We present Prover Agent, a novel AI agent for automated theorem proving that integrates large language models (LLMs) with a formal proof assistant, Lean. Prover Agent coordinates an informal reasoning LLM, a formal prover model, and feedback from Lean while also generating auxiliary lemmas to assist in discovering the overall proof strategy. It achieves an 86.1% success rate on the MiniF2F benchmark, establishing a new state-of-the-art among methods using small language models (SLMs) with a much lower sample budget than previous approaches. We also present case studies illustrating how these generated lemmas contribute to solving challenging problems.
Interactive theorem provers (ITPs) are powerful tools for the formal verification of mathematical proofs down to the axiom level. However, their lack of a natural language interface remains a significant limitation. Recent advancements in large language models (LLMs) have enhanced the understanding of natural language inputs, paving the way for autoformalization - the process of translating natural language proofs into formal proofs that can be verified. Despite these advancements, existing autoformalization approaches are limited to verifying complete proofs and lack the capability for finer, sentence-level verification. To address this gap, we propose StepProof, a novel autoformalization method designed for granular, step-by-step verification. StepProof breaks down complete proofs into multiple verifiable subproofs, enabling sentence-level verification. Experimental results demonstrate that StepProof significantly improves proof success rates and efficiency compared to traditional methods. Additionally, we found that minor manual adjustments to the natural language proofs, tailoring them for step-level verification, further enhanced StepProof's performance in autoformalization.
This paper argues that AI alignment is not merely difficult, but is founded on a fundamental logical contradiction. We first establish The Enumeration Paradox: we use machine learning precisely because we cannot enumerate all necessary safety rules, yet making ML safe requires examples that can only be generated from the very enumeration we admit is impossible. This paradox is then confirmed by a set of five independent mathematical proofs, or "pillars of impossibility." Our main results show that: (1) Geometric Impossibility: The set of safe policies has measure zero, a necessary consequence of projecting infinite-dimensional world-context requirements onto finite-dimensional models. (2) Computational Impossibility: Verifying a policy's safety is coNP-complete, even for non-zero error tolerances. (3) Statistical Impossibility: The training data required for safety (abundant examples of rare disasters) is a logical contradiction and thus unobtainable. (4) Information-Theoretic Impossibility: Safety rules contain more incompressible, arbitrary information than any feasible network can store. (5) Dynamic Impossibility: The optimization process for increasing AI capability is actively hostile to safety, as the gradients for the two objectives are generally anti-aligned. Together, these results demonstrate that the pursuit of safe, highly capable AI is not a matter of overcoming technical hurdles, but of confronting fundamental, interlocking barriers. The paper concludes by presenting a strategic trilemma that these impossibilities force upon the field. A formal verification of the core theorems in Lean4 is currently in progress.
We present LeanTutor, a Large Language Model (LLM)-based tutoring system for math proofs. LeanTutor interacts with the student in natural language, formally verifies student-written math proofs in Lean, generates correct next steps, and provides the appropriate instructional guidance. LeanTutor is composed of three modules: (i) an autoformalizer/proof-checker, (ii) a next-step generator, and (iii) a natural language feedback generator. The first module faithfully autoformalizes student proofs into Lean and verifies proof accuracy via successful code compilation. If the proof has an error, the incorrect step is identified. The next-step generator module outputs a valid next Lean tactic for incorrect proofs via LLM-based candidate generation and proof search. The feedback generator module leverages Lean data to produce a pedagogically-motivated natural language hint for the student user. To evaluate our system, we introduce PeanoBench, a human-written dataset derived from the Natural Numbers Game, consisting of 371 Peano Arithmetic proofs, where each natural language proof step is paired with the corresponding logically equivalent tactic in Lean. The Autoformalizer correctly formalizes 57% of tactics in correct proofs and accurately identifies the incorrect step in 30% of incorrect proofs. In generating natural language hints for erroneous proofs, LeanTutor outperforms a simple baseline on accuracy and relevance metrics.
While large language models (LLMs) with Chain-of-Thought (CoT) reasoning excel in mathematics and coding, their potential for systematic reasoning in chemistry, a domain demanding rigorous structural analysis for real-world tasks like drug design and reaction engineering, remains untapped. Current benchmarks focus on simple knowledge retrieval, neglecting step-by-step reasoning required for complex tasks such as molecular optimization and reaction prediction. To address this, we introduce ChemCoTBench, a reasoning framework that bridges molecular structure understanding with arithmetic-inspired operations, including addition, deletion, and substitution, to formalize chemical problem-solving into transparent, step-by-step workflows. By treating molecular transformations as modular "chemical operations", the framework enables slow-thinking reasoning, mirroring the logic of mathematical proofs while grounding solutions in real-world chemical constraints. We evaluate models on two high-impact tasks: Molecular Property Optimization and Chemical Reaction Prediction. These tasks mirror real-world challenges while providing structured evaluability. By providing annotated datasets, a reasoning taxonomy, and baseline evaluations, ChemCoTBench bridges the gap between abstract reasoning methods and practical chemical discovery, establishing a foundation for advancing LLMs as tools for AI-driven scientific innovation.
Large Language Models have demonstrated remarkable capabilities in natural language processing tasks, including mathematical problem-solving that requires multi-step logical reasoning. However, challenges persist in automating the identification of key mathematical concepts, understanding their interrelations, and formalizing proofs within a rigorous framework. We present a novel framework that leverages knowledge graphs to augment LLMs to construct and formalize mathematical proofs. Our results demonstrate significant performance improvements across multiple datasets, with using knowledge graphs, achieving up to a 34% success rate on the MUSTARDSAUCE dataset on o1-mini and consistently outperforming baseline approaches by 2-11% across different models. We show how this approach bridges the gap between natural language understanding and formal logic proof systems and achieve elevated results for foundation models over baseline.
Probabilities of causation are fundamental to modern decision-making. Pearl first introduced three binary probabilities of causation, and Tian and Pearl later derived tight bounds for them using Balke's linear programming. The theoretical characterization of probabilities of causation with multi-valued treatments and outcomes has remained unresolved for decades, limiting the scope of causality-based decision-making. In this paper, we resolve this foundational gap by proposing a complete set of representative probabilities of causation and proving that they are sufficient to characterize all possible probabilities of causation within the framework of Structural Causal Models (SCMs). We then formally derive tight bounds for these representative quantities using formal mathematical proofs. Finally, we demonstrate the practical relevance of our results through illustrative toy examples.
Formal methods is pivotal for verifying the reliability of critical systems through rigorous mathematical proofs. However, its adoption is hindered by labor-intensive manual proofs and the expertise required to use theorem provers. Recent advancements in large language models (LLMs) offer new opportunities for automated theorem proving. Two promising approaches are generating tactics step by step and generating a whole proof directly with an LLM. However, existing work makes no attempt to combine the two approaches. In this work, we introduce HybridProver, a dual-model proof synthesis framework that combines tactic-based generation and whole-proof synthesis to harness the benefits of both approaches. HybridProver generates whole proof candidates for evaluation directly, then extracts proof sketches from those candidates. It then uses a tactic-based generation model that integrates automated tools to complete the sketches via stepwise refinement. We implement HybridProver for the Isabelle theorem prover and fine-tune LLMs on our optimized Isabelle datasets. Evaluation on the miniF2F dataset illustrates HybridProver's effectiveness. We achieve a 59.4% success rate on miniF2F, where the previous SOTA is 56.1%. Our ablation studies show that this SOTA result is attributable to combining whole-proof and tactic-based generation. Additionally, we show how the dataset quality, training parameters, and sampling diversity affect the final result during automated theorem proving with LLMs. All of our code, datasets, and LLMs are open source.
Ensuring safe and human-like trajectory planning for automated vehicles amidst real-world uncertainties remains a critical challenge. While existing car-following models often struggle to consistently provide rigorous safety proofs alongside human-like acceleration and deceleration patterns, we introduce a novel multi-phase projection-based car-following model. This model is designed to balance safety and performance by incorporating bounded acceleration and deceleration rates while emulating key human driving principles. Building upon a foundation of fundamental driving principles and a multi-phase dynamical systems analysis (detailed in Part 1 of this study \citep{jin2025WA20-02_Part1}), we first highlight the limitations of extending standard models like Newell's with simple bounded deceleration. Inspired by human drivers' anticipatory behavior, we mathematically define and analyze projected braking profiles for both leader and follower vehicles, establishing safety criteria and new phase definitions based on the projected braking lead-vehicle problem. The proposed parsimonious model combines an extended Newell's model for nominal driving with a new control law for scenarios requiring projected braking. Using speed-spacing phase plane analysis, we provide rigorous mathematical proofs of the model's adherence to defined safe and human-like driving principles, including collision-free operation, bounded deceleration, and acceptable safe stopping distance, under reasonable initial conditions. Numerical simulations validate the model's superior performance in achieving both safety and human-like braking profiles for the stationary lead-vehicle problem. Finally, we discuss the model's implications and future research directions.
The generation of high-quality synthetic data presents significant challenges in machine learning research, particularly regarding statistical fidelity and uncertainty quantification. Existing generative models produce compelling synthetic samples but lack rigorous statistical guarantees about their relation to the underlying data distribution, limiting their applicability in critical domains requiring robust error bounds. We address this fundamental limitation by presenting a novel framework that incorporates conformal prediction methodologies into Generative Adversarial Networks (GANs). By integrating multiple conformal prediction paradigms including Inductive Conformal Prediction (ICP), Mondrian Conformal Prediction, Cross-Conformal Prediction, and Venn-Abers Predictors, we establish distribution-free uncertainty quantification in generated samples. This approach, termed Conformalized GAN (cGAN), demonstrates enhanced calibration properties while maintaining the generative power of traditional GANs, producing synthetic data with provable statistical guarantees. We provide rigorous mathematical proofs establishing finite-sample validity guarantees and asymptotic efficiency properties, enabling the reliable application of synthetic data in high-stakes domains including healthcare, finance, and autonomous systems.
Large language models (LLMs) have shown promise in formal theorem proving, but their token-level processing often fails to capture the inherent hierarchical nature of mathematical proofs. We introduce \textbf{Hierarchical Attention}, a regularization method that aligns LLMs' attention mechanisms with mathematical reasoning structures. Our approach establishes a five-level hierarchy from foundational elements to high-level concepts, ensuring structured information flow in proof generation. Experiments demonstrate that our method improves proof success rates by 2.05\% on miniF2F and 1.69\% on ProofNet while reducing proof complexity by 23.81\% and 16.50\% respectively. The code is available at https://github.com/Car-pe/HAGBP.
Compositional generalization is a crucial property in artificial intelligence, enabling models to handle novel combinations of known components. While most deep learning models lack this capability, certain models succeed in specific tasks, suggesting the existence of governing conditions. This paper derives a necessary and sufficient condition for compositional generalization in neural networks. Conceptually, it requires that (i) the computational graph matches the true compositional structure, and (ii) components encode just enough information in training. The condition is supported by mathematical proofs. This criterion combines aspects of architecture design, regularization, and training data properties. A carefully designed minimal example illustrates an intuitive understanding of the condition. We also discuss the potential of the condition for assessing compositional generalization before training. This work is a fundamental theoretical study of compositional generalization in neural networks.
The AI alignment problem, which focusses on ensuring that artificial intelligence (AI), including AGI and ASI, systems act according to human values, presents profound challenges. With the progression from narrow AI to Artificial General Intelligence (AGI) and Superintelligence, fears about control and existential risk have escalated. Here, we investigate whether embracing inevitable AI misalignment can be a contingent strategy to foster a dynamic ecosystem of competing agents as a viable path to steer them in more human-aligned trends and mitigate risks. We explore how misalignment may serve and should be promoted as a counterbalancing mechanism to team up with whichever agents are most aligned to human interests, ensuring that no single system dominates destructively. The main premise of our contribution is that misalignment is inevitable because full AI-human alignment is a mathematical impossibility from Turing-complete systems, which we also offer as a proof in this contribution, a feature then inherited to AGI and ASI systems. We introduce a change-of-opinion attack test based on perturbation and intervention analysis to study how humans and agents may change or neutralise friendly and unfriendly AIs through cooperation and competition. We show that open models are more diverse and that most likely guardrails implemented in proprietary models are successful at controlling some of the agents' range of behaviour with positive and negative consequences while closed systems are more steerable and can also be used against proprietary AI systems. We also show that human and AI intervention has different effects hence suggesting multiple strategies.
This document serves as supplementary material for a journal paper submission, presenting detailed mathematical proofs and derivations that support the results outlined in the main manuscript. In this work, we formulate an Integrated sensing and communication (ISAC) optimization problem aimed at fairly maximizing beampattern gains across multiple directions of interest, while ensuring that the signal-to-interference-plus-noise ratio (SINR) requirements for communication users and the overall power constraint are satisfied. To solve this problem, we propose an optimization algorithm and provide a formal proof of its convergence.
The topological and geometrical features at the boundary zone of planar polygonal networks remain poorly understood. Based on observations and mathematical proofs, we propose that marginal cells in Pyropia haitanensis thalli, a two-dimensional (2D) biological polygonal network, have an average edge number of exactly five. We demonstrate that this number is maintained by specific division patterns. Furthermore, we reveal significant limitations of Lewis law and Aboav-Weaire law by comparing the topological and geometrical parameters of marginal cells and inner cells. We find strong boundary effects that are manifested in the distinct distributions of interior angles and edge lengths in marginal cells. Similar to inner cells, cell division tend to occur in marginal cells with large sizes. Our findings suggest that inner cells should be strictly defined based on their positional relationship to the marginal cells.
Large Language Models (LLMs) have demonstrated significant potential in generating mathematical proofs. However, a persistent challenge is that LLMs occasionally make mistakes, while even a minor mistake can invalidate an entire proof. Proof assistants like Lean offer a great remedy. They are designed for verifying each step of a proof in a formal language, and in recent years researchers have created AI models to generate proofs in their languages. However, the scarcity of large-scale datasets of Lean proofs restrict the performance of such Automated Theorem Proving (ATP) models. We developed LeanNavigator, a novel method for generating a large-scale dataset of Lean theorems and proofs by finding new ways to prove existing Lean theorems. By leveraging an interactive Lean client and an efficient method for proof step generation, LeanNavigator efficiently produces new theorems with corresponding proofs. Applying this approach to Mathlib4, we generated 4.7 million theorems totaling 1 billion tokens, surpassing previous datasets by more than an order of magnitude. Using this extensive dataset, we trained an AI model that outperforms the state-of-the-art ReProver model in theorem-proving tasks. These results confirm our hypothesis and demonstrate the critical role of large datasets in improving the performance of automated theorem provers.
3D anomaly detection (AD) is prominent but difficult due to lacking a unified theoretical foundation for preprocessing design. We establish the Fence Theorem, formalizing preprocessing as a dual-objective semantic isolator: (1) mitigating cross-semantic interference to the greatest extent feasible and (2) confining anomaly judgments to aligned semantic spaces wherever viable, thereby establishing intra-semantic comparability. Any preprocessing approach achieves this goal through a two-stage process of Emantic-Division and Spatial-Constraints stage. Through systematic deconstruction, we theoretically and experimentally subsume existing preprocessing methods under this theorem via tripartite evidence: qualitative analyses, quantitative studies, and mathematical proofs. Guided by the Fence Theorem, we implement Patch3D, consisting of Patch-Cutting and Patch-Matching modules, to segment semantic spaces and consolidate similar ones while independently modeling normal features within each space. Experiments on Anomaly-ShapeNet and Real3D-AD with different settings demonstrate that progressively finer-grained semantic alignment in preprocessing directly enhances point-level AD accuracy, providing inverse validation of the theorem's causal logic.
The demand for synthetic data in mathematical reasoning has increased due to its potential to enhance the mathematical capabilities of large language models (LLMs). However, ensuring the validity of intermediate reasoning steps remains a significant challenge, affecting data quality. While formal verification via theorem provers effectively validates LLM reasoning, the autoformalisation of mathematical proofs remains error-prone. In response, we introduce iterative autoformalisation, an approach that iteratively refines theorem prover formalisation to mitigate errors, thereby increasing the execution rate on the Lean prover from 60% to 87%. Building upon that, we introduce Theorem Prover as a Judge (TP-as-a-Judge), a method that employs theorem prover formalisation to rigorously assess LLM intermediate reasoning, effectively integrating autoformalisation with synthetic data generation. Finally, we present Reinforcement Learning from Theorem Prover Feedback (RLTPF), a framework that replaces human annotation with theorem prover feedback in Reinforcement Learning from Human Feedback (RLHF). Across multiple LLMs, applying TP-as-a-Judge and RLTPF improves benchmarks with only 3,508 samples, achieving 5.56% accuracy gain on Mistral-7B for MultiArith, 6.00% on Llama-2-7B for SVAMP, and 3.55% on Llama-3.1-8B for AQUA.
Large Language Models (LLMs) have demonstrated strong capabilities across various domains, with recent advancements in challenging reasoning tasks such as mathematics and programming. However, solving reasoning tasks often requires long decoding chains (of thoughts), which incur $O(N)$ time and memory consumption, where $N$ is the chain length. To mitigate $O(N)$ time and memory consumption, existing sparsity-based algorithms propose retaining only the most critical token's intermediate data (i.e., key-value cache) and discarding the rest. However, these existing algorithms struggle with the ``impossible trinity'' of accuracy, time, and memory. For example, the state-of-the-art algorithm, Quest, achieves high accuracy with $O(L)$ time but $O(N)$ memory ($L$ is the cache budget, $L \ll N$). To address this issue, in this paper, we identify a new attention pattern during the decode stage of reasoning tasks, where milestone tokens (analogous to lemmas in mathematical proofs) emerge, are utilized, and then become unimportant afterward. Based on this pattern, we propose a new algorithm named RaaS that identifies and retains milestone tokens only until they are no longer needed, achieving high accuracy with $O(L)$ time and $O(L)$ memory complexity.
Developing the logic necessary to solve mathematical problems or write mathematical proofs is one of the more difficult objectives for large language models (LLMS). Currently, the most popular methods in literature consists of fine-tuning the model on written mathematical content such as academic publications and textbooks, so that the model can learn to emulate the style of mathematical writing. In this project, we explore the effectiveness of using retrieval augmented generation (RAG) to address gaps in the mathematical reasoning of LLMs. We develop LemmaHead, a RAG knowledge base that supplements queries to the model with relevant mathematical context, with particular focus on context from published textbooks. To measure our model's performance in mathematical reasoning, our testing paradigm focuses on the task of automated theorem proving via generating proofs to a given mathematical claim in the Lean formal language.