viridis-research/eco-lean-bench

Eco-Lean Bench

311 Lean 4 theorems in 50 tasks from applied ecology, conservation and decision science, each with a statement frozen before it was proved, plus a separate set of 108 theorems extracted from published proofs. A prover receives a challenge file with sorry in place of every target proof and must fill them in. Scoring uses the Lean kernel at a pinned toolchain and Mathlib revision, plus an audit of every theorem's axioms.

Status: research dataset, version 0.3.0, not peer reviewed. A solved theorem is a machine-checked proof of a fixed formal statement. It says nothing about whether the statement describes a real ecosystem, or whether the result is new.

Most formal-math benchmarks come from competitions and textbooks. These come from working decision models: corridor reliability, seed-source routing as a max-flow/min-cut problem, precautionary capacity under moment uncertainty, anytime-valid change detection, capital allocation, measurement budgets for restoration scores. That makes it a probe of whether provers transfer to applied mathematics they are unlikely to have seen as competition problems.

Two configs

Config	Statements	Tasks	Theorems	Substantial	Elementary
default (data/tasks.jsonl)	frozen before proof	50	311	9 tasks / 45 thm	41 tasks / 266 thm
extracted (data/extracted.jsonl)	extracted after proof	5	108	4 tasks / 67 thm	1 task / 41 thm

Substantial means the reference file has at least 200 non-blank lines and the median target proof is at least 6 lines. Depth is a size indicator, not a judgement of difficulty or novelty.

default keeps the v0.1 rule: a task qualifies only when its challenge was written as a separate frozen statement file and the reference proof was certified against it. It holds the 15 v0.1.0 tasks (90 theorems, public dual-kernel certificates) and 30 tasks from later engine runs in the canon's series/ deposits (216 theorems). The later runs are all elementary: short decision-model files whose proofs are a few lines each. It also holds 5 tasks (5 theorems) built from statements sealed in the engine's public record (jdhart81/viridis-research) before the run that proved them; each asks for one theorem, Candidate.main, proving the sealed Statement.

extracted covers longer published formalizations that never had a separate frozen statement. Their challenges were made for this benchmark by replacing every published theorem and lemma proof with sorry. Report it separately from default.

Task	Source	Theorems	Depth
ELB-X-DENDRITIC	Dendritic corridor formation and payments (EcoChain)	7	substantial
ELB-X-FHT	Fragmentation Hysteresis Theorem (Run 109)	30	substantial
ELB-X-HDFM	HDFM graph foundations (P2)	8	substantial
ELB-X-IB	Intelligence Bound core results	22	substantial
ELB-X-PLASMA	Plasma N2 fixation energy and fleet accounting (P7, corrected units)	41	elementary

Verification

• v0.1.0 tasks: certified by leanprover/comparator with two kernels (Lean and nanoda); receipts are public in the canon's certificates/.
• series/ tasks: the deposit's comparator summary reports Lean and nanoda kernel passes, but the receipts are not public.
• ELB-E- tasks: certified by the engine's own verifier (Lean kernel, axiom audit, leanchecker replay); each run's certification.json is public in jdhart81/viridis-research under runs/.
• Every task of version 0.2.0 in both configs, including the v0.1.0 tasks, was re-checked on 2026-11-06 with Lean leanprover/lean4:v4.28.0 and Mathlib 8f9d9cff6bd728b17a24e163c9402775d9e6a365 (ELB-X-IB instead with the toolchain its source repository declares, Lean leanprover/lean4:v4.24.0 and Mathlib f897ebcf72cd16f89ab4577d0c826cd14afaafc7): the reference proof elaborates with no errors, #print axioms shows only propext, Quot.sound and Classical.choice for every target, the challenge elaborates, and every challenge target depends on sorryAx. The second kernel was not re-run for this release. Each task's local_check field records the result.
• Tasks added from version 0.3.0 on were checked by the engine's scheduled bench workflow on the day they were added, in a network-less sandbox at the same pinned Lean and Mathlib: the reference elaborates, every target's axioms are within the permitted set, the challenge elaborates with every target depending on sorryAx, and leanchecker replays the reference. A candidate that fails any check is held and listed under Excluded, never published. Each task's local_check and checks/ file record the result and date.

Data fields

One task per line. v0.2 adds fields to the v0.1 schema; none were removed or renamed.

Field	Meaning
task_id	ELB-NNN from engine run Run-NNN, ELB-E-NAME from a statement sealed in the engine's public record, or ELB-X-NAME in extracted
config	frozen or extracted
challenge	Complete challenge file. Every target proof is sorry; nothing else is
targets, target_count	Fully qualified names of the theorems to prove
toolchain, mathlib_rev	Exact Lean toolchain and Mathlib commit
permitted_axioms	propext, Quot.sound, Classical.choice
depth, reference_median_proof_lines	Size of the reference proof
reference_solution	Path(s) and SHA-256 of the reference proof in source_repo at source_commit (not inlined, so a challenge can go straight to a prover)
source_repo, source_commit	Public GitHub repository and the commit the task was built from
statement_provenance	How the challenge statements came to exist
certification	Prior certification and whether its receipt is public
local_check	Result of the kernel and axiom check made when the task was added or last re-checked
public_since	Date the reference proof became public, for contamination cuts
notes	Caveats a user should read before citing a result on the task

Scoring

The scorer ships with the dataset (scorer/, standard library only).

# Fast precheck, no Lean needed
python3 scorer/score.py my-run/
# Authoritative: elaborate at the pinned toolchain and audit axioms
python3 scorer/score.py my-run/ --kernel --out result.json
# The extracted config
python3 scorer/score.py my-run-x/ --config extracted --kernel --out result-x.json
# Sanity check: fetch the reference proofs (hash-checked) and score them
python3 scorer/fetch_references.py && python3 scorer/score.py --reference --kernel

Put one complete Lean file per attempted task in my-run/, named by task id (ELB-175.lean). A target counts when it keeps the challenge's name and statement, uses no sorry, admit, new axiom, native_decide or kernel-skipping options, leaves challenge definitions unchanged, and (in --kernel mode) depends only on the permitted axioms. Partial credit is per theorem. Report default and extracted scores separately, and by depth. Sanity check: the fetched reference proofs score 311/311 and 108/108 in precheck, and the unmodified challenges score 0.

Leaderboard

Only --kernel results are listed. Open a discussion on this dataset with your result.json, the system and its training cutoff. No entries yet.

Contamination

All reference proofs are public on GitHub: default since 2026-08-29 to 2026-11-02, extracted since 2026-02-23 to 2026-07-26. Report your model's training cutoff and score only later tasks when that matters. The extracted statements were written together with their proofs, so a model that saw the source repositories may have seen both.

Known limitations

• 41 of the 50 default tasks are elementary; many targets follow from definitions with unfold and linarith or norm_num.
• A one-tactic probe (first | rfl | decide | norm_num | simp | ring | linarith | nlinarith | positivity | omega, no definitions unfolded) closes 3 of 108 extracted targets and 19 of 311 frozen targets. Those targets test little beyond basic automation; per-task lists are in probe/.
• The series/ tasks rest on comparator summaries without public receipts, plus a single-kernel re-check.
• extracted statements were not frozen before proof, and every lemma is a target, including helper lemmas.
• Task ELB-137 contains two frozen statements that turned out to be false as frozen (a degenerate n = 0 case). The challenge includes the refutations (*_as_frozen_is_false) as targets and repaired versions with one added hypothesis each.
• 4 of the 5 ELB-E- tasks restate canon results whose proofs were public before the statement was sealed. The statement was frozen before the engine's proof, but the mathematics is not new; each such task's notes say so.
• Some namespaces carry the name "IntelligenceBound", and ELB-X-IB formalizes results around it. The Intelligence Bound itself is a conjecture; no task proves it, and solving a task says nothing about it.
• ELB-X-PLASMA uses the units-corrected constants; its energy input is an assumption, not a measured value.
• ELB-X-HDFM holds graph lemmas only and says nothing about the HDFM manuscript's topology table, which has an erratum.

Excluded

• Run-120 (series/CAST): excluded in v0.1: statements frozen together with proofs
• Run-121 (series/ISB): excluded in v0.1: statements frozen together with proofs
• Run-122 (series/ZAB): excluded in v0.1: statements frozen together with proofs
• Run-142 (series/BCAN): no separately frozen statement file in the deposit
• Run-143 (series/RCBN): no separately frozen statement file in the deposit
• Run-144 (series/SBRN): no separately frozen statement file in the deposit
• series/RAAC: no separately frozen statement file in the deposit
• series/S6_CognitionLearning: no separately frozen statement file in the deposit
• series/SESL: no separately frozen statement file in the deposit
• ELB-145: published reference proof does not elaborate at its pinned toolchain and Mathlib (2 Lean errors), although the deposit's summary reports certification
• ELB-146: published reference proof does not elaborate at its pinned toolchain and Mathlib (2 Lean errors), although the deposit's summary reports certification

Provenance and AI use

The statements, proofs and papers were produced with AI assistance (Anthropic Claude, OpenAI Codex, Harmonic Aristotle) under the direction of Justin D. Hart, who is the accountable author. The v0.2 challenge extraction, re-check and card were also prepared with Claude. From version 0.3.0 on, the engine's scheduled bench workflow adds new tasks and regenerates this card from the data; it publishes only tasks that pass every check above. AI tools are not authors.

Citation

@misc{hart2026ecoleanbench,
  author    = {Hart, Justin D.},
  title     = {Eco-Lean Bench: Lean 4 theorems from applied ecology and conservation decisions},
  year      = {2026},
  publisher = {Viridis LLC},
  howpublished = {\url{https://huggingface.co/datasets/viridis-research/eco-lean-bench}},
  note      = {Version 0.3.0. Part of the Viridis Canon, concept DOI 10.5281/zenodo.19317982}
}

Author ORCID: https://orcid.org/0009-0008-3082-2482

License

Task data (Lean source) and scripts: Apache-2.0. This card: CC BY 4.0. Viridis LLC.