Agentic systems driven by large language models (LLMs) are increasingly deployed in real-world workflows where they act on persistent operational data. Before deployment, these systems need to be verified against business requirements that govern workflow execution and data evolution. However, existing approaches do not provide such system-level guarantees, as they mainly constrain or analyse behaviour at the agent's interface level. We study here the verification of agentic systems comprising a single LLM and a tool orchestration harness over relational operational data. We formalise them as Stateful Tool-Enabled Agentic Deployments (STEADs), give their semantics, define the problem of verifying them against First-Order Computation Tree Logic (FO-CTL) specifications, and show that it is undecidable. We identify sufficient conditions for exact preservation of FO-CTL specifications under a finite-domain restriction, over which verification is PSPACE-complete. The key requirement is that renaming opaque identifiers in the data must correspondingly rename the selected tool calls. We show that LLM-driven agents can violate this condition and introduce a canonical deployment wrapper that guarantees it for arbitrary base agents while preserving already-equivariant behaviour. We prove that computing canonical representations required by this construction is graph-isomorphism-hard. Finally, we illustrate our framework on an LLM agent orchestrating a case-management workflow.
Bohan Chen, Shivam N. Patel, Richard Hoffmann +2cs.AI
Tool calling allows large language models (LLMs) to invoke external computation during problem solving, a useful capability in various fields including AI for mathematics. We study this setting through weighted sum-of-squares (SOS) decomposition, a machine-checkable route to proving polynomial nonnegativity and hence polynomial inequalities. A candidate decomposition can be checked exactly, but finding one requires choosing among non-unique regroupings and coordinating multiple symbolic transformations. We develop an agent that combines algebraic task training, symbolic tools, and verifier-grounded optimization for this task. Rather than training only on the composite SOS task, we construct 1.35 million synthetic examples covering eight supporting polynomial tasks together with weighted-SOS decomposition. We first apply supervised fine-tuning (SFT) to direct algebra problems and simulated symbolic traces, and then use Group Relative Policy Optimization (GRPO) with task-specific symbolic rewards. The SFT corpus contains no native tool-calling messages; at evaluation, the agent uses native SymPy calls for expansion, collection, reordering, and factorization. Every final SOS answer is checked by exact expansion and coefficient comparison. On held-out, same-generator synthetic problems, the full SFT+GRPO+tools system is the strongest of four evaluated configurations, reaching 78.96% verified success on weighted SOS, compared with 44.73% for the base model with the same tools, and 91.75% macro accuracy across nine polynomial tasks. Within this controlled setting, our work provides a case study of combining domain-specific skill training, executable tools, and verifier feedback, and may inform the design of tool-calling agents in other domains with exactly checkable outputs.
To answer a question about a program, move the program to where the question is decidable. Every such move is a translation, and every translation is a place to be wrong. We study translation as a graph -- many languages, a few reasoning targets, independently built routes of honestly different trustworthiness -- and give it a calculus: pairs of languages close commuting squares that are directional (exactness is the identity-embedding special case of over-approximation), checkable per program, and composable, a route's contract being the componentwise meet of its hops' contracts -- assurance class, direction, kept observables, measured cost. One asymmetry organizes trust: witness-carrying answers are self-certifying by replay at the source; universal answers are where grades, independent branches, and re-checked certificates earn their cost. The compositional core, lax telescope included, is mechanized in Lean 4. hurdy-gurdy implements the calculus as two planes meeting in one registry. The use plane reads declarations and produces evidence-carrying answers; its builders and its intended player are both LLMs, untrusted by construction. The evolution plane grows the graph: unmet questions are recorded as demand, pairs are recommended by evidence and registered by humans, and a ratchet keeps every prior verdict standing. Answers never write; growth never answers. Run indefinitely, the loop converges on every reducibly decidable question, at fidelity that only rises. We measure the July 2026 snapshot -- per-construct conjoined coverage, dual-route branch agreement for two ISAs, source-level witness replay, certified unreachability re-validated by a formally verified checker, escape rates for the gate itself -- and report the defects the architecture caught in its own authors' work.
Jihao Liu, Guoxiong Gao, Zeming Sun +8cs.AI cs.CL cs.MA
Recent LLM-based mathematical reasoning agents have begun to tackle research-level problems and, in several cases, have contributed to the resolution of open problems. However, scaling and orchestrating such agents effectively remains challenging, due to the difficulty of coordinating parallel proof search while keeping intermediate claims organized and reliable. In this paper, we propose Danus, an orchestration system for research-level mathematical reasoning centered on a shared fact graph as a global memory-management mechanism. Danus consists of a main agent that performs planning and coordination, multiple worker agents that carry out proof search in parallel, and a stateless verifier that checks proposed mathematical claims before they are admitted into the fact graph. Each verified fact is stored together with its proof and logical dependencies, allowing the system to build long arguments incrementally while keeping the shared proof state organized. The main agent periodically summarizes the evolving proof state, redirects workers across promising directions, and supports interaction with human mathematicians through progress reports. We evaluate Danus through six research-level case studies in algebraic geometry, singularity theory, and combinatorics, illustrating how the fact-graph memory mechanism enables Danus to construct long, detailed mathematical proofs. Our results suggest that fact-graph-based orchestration provides an effective route toward scaling mathematical reasoning agents for long-horizon research problems. Danus is open source at https://github.com/frenzymath/Danus.
Most LP-from-text benchmarks are static datasets of word problems written and labeled by hand. Once such a dataset is released, its size is fixed, its difficulty is fixed, and every problem can leak into the training data of future LLMs. We present \textbf{A$^{2}$utoLPBench}, a benchmark for testing LLM-driven agents on linear programming problems written in plain text. We first pick a feasible point and dual, then write down a problem for which that point is optimal and the objective value is known. The answer is known by construction, with no solver call and no human annotator. The evaluation environment bundles a reference solver-critic baseline and a Docker image whose usage instructions are written for an LLM-driven agent to read. With these in place, any agent can run the benchmark and get a calibrated score with one command. Because the benchmark is a generator rather than a fixed dataset, it has properties no fixed dataset can match: an unlimited supply of fresh problems, a difficulty knob set by $(n,m)$, ground-truth answers correct by construction, low LLM-side cost per problem relative to human authoring, repeatable scores across independent batches, and resistance to training-data leakage when fresh post-cutoff seed ranges are used.
Designing an algorithm from a natural-language problem statement requires identifying the problem structure, reading constraints, choosing a suitable paradigm, checking correctness, and refining complexity. Existing large language model (LLM) methods often rely on direct generation or generic self-refinement, leaving these steps implicit. We propose AlgoSkill, which models algorithm design as sequential decision-making over a typed library of algorithmic skills, including abstraction, constraint analysis, state design, data-structure selection, proof checking, counterexample construction, and complexity refinement. A learned scheduler proposes skills from the current design state, while a Monte Carlo Tree Search (MCTS) controller explores skill sequences using verification feedback from compilation, testing, stress testing, and complexity analysis. Experiments on competitive programming and combinatorial optimization benchmarks show that AlgoSkill improves over direct LLM generation, chain-of-thought prompting, self-refinement, and MCTS without typed skills. Ablations show that typed skills, verification-based repair, and search-based scheduling each contribute to performance. These results support treating automatic algorithm design as verification-guided skill scheduling rather than one-shot code generation.
We present Trellis: an autoformalization system that leverages LLM agents in a deterministically constrained workflow to enforce incremental progress in Lean autoformalization tasks through iterative refinement of natural language proofs. Our approach is motivated by the common mathematician's notion of what it means to have a rigorous proof in the first place: namely, that it would be routine to elaborate any part of the proof in further detail. The result is a system which aims to achieve reliable autoformalization on a modest budget and with generalist agents, with specialization to autoformalization coming not from any task-specific agent training but instead from a meaning-of-rigor inspired workflow enforced by process semantics. We link to an end-to-end Lean formalization of a recent Ramsey theory breakthrough produced by the process.
Large language models (LLMs) are increasingly used in workflows for generating formal proofs in Lean. These workflows often decompose problems into smaller lemmas, sample many proof attempts, and use compiler feedback to guide search. However, they can be prohibitively expensive, often spending substantial compute on attempts that ultimately fail. In this work, we address this problem with an action routing agent that consists of a data plane and a control plane. The data plane generates natural-language lemma decompositions, formalizes them in Lean, and samples proof attempts for the resulting theorem and lemma targets. The control plane observes previous failed Lean attempts, estimates both the likelihood of success and cost of another attempt, and decides whether to continue proving the current target or restart from a new breakdown. On a subset of PutnamBench, our agent decreases the cost by $25.8\%$ over a fixed-step baseline on average, preserving performance while using substantially less compute. These results suggest that failed Lean trajectories provide actionable signals for cost-aware resource allocation in agentic theorem proving.
While Large Language Models (LLMs) have shown strong performance in generating formal proofs, their outputs often remain less readable, modular, maintainable, and reusable than proofs in mature formal mathematics libraries. We argue that this gap stems in part from the compile-first objective implicit in most proof-generation pipelines, which encourages monolithic or ad hoc proof scripts rather than library-quality artifacts. Existing approaches to proof-quality improvement often rely on explicit, computable optimization objectives. In practice, however, the most tractable and experimentally validated objectives are largely length-based, while higher-level qualities such as readability, modularity, maintainability, and reusability are difficult to reduce to reliable automatic metrics. Instead of optimizing proof improvement against a single proxy metric, we take a process-guided approach inspired by human proof-refactoring workflows. We propose an agentic framework $\textbf{Proof-Refactor}$ that decomposes proof refactoring into four phases: extracting candidate proof fragments, designing helper declarations, formally proving the extracted and designed components, and repairing the original proof using the verified components. On generated Lean proofs from PutnamBench and Putnam2025, Proof-Refactor improves rubric-based refactoring scores over a strong Claude Code refactoring baseline, with the largest gains in signature quality and human readability. These results suggest that process-guided refactoring can improve proof structure without treating proof length as the primary objective.
Chenyang An, Qihao Ye, Minghao Pan +1cs.AI math.AP
We explore a central question in AI for mathematics: can AI systems produce original, nontrivial proofs for open research problems? Despite strong benchmark performance, producing genuinely novel proofs remains an outstanding challenge for LLMs. Through systematic experiments with frontier LLMs on research-level proof tasks, we identify seven failure modes that prevent reliable proof generation, including context contamination, citation hallucination, hand-waving on key steps and misallocation of proof effort, unstable proof plans, unfocused verification, problem modification and single-model bottleneck. We argue that the gap between benchmark success and research-level proving is primarily one of system design, due to those failure modes. We present QED, an open-source multi-agent proof system in which each architectural decision directly addresses a specific failure mode. Evaluated on five open problems in applied analysis and PDEs contributed by domain experts, QED produces correct proofs for three problems, each verified by the contributing experts as original and nontrivial. QED is released as open-source software at https://github.com/proofQED/QED.