PRIS — Plausibility Rules for Inorganic Structures

Autonomous discovery of new structure-plausibility laws for explainable and rapid crystal diagnosis and screening

Project page: https://ai4qc.github.io/PRIS/ — the eight laws, the headline results, the five main figures, a worked example of the analyser, and one-page posters of the paper.

An autonomous agent ran 572 numbered investigations over 99,162 experimental crystal structures under a pre-registered protocol, evaluated 2,037,606 candidate laws, and kept eight one-line laws. Those eight are PRIS, and a synthesis score derived from them is PSS. This repository holds the manuscript, the analysis code, the aggregate data every figure is drawn from, and the pre-registrations that fixed the rules of each exercise before it was run.

The eight laws encode five complementary physicochemical mechanisms: short-range repulsion, ionic contact and packing, electrostatic balance, bond-valence conservation, and crystallographic site complexity. A structure that fails is told which mechanism it violated, which is what a bare distance cutoff cannot say.

Every rule is judged on two numbers at once: its satisfaction rate, the fraction of real crystals that satisfy it, and its detection rate, the fraction of deliberately corrupted structures it identifies as implausible. Pauling's rules of 1929 had never been scored on the second axis. Measured on both, the comparison reverses: Pauling's rules 2–5 are jointly satisfied by 6.5% of real crystals and fail to distinguish most pairs of competing structures, while PRIS is satisfied by 82–99% of held-out real crystals and detects up to 91% of damaged ones, where the distance cutoffs deployed in generative pipelines detect 1.6–3.2%.

The laws

Eight one-line laws over quantities computable from a structure file and a radius table. They are applied as nested sets. Set 1 to Set 4 are the chain the paper reports; Set 1' is the guarded band that stands beside the chain rather than inside it, which is why the catalogue holds eight laws while Set 4 applies seven:

	law	Set 1	Set 1′	Set 2	Set 3	Set 4
Law 1	ρ ≥ τ	τ = 0.735	τ = 0.735	τ = 0.804	τ = 0.804	τ = 0.804
Law 2	fi > 0.50 ⇒ ρ ≤ 1.05	–	✓	–	–	–
Law 3	mean anion CN ≤ 3.333 ⇒ mean d/(r₊+r₋) ≤ 1.081	–	–	✓	✓	✓
Law 4	rangei of VM(i)/vi ≤ 31.45 eV	–	–	✓	✓	✓
Law 5	maxi VM(i) ≤ 15.17 eV	–	–	✓	✓	✓
Law 6	fi > 0.55 ⇒ no like-charge bonds	–	–	–	✓	✓
Law 7	inequivalent sites / sites ≤ 2/3	–	–	–	–	✓
Law 8	mean of |BV sum − vi| / vi ≤ 0.7143	–	–	–	–	✓

where ρ is the reduced contact ratio, the shortest cation–anion distance divided by the sum of the two Shannon radii; f_i is Pauling's composition-based estimate of ionic character; V_M(i) is the site Madelung energy from an Ewald sum over formal charges, with v_i the magnitude of the formal charge at site i; and site complexity is inequivalent sites over sites at spglib symprec 0.01. A conditional law is satisfied by any structure whose trigger is not met, so the clauses of Law 2, Law 3 and Law 6 confine each law to its domain.

On the held-out benchmark (5,297 real, 3,612 damaged structures, never seen by any fitting step):

set	laws	satisfaction rate (real)	detection rate (damaged)
Set 1	Law 1 (permissive τ)	0.9919	0.2890
Set 1′	Law 1 (permissive) + Law 2	0.9894	0.3837
Set 2	Law 1, Law 3–Law 5	0.9579	0.6121
Set 3	Law 1, Law 3–Law 6	0.9171	0.7004
Set 4	Law 1, Law 3–Law 8	0.8180	0.9111

Set 4's worst damage class is still detected at 0.7338. Pauling's rules 2–5 applied jointly are satisfied by 0.0651 of experimental structures. On the re-damaged parent benchmark the same sets read 0.991 / 0.268 for Set 1 and 0.830 / 0.879 for Set 4; satisfaction and detection are always quoted from one population, and the population is always named.

The synthesis score

PSS is a PRIS-derived synthesis score. Every term is a quantity one of the laws already measures, refitted to the experimental record of what has been made, with volume per atom dominating (standardised weight −4.90):

• At matched satisfaction on experimental structures, PSS screens 83.7% of hard-to-synthesize structures, 31.8 percentage points more than a computed hull-energy threshold, which reaches 72.0% and additionally requires a relaxation and a phase-hull reference.
• Retained on 80.7% of experimental structures, it screens 51.9% of the candidate set.
• On the most confident fifth of same-composition pairs it reaches 0.944 accuracy against 0.844 for DFT energy above the hull.
• No synthesis label entered the discovery of the laws, so the correlation between PRIS plausibility and synthesizability is a finding rather than a fitting target.

Set 1 to Set 4 are conservative discrete screens that name a violated mechanism; PSS gives continuously tunable control over how strongly the combined evidence shortens a queue.

Judging a structure

Give it a structure file. It reports every law, the measured quantity, the threshold, the verdict, the mechanism the law tests, all five nested sets, and PSS.

python src/pris_analyze.py mystructure.cif
python src/pris_analyze.py --quiet *.cif        # one verdict line per file
python src/pris_analyze.py --json POSCAR        # machine-readable

  MgAl2O4, 14 sites, charges from integer charge balancing, ionic character f_i = 0.759

    law    quantity                                measured  threshold  verdict  applied in       mechanism
    Law 1  reduced contact rho                       0.9865     0.8040     ok    1, 1', 2, 3, 4   short-range repulsion
    Law 2  reduced contact rho                       0.9865     1.0500     ok    1'               ionic contact
    Law 3  mean reduced cation-anion contact         4.0000     1.0810     --    2, 3, 4          packing
    Law 4  range of site Madelung energy / valence    3.6122    31.4500     ok    2, 3, 4          electrostatic balance
    Law 5  largest site Madelung energy            -20.2144    15.1700     ok    2, 3, 4          electrostatic balance
    Law 6  fraction of like-charge bonds             0.0000     0.0001     ok    3, 4             electrostatic balance
    Law 7  inequivalent sites / sites                0.2143     0.6667     ok    4                crystallographic site complexity
    Law 8  mean |BV sum - v_i| / v_i                 0.0384     0.7143     ok    4                bond-valence conservation

    Set 4  crystal chemistry           plausible
    PSS  +3.915        VERDICT  PLAUSIBLE

When a structure fails, the point is not the verdict but the last column:

$ python src/pris_analyze.py --quiet damaged/*.cif
IMPLAUSIBLE  compressed.cif   bond-valence conservation, short-range repulsion
IMPLAUSIBLE  expanded.cif     bond-valence conservation

Thresholds and PSS weights are read from the frozen artefacts in agent_loop/frozen/, so a verdict produced today is the verdict the manuscript reports. src/apply_rules.py remains as a lighter alternative that needs neither spglib nor bond-valence parameters and therefore covers Set 1 to Set 3 only.

Charges are formal oxidation states inferred from composition. BVAnalyzer is never called: it infers valence from bond lengths, so testing a bond-length law on its output would use the conclusion as the premise. Roughly 19% of structures cannot be judged (multiple anions, complex molecular groups, no integer or fractional charge assignment). "Skipped" does not mean "passed."

Layout

PREREG.md            pre-registration: criteria, split and vocabulary, frozen before any evaluation
src/                 the laws, the score, and the figure code
figures/             every figure of the manuscript -> the script that draws it
agent_loop/          the 572 numbered investigations, their plans, tests and frozen chains
paper/data/          aggregate statistics each main figure is drawn from
paper/si_data/       aggregate statistics for the supplementary figures
data/                small reference tables (bond-valence parameters, Lewis acidity)
dft/                 the first-principles campaign: protocol, inputs and collected results
experiments/         the analyses behind Fig. 4 and five supplementary figures
pipeline/            data acquisition
tests/               checks on the laws, the score and the figures
manuscript/          submission sources and the two compiled PDFs
docs/                the project page served at https://ai4qc.github.io/PRIS/

manuscript/main.pdf and manuscript/si.pdf are the compiled manuscript and supplementary information. agent_loop/README.md explains which investigation lines are the main line and which were later corrected.

Reproducing

Figures from committed data — no external data needed:

pip install -r requirements.txt
python figures/make.py --list      # which script draws which figure
python figures/make.py "Fig. 1"    # one figure
python figures/make.py --all       # every generator once
cd manuscript && ./build.sh        # needs tectonic; writes main.pdf and si.pdf

Thirty figures, thirteen generators. Three script names predate the final numbering and disagree with the figure they draw — figures/manifest.json is the authority and figures/README.md spells the mismatches out. Four figures (Fig. 4, S17, S19, S22) additionally need the positive–unlabelled score shards, which are not redistributed; figures/README.md says so and gives the aggregate numbers instead.

Analyses from structures — these need the derived feature store, which is gigabytes of parquet and is not in this repository. Point the scripts at it:

export PRIS_FEATURES=/path/to/features/
python src/validity_rulesets.py   # rule sets vs deployed generative-model validity filters
python src/rank_rulesets.py       # rule sets on the same-composition ranking task

Both scripts reproduce previously published rows to four decimal places before emitting any new number, and rank_rulesets.py refuses to write output if that check fails.

The scripts that fitted the law sets and the score (src/l4_*.py, src/f2r*.py, src/f3_*.py) refuse to rerun their sealed evaluations; the frozen results are the ones quoted above. f2r and f3 are the original chain identifiers for what the manuscript now calls PSS and its predecessors. Most other scripts still contain machine-specific absolute paths from the original run. They are kept as the audit trail of what was actually executed, not as a portable library.

Data licensing — read before redistributing anything

• ICSD is not redistributable: FIZ Karlsruhe copyright plus the European Union sui generis database right.
• ELEMENTA is CC-BY-NC-4.0.
• COD is CC0.
• data/bvparm2020.cif is I. D. Brown's bond-valence parameter table (McMaster University); it may be redistributed free of charge for non-commercial use with its copyright notice intact, which the file header carries.

Only aggregate statistics are committed here. Per-structure intermediates keyed to ICSD identifiers are excluded by .gitignore (paper/data/*_raw.csv) and must stay excluded. Any public benchmark released from this work is built on COD alone.

Two things worth knowing before trusting a number

The split discipline. Thresholds were fitted only on the discovery split (12,632 real / 8,590 corrupted). calibration (5,297 / 3,612) was never seen by a fitting step. Satisfaction and detection rates are always quoted from the same split and the split is always named. Assignment is by a seeded hash of each structure identifier, not grouped by composition, so structures sharing a composition may cross partitions. A lockbox of 5,748 structures was sealed with a quota of three permitted openings. One opening has been used (2026-08-01, a fitted-score validation that failed its pre-registered gate; the full record is SI Note S8); two remain. Separately, when the analysis set was widened after the freeze, 3,215 lockbox rows carried no split label and entered one full-sample quantile fit. The thresholds have therefore seen part of the sealed set, and its value as a final test should be discounted accordingly.

The refutation ledger. Eleven conclusions were written down as results and then overturned by the agent that produced them. They are published with the surviving results (agent_loop/refutation_ledger.csv, and Supplementary Figs. S1 and S2), each paired with the cheap diagnostic that exposed it, because those diagnostics outlive the claims that motivated them.

The manuscript and its Supplementary Information are the record for every number. Where a figure and the text disagree, the text wins; manuscript/main.pdf and manuscript/si.pdf are the compiled versions this repository was released with.

Citation

If you use PRIS, PSS, or the code in this repository, please cite the preprint, arXiv:2609.01209:

@article{song2026pris,
  title         = {Autonomous discovery of new structure-plausibility laws for explainable
                   and rapid crystal diagnosis and screening},
  author        = {Song, Zhilong and Cheng, Lixue},
  year          = {2026},
  eprint        = {2609.01209},
  archivePrefix = {arXiv},
  primaryClass  = {cond-mat.mtrl-sci},
  url           = {https://arxiv.org/abs/2609.01209},
}

To cite the software and the frozen law definitions specifically:

@software{pris2026software,
  title   = {{PRIS}: Plausibility Rules for Inorganic Structures},
  author  = {Song, Zhilong and Cheng, Lixue},
  year    = {2026},
  url     = {https://github.com/AI4QC/PRIS},
  license = {MIT},
}

CITATION.cff carries the same metadata in machine-readable form, so GitHub's "Cite this repository" button stays in step with this section.

Licence

Code and manuscript text: MIT (see LICENSE). This does not extend to the third-party structural databases described above, nor to the Springer Nature LaTeX class files (tex/sn-jnl.cls, tex/sn-nature.bst), which carry their own terms.