4.4 points lower than adaptive novelty alone; the policy proposes 97,599 distinct reduced formulas among 101,237 valid structures.Source: §4 · p. 6
  • Replacing DNG compositions with ANCHOR’s policy raises MSUN by 32–36 points and SUN by 11–20 times, while swapping only the structure generator changes either metric by at most 1.0 point.Uniformly sampled charge-neutral formulas reach 8.8% MSUN, versus 1.5% for random formulas, so charge neutrality alone does not account for the policy’s discovery rate.Source: §4 · p. 6, Table 2 · p. 7
  • Under the KLDM-trained policy, substituting Crystalite-CSP or OMatG-CSP preserves the same high discovery rate, showing transfer across structural priors without retraining; DNG-composition rows remain at or below 2.5%.Source: §4 · p. 7
  • Reward fine-tuning raises Crystalite-DNG validity and Meta without improving its metastable fraction, while reducing on-hull stability and formula uniqueness.Under MatterGen, both fine-tunes lower Meta to 51.9% from 57.0% and Stable to 4.3–4.4% from 7.2%; frozen-CSP tests attribute the stability loss primarily to compositions.Source: §4 · pp. 7–8, Table 3 · p. 8
  • Stability-only reward updates to the CSP backbone reach 29.5% MSUN and 10.7% SUN, still roughly half the frozen-backbone SUN of 22.1%.Likelihood training with structures found during search raises Crystalite-CSP internal MSUN/SUN from 46.7/20.4% to 52.8/26.0%, but leaves KLDM-CSP unchanged in both metrics.Source: Table 2 · p. 7, §4 · pp. 8–9
  • On-hull ranking reverses across evaluators: ANCHOR→KLDM-CSP falls from 13.6% SUN in the internal pipeline to 0.4% under LeMat-GenBench.The potential, relaxation, hull, and validity gate alter measured outcomes, so on-hull performance depends on the scorer used.Source: §4 · p. 9

5 Discussion

The discussion argues for separating composition search from structural learning: rewards guide composition proposals, while likelihood training improves the structural prior. The authors also note that stability estimates depend on the potential used.

5 Discussion

  • Rewards, including history-dependent novelty, belong in the composition policy, while likelihood training on discovered structures can improve the structural prior.The discussion reports that reward gradients through the denoising chain do not improve structures in these experiments, even when the reward measures only stability.Source: §5 · p. 9
  • Stability estimates rely on MatterSim, and evaluation with a different potential ensemble and validity gate is only a partial check; DFT validation remains a natural extension.The authors state that better structural priors can be substituted directly as potentials and CSP models improve.Source: §5 · p. 10

Software and Data

  • The paper states that ANCHOR code and associated experiments will be released with the article when accepted for publication.Backbones, reference datasets, and interatomic potentials used in the paper are publicly available and cited.Source: Software and Data · p. 11

A Notation

  • Table 4 defines symbols for the composition policy, reward, and evaluation; rates are fractions of an evaluation batch of 1,000 structures unless stated otherwise.Generative-model notation is defined in Section 2.Source: Table 4 · p. 15

B Additional Related Work

Related work spans crystal-generation models, reward-guided optimization, and history-dependent exploration. The paper distinguishes ANCHOR by using a composition policy as an adaptive search over a separately pretrained, frozen structural generator.

Crystal-generation tasks and benchmark datasets

  • CSP searches for structures at a fixed composition, whereas de novo crystal generation explores composition and structure jointly.Crystal representations use atomic species, fractional coordinates, and lattice vectors in a periodically repeated unit cell.Source: B Additional Related Work · p. 16

Generative models for crystalline materials

  • The related work describes crystal-generation advances from periodic-geometry models such as CDVAE and DiffCSP to property-conditioned, controllable, flow-based, and Transformer-based approaches.KLDM models fractional coordinates on the hypertorus through kinetic Langevin dynamics and serves as this work’s primary structural backbone.Source: B Additional Related Work · p. 16

Novelty and structural similarity

  • Novelty asks whether a structure is absent from a fixed reference collection, while uniqueness measures repetition among generated samples; neither captures discoveries made during continuing search.Larger reference collections can change whether a candidate counts as novel, and continuous similarity measures distinguish near-repeats from more different structures.Source: B Additional Related Work · p. 16

Related work

  • A concurrent approach reports higher novelty alongside lower stability after recombining local-environment codes, but its scores use MP-2023 and are not directly comparable to MatterGen-pipeline results.Its intervention is a structural conditioning signal rather than a reward.Source: B Additional Related Work · pp. 16–17
  • Prior materials-search work uses search history to discourage revisiting candidates, and reinforcement learning has history-dependent exploration mechanisms such as count bonuses and curiosity.These approaches allow exploration objectives to depend on search state rather than only on a fixed external dataset.Source: B Additional Related Work · p. 17
  • Reward-guided crystal-generation methods modify the generator responsible for producing structures, whereas ANCHOR adapts only composition search and keeps the conditional structural distribution fixed.Composition-level reinforcement learning itself is not the paper’s central distinction; the policy’s role as an outer search over a frozen generator is.Source: B Additional Related Work · p. 17

C Composition Search, Novelty, and the Structural Prior

This section formalizes composition-only adaptation around a fixed conditional structural prior, explaining how joint reward-guided fine-tuning can alter both components and specifying the composition search and structure-evaluation path. It then describes ANCHOR’s validity, stability, novelty, and diversity scores and its training around a frozen CSP backbone.

C Composition Search, Novelty, and the Structural Prior

  • ANCHOR adapts composition queries while keeping the CSP conditional fixed, directing search toward productive chemistry even when its likelihood-trained composition marginal assigns it little probability.Repeated queries can still sample different structures from the frozen CSP distribution.Source: C Composition Search, Novelty, and the… · pp. 18, 22
  • Joint reward fine-tuning can change both composition and structure, while KL regularization couples their adaptation and can only reweight chemistry already supported by the pretrained marginal.For bounded rewards, the marginal retains the pretrained support and each composition's reweighting is bounded by e^ΔR/β; stronger joint KL also restricts compositional movement.Source: C Composition Search, Novelty, and the… · pp. 20–21
  • Novelty is a search-utility signal, not a measure of structural plausibility: history can make an already discovered structure less useful to resample without changing its physical quality.An illustrative likelihood-based novelty reward can flatten or reverse the pretrained structural distribution, and low likelihood cannot distinguish a useful polymorph from an implausible structure.Source: C Composition Search, Novelty, and the… · pp. 19, 21

D.1 Element Vocabulary and Action Space

  • The policy uses 84 elements and searches roughly 10^9 charge-neutral compositions, while a training run evaluates about 10^5 candidates.Actions contain 2–4 distinct elements, with cell size from k to 20 atoms and at most 12 atoms per element; the five noble gases are excluded.Source: §D.1 · p. 23

D.2 Structure Generation, Relaxation, and the Validity Gate

  • Each composition query samples from the frozen KLDM CSP backbone, then relaxes positions with BFGS under MatterSim at fixed lattice; neither model receives gradient updates.The validity gate includes a minimum-bond-length check, rejecting any structure with a pair closer than its element-pair threshold.Source: §D.2 · pp. 23–24
  • Invalid structures receive a fixed penalty and skip stability and novelty, but remain in group-diversity calculations and can lower valid siblings’ diversity scores.Source: §D.2 · p. 24

D.3 Stability Score

  • The stability score maps Ehull to a bounded higher-is-better scale: 1 on the hull and 0 at Ehull ≥1 eV/atom.Ehull is computed after fixed-lattice MatterSim relaxation and MP2020 compatibility against the MatterGen Alex-MP reference hull; most ANCHOR relaxations hit the 100-step cap, making stability a within-group ranking signal rather than a converged Ehull.Source: §D.3 · p. 24

D.4 Continuous Adaptive Novelty and the Count Bonus

  • CAN gives each valid structure graded within-formula novelty against MP-20 and the agent’s growing history, while a count bonus shifts effort away from repeatedly proposed formulas.The count bonus and CAN address formula-level repetition and structural rediscovery, respectively; the separate uniqueness term is omitted. First-seen formulas receive a 0.75 bootstrap novelty score, leaving headroom for later distinct structures.Source: §D.4 · pp. 25–26

D.5 Within-group Diversity Kernels

  • The within-group diversity reward favors structures differing from rollout siblings using RDF structural similarity and Jaccard similarity of coarse chemical-role sets.Unlike CAN, these diversity terms compare only within the rollout and carry no memory across steps.Source: §D.5 · pp. 26–27

D.6 Policy Parametrization and Optimization Details

  • ANCHOR updates an autoregressive composition policy conditioned on shared objective weights and a sampled element subgroup, while the CSP backbone and reward evaluator remain frozen.The entropy schedule is keyed to cumulative structures generated, not optimizer steps, so runs with different group sizes receive the same entropy pressure at the same sample budget.Source: §D.6 · pp. 27–29

D.7 Training Algorithm

  • Training alternates shared-context composition rollouts, frozen-backbone generation and scoring, then K clipped policy-gradient epochs using group-relative advantages.Only the update phase changes the policy parameters; gradients do not pass through the backbone, relaxer, or reward.Source: §D.7 · p. 29

D.8 Training Setup and Hyperparameters

The setup fixes the CSP backbone and distinguishes the training potential from the evaluation potential. Runs share a seven-day A100 limit, with generated structures used to compare training budgets.

D.8 Training Setup and Hyperparameters

  • Training uses a frozen KLDM CSP checkpoint, 50-step Euler–Maruyama sampling, and MatterSim-1M relaxation over atomic positions for up to 100 BFGS steps.Each formula produces one structure.Source: §D.8 · p. 29
  • MatterSim-1M supplies the training stability reward, while MatterSim-5M evaluates internal tables; LeMat-GenBench and MatterGen use their respective default pipelines.The evaluation potential remains fixed across internal tables.Source: §D.8 · p. 29
  • Each run used one A100 for a seven-day wall, so runs are compared by structures generated rather than wall-clock time or optimizer steps.The selected configuration generated about 123k structures, while most other cells generated 102–131k.Source: §D.8 · p. 29
  • The reported accounting includes 19 reinforcement-learning runs, likelihood training, fine-tunes, corpus generation, and evaluation, totaling roughly 4,200 A100 GPU-hours.The total excludes unreported exploratory runs.Source: §D.8 · p. 30
  • The entropy schedule targets 105 structures, but the seven-day cutoff means actual consumption varies; Table 8 reports each policy’s generated-structure count.The selected base model is G=32 / MatterSim-1M.Source: §D.8 · p. 29, Table 8 · p. 31

E.1 Internal Evaluation Protocol

The protocol evaluates fresh stochastic samples with a fixed evaluator, while training-time novelty uses a growing search history. Shared row conventions distinguish composition policies, structure generators, and reward-fine-tuned models.

E.1 Internal Evaluation Protocol

  • Headline internal values come from fresh samples scored after training with a common MatterSim-5M evaluator, rather than from path-dependent training logs.Main-text internal tables relax positions and lattice; lattice-fixed appendix tables hold the lattice fixed, and neither protocol matches the MatterSim-1M, 100-step training reward.Source: §E.1 · pp. 31–32
  • Each policy samples 1,000 structures per seed stochastically, using mean training objective weights and the full 84-element vocabulary.Main-text lattice-free results average three sampling seeds; lattice-fixed appendix results use seed 0.Source: §E.1 · p. 31
  • Unique is distinct reduced formulas divided by N; Novel requires validity and ν(x) > 0.5; Stable requires validity and Ehull ≤ 0; Meta requires validity and Ehull ≤ 0.1 eV/atom.MP-20 hits count structures whose reduced formula occurs in MP-20.Source: §E.1 · p. 32
  • At evaluation, novelty is the graded kNN-RBF score against frozen MP-20 alone; the adaptive search-history index is switched off.With history included, an ablation that repeats a handful of formulas is 99.9% novel against MP-20 alone but only 0.9% novel against MP-20 plus history.Source: §E.1 · p. 32

E.1.1 Shared rows and reading conventions

  • ANCHOR→X denotes one composition policy queried through generator X, whereas trainable-backbone rows update the CSP model and FT rows reward-fine-tune DNG generators.FT→CSP rows use the fine-tuned model’s compositions with a frozen CSP model of the same family; missing or non-final entries are not ranked.Source: §E.1.1 · pp. 32–33

E.1.2 Internal results

Across internal protocols, frozen-backbone composition search outperforms reward-trained backbones and DNG fine-tunes on on-hull discovery. Validity gates and relaxation choices change measured values, while relaxation budgets impose a cost and convergence trade-off.

E.1.2 Internal results

  • 38.5% MSUN for the frozen-backbone policy exceeds 19.4% and 15.0% when the KLDM-CSP backbone receives the shared reward or stability-only reward, respectively, with lattice fixed.Reward-fine-tuned DNGs reach 96.4% validity but only 40.1–50.0% formula uniqueness and 0.2–0.4% SUN; conclusions persist with lattice relaxation.Source: §E.1.2 · p. 33

E.1.3 Sensitivity to the validity gate

  • Changing only the validity gate shifts ANCHOR→KLDM-CSP MSUN from 38.5% to 25.0%, while Crystalite-DNG→CSP rises from 12.4% to 23.6%.The policy trained against the LeMat gate reaches 98.3% validity under that gate but 0% SUN under the unchanged MatterSim hull.Source: §E.1.3 · pp. 33–34, Table 10 · p. 34

E.1.4 Sensitivity to lattice relaxation

  • Freeing the lattice raises stability-derived values without changing the main conclusions; the frozen backbone still outperforms reward-trained backbones, and every reward fine-tune lowers its DNG model’s on-hull fraction.The lattice-fixed comparison is 38.5 > 19.4 > 15.0% MSUN; the free-lattice comparison is 47.6 > 29.5 > 22.6%.Source: §E.1.4 · pp. 34–35

E.1.5 Relaxation budget

  • Freeing the lattice costs 1.28× the force calls and 1.6–1.9× the wall clock at 400 steps; at 100 steps, ANCHOR convergence falls from 16.4% to 6.5% on KLDM and from 27.7% to 9.4% on Crystalite.Restoring convergence requires 400 steps, at which 99.4–99.5% of cell-free ANCHOR relaxations converge; on ANCHOR→Crystalite, that takes 2.4× the positions-only wall clock.Source: §E.1.5 · p. 35, Table 11 · p. 35
  • The 100-step positions-only training reward is truncated: only 16.4% of ANCHOR→KLDM relaxations converge, so its stability term ranks candidates using energy along incomplete relaxations.Reported evaluation values use longer relaxation, where at least 98.8% of structures converge.Source: §E.1.5 · p. 35

E.2 MatterGen Evaluation Pipeline

MatterGen’s pipeline reproduces the qualitative pattern that ANCHOR’s discovery advantage is not a higher raw metastable fraction. Its metrics and validity rules differ from the internal protocol, so values are pipeline-specific.

E.2 MatterGen Evaluation Pipeline

  • ANCHOR’s 42.3–43.2% Meta is below Crystalite-DNG’s 54.8–57.3%, yet its 100% Unique and Novel rates make MSUN equal its raw metastable rate.Only 22.9–23.6% of Crystalite-DNG samples survive the unique-and-novel conjunction; reward fine-tuning lowers DNG Stable and MSUN, and CSP replacement changes MSUN by at most 0.8 points.Source: §E.2 · p. 35
  • MatterGen evaluates composition and structure validity independently, using a SMACT charge-neutrality and electronegativity check for compositions and minimum-distance and cell-volume checks for structures.The composition check exempts unary cells and all-metal alloys.Source: §E.2 · p. 36
  • MatterGen’s Unique and Novel flags are structure-level measures applied to every scored cell, and its S.U.N. is unique ∧ novel ∧ Ehull ≤0.1 eV/atom rather than a validity-filtered count.The pipeline uses MatterSim-v1-1M, FIRE with ExpCellFilter, and the Alex-MP reference_MP2020correction hull.Source: §E.2 · p. 36, Table 12 · p. 36
  • Cells containing elements without terminal entries in the reference hull are removed before scoring, so the pipeline may score fewer than 1,000 structures.The same discard precedes published results on this pipeline.Source: §E.2 · p. 36, Table 12 · p. 36

E.3 LeMat-GenBench Evaluation

LeMat-GenBench uses its own validity checks, reference hulls, novelty definition, and structure-level deduplication, so its absolute stability and validity rates are not directly interchangeable with internal results. Changing the hull alone substantially shifts stability rates.

E.3 LeMat-GenBench Evaluation

  • Against the Bulk hull, frozen-backbone ANCHOR variants have 0% SUN and 2.9–3.1% MSUN, while Crystalite-DNG variants have 1.4–2.3% SUN and 15.8–20.4% MSUN.Training against the LeMat validity gate raises MSUN to 26.0% but still yields 0% SUN.Source: §E.3 · p. 36, Table 14 · p. 38
  • Switching only to the MP-20 hull changes ANCHOR→KLDM-CSP from 2.9% to 7.8% MSUN and from 0% to 0.4% SUN; Crystalite-DNG changes from 19.6% to 30.0% MSUN and from 1.4% to 9.8% SUN.Under the published MP-20 protocol, DNG SUN is 9.8–10.9% versus 0.4–0.8% for ANCHOR trained under the internal evaluator.Source: §E.3 · p. 37
  • LeMat validity requires three independent checks, with a charge-neutrality cascade that accounts for essentially all rejected structures in these submissions.Its distance threshold varies by atom pair, and its plausibility screen covers bulk and cell quantities.Source: §E.3 · p. 37
  • LeMat deduplicates structures rather than reduced formulas and computes Ehull from a three-potential ensemble against its own hull, unlike the internal single-potential evaluation.Its novelty is also a structure-level match against its reference corpus; the metallicity exemption could bias RL toward the exempt set rather than charge-neutral formulas.Source: §E.3 · pp. 37–38
  • The benchmark supports both the LeMat-Bulk and MP-20 reference hulls, which are not interchangeable.Table 14 uses the Bulk hull and Table 15 the MP-20 hull.Source: §E.3 · p. 38

E.3.1 Training against the LeMat validity gate

Training against the LeMat validity gate produces high scores under that gate but few structures pass the paper’s own gate, while the reported model ranking depends on the gate and stability threshold. The base-model comparison also cautions that late-window scores from different training-time potentials are not independent corroboration.

E.3.1 Training against the LeMat validity gate

  • 98.3% validity under the LeMat gate falls to 5.6% under the paper’s gate on the same structures.The benchmark’s metallicity heuristic exempts compositions classified as metallic by assigning every atom oxidation state zero rather than testing neutrality.Source: §E.3.1 · p. 39
  • 28.1% MSUN under the LeMat gate in our pipeline and 32.5% on LeMat-GenBench exceed every DNG model, but produce almost no on-hull structures under either evaluator.The authors therefore use SUN at Ehull ≤0 as the headline and treat MSUN as secondary.Source: §E.3.1 · p. 39
  • Crystalite-DNG reaches 9.8% SUN and 30.0% MSUN against the MP-20 hull, while ANCHOR reaches 0.4% SUN and 7.8% MSUN at 73.0% validity under a different gate, relaxer, and hull.The authors conclude that which model leads depends on the benchmark’s gate and threshold, including their own.Source: §E.3.1 · p. 39

F.1 Base-model Selection

  • Table 18’s late-window scores use each run’s own training-time potential, confounding the 1M-versus-5M comparison with grader leniency; the authors rely on shared-potential evaluation for selection.Table 15 reports LeMat-GenBench evaluation against the MP-20 ensemble hull, with only the reference hull changed from Table 14.Source: Table 15 · p. 39, §F.1 · p. 41

F.2 Invalid-structure Penalty Sweep

The invalid-structure penalty acts less like a validity-rate knob than an exchange rate between validity and discovery objectives. Across the sweep, validity and uniqueness remain high, while SUN and MSUN vary more and the authors select −0.2 without claiming an interior optimum.

F.2 Invalid-structure Penalty Sweep

  • A more negative penalty inflates within-group reward variance, compressing normalized differences among valid structures and spending update signal on validity rather than discovery.Invalid structures score below zero while valid structures score in [0, 1].Source: §F.2 · p. 42
  • Across a five-fold penalty range, validity moves only 1.3 percentage points (96.6–97.9%) and non-monotonically, while charge neutrality passes at 99.5–99.9% and bond-length plausibility at 96.7–98.4%.The text attributes this insensitivity to the frozen backbone and validity gate, with nearly all failures geometric.Source: §F.2 · p. 42
  • The sweep shows no retreat into familiar chemistry: formula uniqueness stays 100.0% across all three penalties, although removing a load-bearing exploration mechanism collapses it to 1.0%.A penalty set too high could plausibly suppress exploration into unfamiliar chemistries without an obvious validity-rate warning.Source: §F.2 · p. 42
  • SUN and MSUN vary more than validity or uniqueness across the penalty sweep, ranging from 8.7–18.1% and 22.0–38.5%, respectively, with both peaking at −0.2.The authors use −0.2 but do not interpret three single-seed points as an interior optimum.Source: §F.2 · p. 43

F.3 Exploration-mechanism Ablations

The ablations distinguish exploration mechanisms, learned composition preferences, and reward fine-tuning of DNG models. They show that graded, adaptive novelty and composition selection support discovery, while optimizing a generator directly can trade away formula uniqueness.

F.3 Exploration-mechanism Ablations

  • The all-off cell has 68.7% of samples at Ehull ≤0.1 but proposes only 10 distinct reduced formulas in 1,000 samples.Its stability at Ehull ≤0 is 60.0%, against 18.1% for the full method; per-structure stability must be read alongside diversity.Source: §F.3 · p. 43
  • MSUN is 37.1% with adaptive novelty alone, versus 28.3% with the count bonus and 2.0% with the sub-group mask alone.The count bonus alone preserves 100.0% compositional uniqueness, while the mask alone yields 6.3%; the two depth mechanisms can partly substitute for each other when combined.Source: §F.3 · pp. 43–44
  • The ablation runs share a wall-clock stop, but collapsed cells consumed more structures than the full method: 648k for all-off versus 102–131k elsewhere.Despite more training, collapsed cells produced two orders of magnitude fewer distinct formulas.Source: §F.3 · p. 43
  • Across one 123,968-structure run, collisions with the agent’s own history fell from 1,078 in the first fifth to 635 in the last as the history grew.These descriptive counts show the adaptive branch was active, not that it caused downstream gains.Source: §F.3 · p. 44
  • When binary novelty is constant across a GRPO group, it cancels from the group-relative advantage, leaving validity and the count bonus to train the policy.In this search outside MP-20, same-formula novelty tests cannot fail once the reference list is empty; the resulting cell retains 99.9% uniqueness but scores below configurations retaining continuous novelty.Source: §F.3 · p. 44

F.4 Learned Composition Preferences

  • ANCHOR reaches 97.9% Valid and 38.5% MSUN, versus 70.1% Valid and 3.1% MSUN for charge-neutral enumeration.All three composition sources remain 100% unique, so the gain is not explained by collapse or charge-neutrality learning alone.Source: §F.4 · p. 45, Table 21 · p. 45

F.5 Reward Fine-tuning Baselines

  • Reward fine-tuning reduced Crystalite-DNG uniqueness from 98.8% to 40.5% (mean) and 49.5% (bal.), while KLDM-DNG stayed above 99%.DDPO assigns the terminal reward to scored denoising steps; the reported samples are from a pinned snapshot after about 4.1k episodes.Source: §F.5 · pp. 45–46
  • For reward-trained CSP rows, GRPO updates all 14.9M KLDM parameters alongside the composition policy, without a divergence penalty to pretrained weights.The backbone is held fixed for the first 50 policy steps, then updated within the nested per-step objective.Source: §F.5 · pp. 45–46

F.6 Training the CSP Backbone on ANCHOR Structures

The authors test whether training CSP backbones on structures found by ANCHOR improves their discovery performance. The corpus raises SUN for Crystalite-CSP but not for KLDM-CSP, with geometry source also affecting the outcome.

F.6 Training the CSP Backbone on ANCHOR Structures

  • The 27,136-structure ANCHOR corpus is added to MP-20 and compared with step-matched controls trained on pristine MP-20.Each backbone is evaluated on the same 1,000 held-out ANCHOR compositions; the matched design isolates the corpus from extra optimization.Source: Table 23 · p. 47, §F.6 · p. 47
  • The corpus preserves the trained policy’s composition distribution, not the individual cells seen during training: logged compositions are replayed through the frozen backbone to obtain cells.The sampled corpus uses 27,136 valid, deduplicated candidates drawn uniformly from the run, without stability or novelty thresholds; this dilutes it relative to late-run sampling.Source: §F.6 · p. 47
  • Crystalite-CSP gains 10.1 SUN points with finetuning (14.4 →24.5) and 7.6 with retraining (13.4 →21.0), while KLDM-CSP is flat to slightly negative.On KLDM-CSP, finetuning changes SUN from 17.1 to 16.5 and retraining from 17.9 to 17.1; rendering the compositions with Crystalite instead moves it from 16.5 to 18.9, while removing relaxation changes it from 16.5 to 16.7.Source: §F.6 · pp. 47–48, Table 23 · p. 47
  • The diagnostic comparison indicates that which model produced the geometries matters more than whether the cells were relaxed.The authors note that the arm that fails to improve is also the only one trained on cells from its own backbone, but the backbones differ enough that the Crystalite gain cannot be attributed to that alone.Source: §F.6 · pp. 47–48

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