PhysLeanData: Verified Theorems & Likelihoods

• PhysLeanData is a dual-purpose framework comprising a Lean 4 corpus of verified physics proofs and simplified likelihood data for statistical inference.
• Its formal section rigorously verifies physics theorems through hybrid proof generation and multi-prover validation to ensure high accuracy.
• Its experimental section employs the Simplified Likelihood Framework to produce compact, statistically complete data products that promote reproducibility in physics.

PhysLeanData refers collectively to two distinct, domain-specific frameworks for rigorous and reproducible data sharing in modern physics research. In the context of formalized theoretical physics, PhysLeanData is a high-quality corpus of Lean 4 theorems and verifiable proof scripts enabling automatic theorem proving for advanced physics topics. In experimental and phenomenological physics, PhysLeanData designates compact, statistically complete data products based on the Simplified Likelihood (SL) Framework, facilitating the sharing of approximate—but information-rich—likelihoods in a computationally tractable format. Both uses serve the goal of increasing reproducibility and accelerating inference or proof automation in physics, targeting gaps left by prevailing mathematics-centric resources or opaque experimental result summaries (Zhang et al., 22 Jan 2026, Buckley et al., 2018).

1. Scope and Definitions

There are two principal meanings for PhysLeanData within recent physics research literature:

• Formal Theorem-Proving Corpus: A structured, verified dataset of formal physics theorems and proofs in Lean 4, designed to train and evaluate automated theorem provers on physics rather than pure mathematics (Zhang et al., 22 Jan 2026).
• Experimental PhysLeanData Products: Approximate, publication-ready likelihood representations derived from the Simplified Likelihood Framework, distilling hundreds of nuisance parameters and their correlations into a compact, public data package without significant loss of statistical information (Buckley et al., 2018).

Each usage is tailored to the methodological requirements of its domain, with shared emphases on verifiability, compactness, and community accessibility.

2. Construction and Methodology

2.1. PhysLeanData for Formal Theorem Proving

The corpus construction involves a multi-stage hybrid process:

• Seed Extraction: Source theorems and proofs are drawn from the PhysLean (also referred to as "HepLean," Tooby-Smith et al. 2025) repository, comprising fully formalized lemmas in areas such as classical mechanics, electromagnetism, quantum mechanics, relativity, quantum field theory, and string theory. After file-length filtering, 2,933 training and 250 test seed examples are obtained.
• Synthetic Conjecture Generation: For each seed, 10 variant conjectures are generated in Lean syntax using the Claude-4.5 Sonnet model, yielding 29,330 candidates.
• Rigorous Filtering:
  • Stage 1: Lean’s elaborator discards ill-formed statements, yielding 6,971 syntactically valid conjectures.
  • Stage 2: For each valid conjecture, three open-source provers (DeepSeek-Prover-V2-7B, Kimina-Prover-8B, Goedel-Prover-8B) and Lean’s kernel check provability; only conjectures for which at least one proof is generated and verified are retained, resulting in 2,608 proofs. The total training set is thus 5,541 unique, fully verified theorems and proofs (Zhang et al., 22 Jan 2026).

2.2. PhysLeanData via Simplified Likelihood

The process for SL-based PhysLeanData involves:

• Theoretical Foundation: The SL formalism approximates the full likelihood by summarizing the effect of $N \gg 1$ elementary nuisance parameters ($\delta_i$) through a smaller set of Gaussian nuisance parameters ($\theta_I$), capturing both covariance and leading non-Gaussian (skewness) corrections. The likelihood for observed data $n_I^{\text{obs}}$ in each observable (e.g., bins, channels) is written:

$$n_I(\boldsymbol\alpha,\boldsymbol\delta) \rightarrow a_I(\boldsymbol\alpha) + b_I(\boldsymbol\alpha)\theta_I + c_I(\boldsymbol\alpha)\theta_I^2$$

with $\boldsymbol\theta \sim \mathcal{N}(0, \rho)$, and $c_I$ encoding asymmetry.

• Parameter Extraction: Monte Carlo samples ("toys") with all uncertainties varied yield estimates of bin means ($m_1$), covariances ($m_2$), and skewness ($m_3$). These moments are algebraically mapped to SL parameters ($\delta_i$0).
• Data Packaging: The derived parameters are organized into JSON or YAML records for deposition in HepData following the error source schema.
• Validation: Performance is assessed against full likelihoods in LHC-style analyses, with sub-percent agreement near confidence limits when non-Gaussian terms are included. Convergence and validity are tied to theory-based conditions (e.g., CLT, skewness thresholds) (Buckley et al., 2018).

3. Internal Structure and Data Formats

3.1. Formal Proof Dataset (Lean 4)

Each entry in the PhysLeanData formal corpus is structured as: | Field | Description | Format Example | |------------------|--------------------------------------------------------|-------------------------------------------------| | header | Imports, namespace, variable declarations | Lean 4 imports and namespace block | | lemma_statement | Lean lemma (with optional attributes, docstring) | lemma … : … := | | proof_script | Tactic script (Lean 4) completing the formal proof | Sequence of tactic invocations ending with by |

Example snippet: $\delta_i$1 (Zhang et al., 22 Jan 2026)

3.2. Simplified Likelihood Data Product

A minimal example in JSON: $\delta_i$2 (Buckley et al., 2018)

4. Applications and Usage

• Automatic Theorem Proving in Physics: PhysLeanData is used to train and fine-tune theorem-proving models such as PhysProver. The dataset enables efficient bootstrapping and curriculum learning, permitting small expert models (~7B parameters) to master diverse proof strategies in physics with limited data (Zhang et al., 22 Jan 2026).
• Few-Shot Inference and Verification: Via Lean 4, PhysLeanData examples can be loaded for ad hoc verification or used as few-shot prompts for proving new conjectures.
• Statistically Faithful Experimental Results: In LHC and similar contexts, PhysLeanData (in the SL sense) allows collaborations to release lean, accurate likelihoods. This enables re-interpretation, combination, and robust statistical inference by external groups, reproducing the conclusions of full-complexity analyses.
• Cross-Domain Utility: Curriculum learning with PhysLeanData for physics theorem-proving yields observable (“1.3% gain”) improvements on out-of-domain formal mathematics benchmarks (MiniF2F), indicating positive transfer between domain-specific and general formal reasoning (Zhang et al., 22 Jan 2026).
• Reproducible Publication: The SL-based packaging standardizes data release, closing the reproducibility gap in high-energy physics by providing manageable and sufficiently detailed experimental results (Buckley et al., 2018).

5. Quality, Validation, and Community Impact

• Quality Assurance: All formal theorems and proofs are subject to Lean 4 kernel verification, guaranteeing correctness and eliminating hallucinated proofs. Only conjectures with verified proofs are included.
• Synthetic Diversity: The inclusion of synthetically generated and verified conjectures broadens the diversity of proof patterns, extending exposure far beyond the initial human-authored corpus.
• Statistical Robustness: The SL approximation reproduces full likelihood test statistics to sub-percent agreement near relevant confidence thresholds, provided bins are suitably merged and the CLT limit holds. Omission of non-Gaussian terms can result in several percent bias (Buckley et al., 2018).
• Domain Coverage: The formal PhysLeanData covers advanced topics: classical and analytical mechanics, electromagnetism, statistical physics, quantum mechanics, relativity, quantum field theory, and string theory.
• First in Physics: PhysLeanData is the first publicly available Lean 4 corpus specifically for physics theorem proving, setting a precedent in the field (Zhang et al., 22 Jan 2026).

6. Practical Guidance and Extensions

• Loading and Integration: Examples for loading the formal dataset (Python, Lean 4) and the SL-based data (JSON/YAML, HepData) are provided. Practitioners are guided to compute moments and extract SL parameters using available tools (e.g., SLtools Python package).
• Parameter Guidelines: For accurate SL parameters, 5,000–10,000 Monte Carlo toy throws are recommended for mean and width estimation, somewhat fewer for skewness if bin yields are high.
• Extensions: The SL formalism is extensible to unbinned analyses (via parametric PDFs) and to fields beyond HEP, such as astronomy or cosmology, by minor redefinition of “bins.” When systematics depend on signal hypothesis, parameter recalculation is required.
• Best Practices: For both formal and experimental PhysLeanData, documentation and example scripts should be distributed, and the conditions under which approximations are valid should be made explicit.

7. Significance and Future Directions

PhysLeanData enables physics-specific automation and reproducibility by combining domain-specialized, formally verified theorem-proving benchmarks with statistically complete, algorithmically accessible data from complex experiments. Its design bridges the methodological gap between pure mathematics and physics, as well as between experimental analysis and public result dissemination. A plausible implication is the acceleration of formal and statistical inference in physical sciences by providing shared, high-quality, and verifiable resources to the research community (Zhang et al., 22 Jan 2026, Buckley et al., 2018).