{
"markdown": "\n \n \n\n\n# hea-bench\n\n\n\n[](https://doi.org/10.3390/ma19143075)\n[](https://doi.org/10.5281/zenodo.20346287)\n[](https://pypi.org/project/hea-bench/)\n[](https://pypi.org/project/hea-bench/)\n[](https://github.com/dfieser/hea-bench/actions/workflows/ci.yml)\n[](./LICENSE)\n\nOpen, interpretable tools for computing the standard **high-entropy-alloy\n(HEA) and high-entropy-oxide (HEO) thermodynamic and geometric\ndescriptors** and the classic empirical **phase-prediction rules**, from\nany composition, with no fitted model and no black box. Every number is a\ntransparent closed-form expression over a curated element-property table,\nvalidated against the primary literature.\n\n**Try it now:** . No install, it runs\nentirely in your browser.\n\n\n \n \n\n\nThe equimolar Cantor alloy CoCrFeMnNi, as the browser app reports it.\nThe Python library, the desktop app and this page print the same digits, and\na parity suite keeps it that way.\n\n> **Using an AI coding agent to integrate this?** See\n> [AGENTS.md](./AGENTS.md) for a machine-oriented guide to the API,\n> exact return types and units, the fastest path to each task, and the\n> mistakes to avoid.\n\n## What it computes\n\nFor any composition it reports:\n\n- **Core descriptors:** mixing entropy ΔSmix, atomic-size\n mismatch δ, mean melting temperature Tm, Miedema mixing\n enthalpy ΔHmix, valence-electron concentration VEC,\n Yang–Zhang Ω, Pauling electronegativity mismatch Δχ, Mansoori excess\n entropy SE, ΔGss, ΔGmax, King Φ, Ye φ.\n- **Phase-prediction rules:** Yeh entropy, Zhang δ, Guo–Liu VEC,\n Yang–Zhang Ω, King Φ, Ye φ.\n- **Miedema formation enthalpies** (browser/desktop apps): compound /\n solid-solution / amorphous, decomposed into chemical, elastic,\n structural, and topological terms.\n- **High-entropy oxides** (`hea_bench.oxides` + the apps' Oxides mode):\n rock-salt, perovskite, fluorite, and pyrochlore formability\n descriptors over Shannon ionic radii with automatic charge-balance\n oxidation-state assignment: per-sublattice configurational entropy,\n cation size disorder, Goldschmidt t / octahedral μ / Bartel τ, the\n fluorite radius-dispersion rule, and the pyrochlore radius-ratio\n window.\n\nElement coverage: 55 elements for alloys (Ag Al Au Be Bi Ca Ce Co Cr\nCu Dy Er Fe Ga Gd Ge Hf Ho In Ir La Li Lu Mg Mn Mo Nb Nd Ni Os Pb Pd\nPr Pt Re Rh Ru Sb Sc Si Sm Sn Sr Ta Tb Th Ti Tm U V W Y Yb Zn Zr,\ncovering the full experimentally active rare-earth HEA palette plus the\nnuclear, solder, and HE-BMG corners); the Miedema pair table covers\n75 (1484 of our 1485 pairs; the lone Th-U gap is reported, never\nzeroed); the oxide module's Shannon table covers 94.\n\n## How a number gets made\n\nNo fitted model sits anywhere in this chain. Each descriptor is a\nclosed-form expression over curated tables, and the report carries the\nliterature source of every input alongside the value.\n\n```mermaid\nflowchart LR\n A[\"Composition CoCrFeMnNi\"] --> B[\"Curated element tables 55 elements, 1484 Miedema pairs\"]\n B --> C[\"Closed-form descriptors ΔS, δ, VEC, ΔH, Ω, Φ, φ, Λ, γ, κ\"]\n C --> D[\"Empirical phase rules Yeh, Zhang, Guo-Liu, Yang-Zhang, King, Ye\"]\n C --> E[\"Report per-value provenance, content-hashed result ID\"]\n D --> E\n```\n\n## Four ways to run it\n\n| Surface | Where | Status |\n|---|---|---|\n| **Python library + CLI** | `pip install hea-bench` | done, tested |\n| **Zero-install browser app** | · `web/index.html` | done, Python-parity-tested |\n| **Native desktop app** | a single portable `.exe`, [download (no install)](https://github.com/dfieser/hea-bench/releases/latest/download/HEA-Bench.exe) (Tauri wrapper of the same page) | done, built from the same parity-tested core |\n| **MCP server for AI agents** | `pip install \"hea-bench[mcp]\"`, then `hea-bench-mcp` | done, seven tools over the same core |\n\nThe three surfaces share **one calculation core**. The browser/desktop\ncore (`web/hea-calculator-core.js`) is a pure-JS port of the Python\nlibrary, and `tests/test_web_parity.py` guarantees the two match on all\n1484 binary pairs and the canonical multi-element fixtures, while\n`tests/test_web_oxides_parity.py` does the same for the oxide module,\ndown to identical warning messages.\n\n## Quick start (Python)\n\n```bash\npip install hea-bench\n```\n\n```python\nimport hea_bench as hb\n\ncantor = {\"Co\": 0.2, \"Cr\": 0.2, \"Fe\": 0.2, \"Mn\": 0.2, \"Ni\": 0.2}\n\nhb.smix(cantor) # 13.381 J/(mol·K) = R · ln 5\nhb.delta(cantor) # 3.164 % atomic-size mismatch\nhb.vec(cantor) # 8.0 valence electrons\nhb.mixing_enthalpy(cantor) # -4.16 kJ/mol (Miedema)\nhb.omega(cantor) # 5.79 (Yang–Zhang)\nhb.delta_chi(cantor) # 0.138 Pauling electronegativity mismatch\nhb.s_excess(cantor) # 0.318 J/(mol·K) (Mansoori excess entropy)\nhb.delta_g_max(cantor) # -8.00 kJ/mol (most-negative Miedema pair)\nhb.phi_king(cantor) # 3.533 (King 2016 proxy)\nhb.phi_ye(cantor) # 34.82 (Ye 2015 proxy)\n\n# Apply the canonical rules\nfrom hea_bench.rules import guo_vec, king_phi, yang_omega, ye_phi, zhang_delta\nzhang_delta.predict(cantor) # 'single-phase'\nyang_omega.predict(cantor) # 'single-phase'\nguo_vec.predict(cantor) # 'FCC'\nking_phi.predict(cantor) # 'solid_solution'\nye_phi.predict(cantor) # 'solid_solution'\n```\n\nThese Cantor-alloy values are pinned in the test suite as the canonical\nsanity check. The rules are simple empirical surrogates, fast screens\nrather than predictions, so treat their output accordingly.\n\n### Descriptor backends (optional interop)\n\nDescriptors can also be computed through a pluggable backend. The\ndefault (`native`) is this package's own stdlib implementation; with\n`pip install \"hea-bench[interop]\"` the same interface drives an\ninstalled [HEACalculator](https://github.com/dogusariturk/HEACalculator)\n(GPLv3, installed at the user's choice), so a workflow standardized on\nits numbers can keep them while using everything downstream here:\n\n```python\nfrom hea_bench.descriptors.backend import get_backend\nget_backend(\"heacalculator\").compute(cantor) # same names, their reference data\n```\n\n```bash\nhea-bench describe Al0.3CoCrFeNi --backend native\n```\n\nThe two backends vendor different reference data (radius conventions\ndiffer most), so same-named values legitimately differ; the measured,\nper-descriptor comparison lives in\n[docs/backend-agreement.md](docs/backend-agreement.md). Quantities\nwhose implementations differ structurally are deliberately not mapped\nonto each other, and the benchmark's published baselines use the\nnative backend unchanged.\n\n## Quick start (oxides)\n\n```python\nfrom hea_bench import oxides\n\n# Rost 2015 \"J14\" entropy-stabilized rock salt\nj14 = oxides.describe_rock_salt({\"Mg\": 1, \"Co\": 1, \"Ni\": 1, \"Cu\": 1, \"Zn\": 1})\nj14[\"descriptors\"][\"s_config\"] # 13.382 J/(mol·K) = R·ln 5\nj14[\"oxidation_states\"] # all 2+ by charge balance\n\n# Jiang 2018 single-phase high-entropy perovskite\npvk = oxides.describe_perovskite({\"Sr\": 1}, {\"Zr\": 1, \"Sn\": 1, \"Ti\": 1, \"Hf\": 1, \"Mn\": 1})\npvk[\"descriptors\"][\"goldschmidt_t\"] # 0.979, inside the 0.92–1.04 window\npvk[\"verdicts\"][\"bartel\"] # 'perovskite' (τ = 3.72 < 4.18)\n```\n\nEach `describe_*` report carries the solved oxidation states, the\nShannon radii actually used, every descriptor, the formability\nverdicts with their windows, and any warnings. See\n[`examples/02_oxides_walkthrough.py`](./examples/02_oxides_walkthrough.py)\nfor the full tour, including the fluorite and pyrochlore screens and\noxidation-state overrides.\n\n## Quick start (ceramics, experimental)\n\n`hea_bench.ceramics` extends the calculator to rock-salt carbides and\nnitrides and AlB2-type diborides, composition-only and honest about\nwhat that buys:\n\n```python\nfrom hea_bench import ceramics\n\nhec = ceramics.describe_rock_salt_carbide({\"Ti\": 1, \"Zr\": 1, \"Hf\": 1, \"Nb\": 1, \"Ta\": 1})\nhec[\"vec_per_formula_unit\"] # 8.4, with annotated literature reference points\nhec[\"entropy\"] # all normalization conventions, labelled\n```\n\nReports carry the metal-sublattice entropy in every published\nnormalization convention (papers switch between them without warning),\nVEC with annotated reference points rather than a verdict (the\nliterature marks points, not one window), and explicit notes on what\nis deferred: the size-mismatch descriptor (the field computes it from\nDFT binary-cell bond lengths, and adopting a cited table is real\ncuration work), and entropy-forming-ability or DEED, which are\nDFT-ensemble quantities this package cannot and does not claim to\nreproduce. Background, citations, and a license audit of candidate\nceramics datasets: [docs/ceramics.md](docs/ceramics.md).\n\n## Quick start (AI agents, MCP)\n\nLLM agents hallucinate descriptor values; this server grounds them.\n`hea_bench.mcp_server` exposes the whole workflow over the\n[Model Context Protocol](https://modelcontextprotocol.io/) as thirteen\ndeterministic tools: the original calculator seven\n(`parse_composition`, batch `alloy_descriptors` and `alloy_rules`,\n`omega_sensitivity`, `oxide_report`, `element_coverage`, `about`) plus\nthe capability layers (`corpus_query` and `corpus_describe` over the\nprovenance-tracked experimental corpus, `predict_properties` with\nintervals and domain flags at the top level of every payload,\n`check_applicability` for the novelty components, `design_search` with\nhard caps on palette, step, and candidate count, and\n`campaign_suggest` operating on a campaign file the user supplies).\nEvery response carries units or uncertainty fields, citation keys\nwhere a parametrization is involved, and the library version, so an\nagent's reasoning trace contains auditable receipts rather than bare\nfloats; `about()` reports which capabilities are available in the\nrunning environment, and missing optional extras come back as a clear\nmessage naming the exact install.\n\n```bash\npip install \"hea-bench[mcp]\"\n```\n\nRegister it with any MCP client (Claude Desktop, Cursor, ...), for\nexample in `claude_desktop_config.json`:\n\n```json\n{ \"mcpServers\": { \"hea-bench\": { \"command\": \"hea-bench-mcp\" } } }\n```\n\nThe `omega_sensitivity` tool is worth singling out: it reports the\nper-pair Miedema contributions and how far Ω moves when the dominant\nelement's pair enthalpies are shifted within the spread of published\ncompilations, so an agent can ask not just for a number but for how\nmuch to trust it.\n\n## Quick start (browser, no install)\n\nA self-contained HTML calculator computes every descriptor, applies all\nsix rules, runs the Miedema decompositions, and covers the oxide mode,\nentirely client-side. Two equivalent paths:\n\n- Open the hosted site: ****. The\n page is the calculator.\n- Or clone the repo and open `web/index.html`. No install, no\n terminal, no server.\n\nThe calculator ships its own documentation: a **Theory** view deriving\nevery alloy and oxide formula with citations, a grouped, filterable\n**Equations** reference, and a grouped **References** bibliography.\nDeep links open a view directly (`index.html#theory`,\n`#equations`, `#refs`). The parity-critical math lives in\n`web/hea-calculator-core.js` and is regression-checked against Python\nby the two parity test suites.\n\n## Paired evaluation: the phase-prediction benchmark (experimental)\n\nPublished HEA phase-prediction accuracies are mostly measured with\nrandom train/test splits over corpora full of stoichiometric series, so\nmodels are tested on close variants of alloys they trained on. That\nlargely measures interpolation within known systems.\n`hea_bench.benchmark` ships frozen, family-grouped and random paired\nsplits over a consolidated experimental corpus (~7,700 alloys) and an\nevaluator that reports both **side by side**:\n\n```python\nfrom hea_bench.benchmark import evaluate, load_benchmark\nprint(evaluate(my_model, load_benchmark(task=\"phase4\")).table())\n```\n\nA stock random forest over this package's own descriptors scores 0.941\nbalanced accuracy under the random split and 0.734 under the grouped\none. Neither number is wrong; they answer different questions (new\nstoichiometries of known systems versus unseen element systems), and\nthe gap between them quantifies how much of the random-split score\ncomes from testing on close relatives of training alloys. For this\ninterpolation-versus-extrapolation reading of grouped evaluation, see\nLi et al., *Commun. Mater.* **6**:9 (2025),\n[doi:10.1038/s43246-024-00731-w](https://doi.org/10.1038/s43246-024-00731-w).\nBaselines, split digests, and full provenance:\n[`docs/benchmark-baselines.md`](./docs/benchmark-baselines.md).\n\nThis surface currently works from a repository checkout only: the\ncorpus's largest source dataset declares no license, so the corpus is\nrebuilt locally from a fetch script and pinned hashes rather than\nredistributed (see [`data/raw/README.md`](./data/raw/README.md)).\n\n## The corpus as a standalone product\n\nThe consolidated experimental corpus behind the benchmark is also\naddressable directly, with no task or split machinery involved:\n\n```python\nfrom hea_bench.corpus import load_corpus\n\ncorpus = load_corpus() # v0.1.0, every row, full provenance\ncorpus.describe() # counts, families, agreement rate\nal_bcc = corpus.query(contains=[\"Al\"], phase=\"BCC\", descriptor_ready=True)\nal_bcc.rows[0].raw_labels # each source's verbatim reported phase\nal_bcc.to_csv(\"al-bcc.csv\")\n```\n\nEvery row carries per-source canonical and verbatim labels, Borg's\nprocessing route and primary-literature DOI where available, and\nupstream record identifiers, so a label can be audited without leaving\nthe package. Provenance chains, per-source license status,\nharmonization rules, and known limitations are documented in the\n[corpus card](docs/corpus-card.md). The corpus data is still built\nlocally from the recipe above, for the same licensing reason.\n\n## Uncertainty and domain of applicability\n\n`hea_bench.uncertainty` is the trust layer for anything fitted on the\ncorpus. Split conformal prediction wraps any sklearn-style model with\nsets or intervals carrying a distribution-free finite-sample coverage\nguarantee, and a domain-of-applicability model says whether that\nguarantee's exchangeability assumption plausibly holds for your query:\n\n```python\nfrom hea_bench.corpus import load_corpus\nfrom hea_bench.uncertainty import ConformalClassifier, fit_domain\n\ndomain = fit_domain(load_corpus())\ndomain.novelty({\"Hf\": 0.2, \"Nb\": 0.2, \"Ta\": 0.2, \"Ti\": 0.2, \"Zr\": 0.2})\n# {'element_set_seen': True, 'family_count': ..., 'nearest_family_distance': 0.0,\n# 'descriptor_distance': ..., 'element_coverage': True, 'in_domain': True, ...}\n```\n\nThe novelty output is several deliberately orthogonal signals plus one\nconservative `in_domain` flag, because the signals fail differently and\na single scalar invites misreading. Empirical coverage of the conformal\nsets on the frozen grouped folds, in and out of domain, is measured in\n[docs/uncertainty-coverage.md](docs/uncertainty-coverage.md). The\nmeasured pattern is worth internalizing: in this corpus the flagged\nout-of-domain queries are almost entirely far-from-HEA binaries the\nmodel handles confidently, while the residual risk concentrates in\nunseen families that look descriptor-close to the training data, so\nread the flag together with the set size rather than either alone.\nThese tools describe this package's confidence about your composition\non this corpus, nothing else.\n\n## Property predictions, in explicit tiers\n\n`hea_bench.properties` predicts what experimentalists ask about first,\nwith the data quality stated in the API rather than implied:\n\n```python\nfrom hea_bench.properties import predict_property\n\npredict_property({\"Al\": 0.2, \"Co\": 0.2, \"Cr\": 0.2, \"Fe\": 0.2, \"Ni\": 0.2}, \"hardness\")\n# PropertyPrediction(prop='hardness', value=..., unit='HV',\n# interval=(low, high), alpha=0.1, tier='B',\n# in_domain=True, n_training=..., ...)\n```\n\nTier A (`density`, `melting_temperature`, and an explicitly indicative\n`cost_per_kg` over a date-stamped, per-element-sourced price table) is\nclosed-form arithmetic over cited tables, validated where experiment\nexists ([docs/property-tier-a.md](docs/property-tier-a.md)).\nTier B (`hardness`, behind `pip install \"hea-bench[properties]\"`) is a\nseeded random forest over this package's descriptors wrapped in a\nfamily-grouped conformal interval and a domain flag; its held-out\nerror, interval calibration, and the decisions that error does and\ndoes not support are stated in\n[docs/property-hardness.md](docs/property-hardness.md). Intervals are\nwide because the public data is small and heterogeneous; that is the\nhonest outcome, shown rather than hidden. Properties whose public data\ncannot support a defensible held-out error (yield strength across\nuncontrolled test temperatures, ductility, corrosion) are deliberately\nnot shipped, and the model card says why.\n\n## Constrained composition search\n\n`hea_bench.design.search` answers \"what should I make\" as a screening\naid: a deterministic composition lattice over your palette, filtered by\nrule, property, composition, and domain constraints, returning a Pareto\nfront where every candidate carries its full receipt:\n\n```python\nfrom hea_bench.design import Maximize, Minimize, PropertyConstraint, search\n\nresult = search(\n elements=[\"Al\", \"Co\", \"Cr\", \"Fe\", \"Ni\"],\n n_elements=(4, 5),\n constraints=(PropertyConstraint(\"density\", max=8.0),),\n objectives=(Maximize(\"hardness\"), Minimize(\"cost_per_kg\")),\n step=0.05,\n)\nresult.candidates[0].properties[\"hardness\"].interval # every number has one\n```\n\nThe domain constraint is on by default (optimizers exploit model error\nhardest where data runs out; opting out is explicit), and\n`optimize_bound=\"lower\"` ranks fitted objectives by the conservative\nend of their intervals. The search is exhaustive within a hard budget\nand refuses loudly rather than sampling silently, so a result is\nreproducible by construction. Where measured alloys land relative to a\nrecovered front is studied honestly in\n[docs/design-recovery.md](docs/design-recovery.md); the front is a\nprioritization aid, not a set of answers.\n\n## Active-learning campaigns (bring your own measurements)\n\n`hea_bench.design.campaign.Campaign` runs the loop that creates repeat\nusage: observe your own measurements, get a ranked next batch, keep\neverything in a plain JSON file on your disk (no accounts, no server,\nno telemetry):\n\n```python\nfrom hea_bench.design.campaign import Campaign\n\ncampaign = Campaign(\"hardness\", [\"Al\", \"Co\", \"Cr\", \"Fe\", \"Ni\"])\ncampaign.observe({\"Al\": 0.1, \"Co\": 0.25, \"Cr\": 0.2, \"Fe\": 0.25, \"Ni\": 0.2}, 430.0)\ncampaign.suggest(n=5) # each Suggestion prints its interval and domain flag\ncampaign.save(\"my-campaign.json\")\n```\n\nThe surrogate is a seeded random-forest ensemble whose uncertainty is\ntree disagreement (a model-disagreement band, deliberately not sold as\na coverage guarantee), acquisition is expected improvement or UCB with\nbatched picks via the believer heuristic, and hardness campaigns warm\nstart from the Borg records inside your palette so the loop is useful\nbefore your tenth sample. Below 10 informative rows it refuses rather\nthan pretending. A year-ordered replay of the loop on the\nAl-Co-Cr-Fe-Ni hardness record is reported honestly in\n[docs/campaign-replay.md](docs/campaign-replay.md).\n\n## A note on Ω near ΔHmix ≈ 0\n\n`Ω = Tm·ΔSmix / |ΔHmix|` diverges as ΔHmix → 0, so for\nnear-ideal alloys (|ΔHmix| ≲ 1–2 kJ/mol) the Ω *magnitude* is\nextremely sensitive to the choice of Miedema pair table (sources\ndisagree most on Mn). The phase verdict (Ω ≫ 1.1) stays robust even when\nthe number does not, so read Ω qualitatively in that regime.\n\n## Project layout\n\n```\nhea-bench/\n├── src/hea_bench/\n│ ├── descriptors/ ΔS_mix, δ, VEC, T_m, ΔH_mix, Ω, S_E, φ + data tables\n│ ├── rules/ the six empirical phase-prediction rules\n│ ├── oxides/ HEO module: families, oxidation-state solver,\n│ │ Shannon radii (94 elements, vendored from pymatgen)\n│ ├── benchmark/ frozen family-grouped + random paired splits and evaluation\n│ │ (repo-only; corpus is built locally, see data/raw/)\n│ ├── composition.py formula parser, normalizer\n│ ├── constants.py R = 8.314\n│ └── cli.py command-line entry point\n├── tests/ unit tests + BOTH Python↔JS parity suites\n├── web/ landing page + self-contained calculator (+ MathJax)\n├── src-tauri/ native desktop wrapper (Rust/Tauri)\n├── examples/ Cantor-alloy and oxides walkthroughs (.py + .ipynb)\n└── pyproject.toml\n```\n\n## Development\n\n```bash\ngit clone https://github.com/dfieser/hea-bench\ncd hea-bench\npip install -e \".[dev]\"\npython -m pytest tests/ -q # includes the Python↔JS parity test (needs Node)\n```\n\nThe HTML calculator (`web/index.html` over\n`web/hea-calculator-core.js`) is an independent JavaScript\nimplementation of the same descriptors and rules. When you modify the\nPython descriptor code, update the JS core to match and re-run\n`tests/test_web_parity.py` and `tests/test_web_oxides_parity.py` so the\nsurfaces don't drift. The element data tables inside the JS core are\ngenerated from the Python library by `tests/data/_sync_js_tables.py`\nand `tests/data/_sync_js_oxide_tables.py`. Regenerate them, never\nhand-edit them.\n\n## License\n\n[MIT](./LICENSE). The vendored\n[matminer Miedema data files](./src/hea_bench/descriptors/data/) remain\nunder their upstream BSD-3-Clause license, preserved at\n[`descriptors/data/LICENSE.matminer.txt`](./src/hea_bench/descriptors/data/LICENSE.matminer.txt).\n\n## Contributing and support\n\nContributions and bug reports are welcome. See\n[CONTRIBUTING.md](./CONTRIBUTING.md) for development setup and the\ntesting convention.\n\nReport a bug or request a feature in the\n[issue tracker](https://github.com/dfieser/hea-bench/issues). Ask a\nquestion or float an idea in\n[Discussions](https://github.com/dfieser/hea-bench/discussions), where\nthe Q&A category is the right place for how a descriptor is defined,\nwhich rule applies to a composition, or why two sources disagree.\nAnswers there stay findable for the next person with the same question.\nFor direct contact, email the maintainer at `davjfies@gmail.com`.\nParticipation is governed by the [Code of Conduct](./CODE_OF_CONDUCT.md).\n\n## Acknowledgements\n\n**Yen-Ming Horng** ([@infinitus01](https://github.com/infinitus01)),\nIndependent Researcher, Taiwan. External reproducibility and\ndocumentation review. Reported the `delta_g_max` documentation contract\nmismatch corrected in v2.1.4.\n\nExternal reviews of this kind cover reproducibility and\ndocumentation-to-implementation consistency. They are not a validation\nor endorsement of the underlying scientific conclusions.\n\n## Citation\n\nIf you use hea-bench in your work, please cite the paper that\ndescribes it:\n\n> Fieser, D.; Dewanjee, U.; Hu, A. HEA-Bench: An AI-Agent-Optimized\n> Calculator of High-Entropy Alloy and Oxide Descriptors and\n> Phase-Prediction Rules. *Materials* **2026**, *19*, 3075.\n> \n\n```bibtex\n@article{ma19143075,\n author = {Fieser, David and Dewanjee, Unmanaa and Hu, Anming},\n title = {{HEA-Bench}: An {AI}-Agent-Optimized Calculator of High-Entropy Alloy and Oxide Descriptors and Phase-Prediction Rules},\n journal = {Materials},\n volume = {19},\n year = {2026},\n number = {14},\n article-number = {3075},\n issn = {1996-1944},\n doi = {10.3390/ma19143075},\n url = {https://www.mdpi.com/1996-1944/19/14/3075},\n}\n```\n\nMachine-readable metadata, including this preferred citation, is in\n[`CITATION.cff`](./CITATION.cff) (GitHub's \"Cite this repository\"\nbutton uses it). To reference the exact software version you used,\nadditionally cite the Zenodo archive: the concept DOI\n[10.5281/zenodo.20346287](https://doi.org/10.5281/zenodo.20346287)\nalways resolves to the latest version.\n\nWhen citing hea-bench, please also cite the primary sources for the\nparametrizations it implements: de Boer et al. 1988 for the Miedema\nmodel, the rule papers (Yeh 2004, Zhang 2008, Guo–Liu 2011, Yang–Zhang\n2012, King 2016, Ye 2015), the oxide primaries (Shannon 1976,\nGoldschmidt 1926, Bartel 2019, Spiridigliozzi 2021, Subramanian 1983),\nmatminer for the vendored pair table, and pymatgen for the\nShannon-radius digitization. The full grouped bibliography is in the\ncalculator's References view.\n\n## Disclaimer\n\nDescriptor values and rule predictions reported by hea-bench are\n**empirical estimates** for research and informational purposes only.\nThe rules and descriptors are semi-empirical surrogates with known\nlimitations. No warranty is made as to accuracy, completeness, fitness\nfor any particular purpose, or suitability for material qualification.\nDo not use these outputs as the sole basis for engineering design or\nmaterial qualification without independent verification by validated\nthermodynamic methods (e.g. CALPHAD or DFT).\n\nSoftware is provided **\"as is\"** under the [MIT License](LICENSE).\nVendored Miedema elemental parameters from\n[matminer](https://github.com/hackingmaterials/matminer) remain under\ntheir upstream BSD-3-Clause license; see\n[`src/hea_bench/descriptors/data/LICENSE.matminer.txt`](./src/hea_bench/descriptors/data/LICENSE.matminer.txt).\n",
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