Why3-py: A Tool for Formal Verification of Hypothesis Testing and Meta-Analysis in Python

Reliable hypothesis testing is the foundation of many empirical scientific claims. Large language model (LLM) agents are increasingly used to automate this process, as they can inspect datasets, generate code, and produce analyses end-to-end. However, we show that they frequently make subtle inferential errors that lead to incorrect conclusions despite correctly executed analyses. Existing benchmarks fail to capture this failure mode, as they rarely assess whether a reported p-value is statistically valid given the assumptions underlying the data. We address this gap by building P-Bench, a benchmark comprising 425 open-ended, realistic hypothesis-testing tasks spanning economics, biology, and medicine. Each task requires an agent to select a statistical method, compute a p-value, and draw a conclusion given only a scientific hypothesis and a dataset. We further introduce Fisher-R1, an open-weight LLM agent trained for rigorous hypothesis testing using synthetic tasks and reinforcement learning. On P-Bench, Fisher-R1-14B substantially improves over its backbone and outperforms strong proprietary and open-source baselines, including GPT-5.4 and DeepSeekV4-Pro, achieving a 21% average relative improvement in single-trial success over DeepSeek-V4-Pro, with gains up to 26% on the most challenging tasks. Our results demonstrate that current LLM agents lack reliable statistical reasoning for hypothesis testing and that reinforcement learning on tasks with verified statistical reward substantially improves reliability.

Meta-analysis synthesis highlights a fundamental challenge in knowledge-based scientific analysis: structured evidence does not by itself represent the analytical knowledge required for executable computation. Decisions about evidence assignment, analytical contrasts, outcome and time-point alignment, effect-size formulation, and methodological admissibility must be explicit before statistical execution. Existing automated approaches often embed these decisions in model outputs, generated code, or workflow traces rather than representing them as independently verifiable knowledge. We introduce the Executable Analytical Knowledge Representation (EAKR), a machine-actionable representation of the knowledge required to transform structured evidence into executable meta-analysis. An EAKR represents evidence, relations, numerical inputs, constraints, provenance, and unresolved issues. We operationalise EAKR in MetaSynDec, an agentic harness in which large language models propose structured updates and deterministic services govern schema- and contract-based validation and execution. Across 58 synthesis units, MetaSynDec constructed all EAKRs, with 57 proceeding to statistical execution. Of 56 units with sufficient information to define a reference analysis object, 38 (67.9%) achieved complete object fidelity and 42 (75.0%) exact evidence-set agreement, with a mean Jaccard similarity of 0.909. Generated and published confidence intervals overlapped in 54 of 55 units (98.2%). MetaSynDec outperformed direct LLM generation in reference synthesis-structure agreement (57/58 versus 23/58; p<0.001) and among 23 jointly completed units, exact reference-formulation agreement (23/23 versus 1/23; p<0.001). These findings provide feasibility evidence that EAKR supports formal validation, traceability, statistical execution, and improved methodological agreement relative to direct LLM generation.

Asymptotic statistical theory is a challenging domain for AI-assisted formalization: its central results mix convergence statements, asymptotic expansions, functional analysis, and regularity conditions that have a large gap from existing infrastructure in Lean 4 formalization. To address these challenges, we propose a hypothesis-disciplined Lean 4 formalization pipeline built from multiple agents: a manager that coordinates seven specialist roles for proof planning, skeleton scaffolding, Mathlib reconnaissance, proof construction, integration, independent review, and audit. The main methodological discipline is the hypothesis-disciplined audit, implemented by the Auditor agent: every main-theorem hypothesis and concept-layer field must be anchored in the source mathematical prose, justified as a Lean encoding adapter, marked as source-implied, or rejected as an unsupported strengthening. Using this workflow, we build a systematic formalization of asymptotic statistical theory, especially the parametric and semi-parametric models' asymptotic distribution and efficiency results. The resulting Lean development is axiom-clean and source-faithful, with Lean-checked and human-audited proofs of core parametric and semi-parametric theorems organized so that theorem-agnostic infrastructure and statistical concept definitions are separated from theorem-specific assembly. The formalization results are available at https://github.com/junwei-lu/Lean-Asymptotic-Statistical-Theory.