Daily Papers

What makes a short story gripping; a news article newsworthy; or a math proof elegant? These constructs resist articulation or verification; their meaning is at least partially tacit. However, modern AI models are improved primarily via articulated constitutions, rubrics and verifiers (i.e. in RLAIF and RLVR); tacit components of preferences are typically understudied. We introduce a large, labeled preference dataset CreativePreferences, containing 2.8M texts labeled by 317M human preference judgments across 7 creative domains, with 42 benchmark tasks. We model these labels with executable programs, rubric banks and densely trained models (V, A and VAT, respectively). We observe robust articulability gaps, VAT-VA; and verifiability gaps, VAT-V; we estimate upper and lower bounds for each gap with a novel measurement approach that discovers articulable and verifiable metrics, identifies spurious variables and estimates the value of undiscovered metrics using capture-recapture. These gaps occur across all domains, even in domains traditionally treated as fully verifiable: correctness-centered domains (i.e. mathematics and software engineering) and claim- and novelty-centric domains (i.e. news, patents, peer review). The size of the gap varies based on domain (e.g. peer review and creative writing have the largest articulability gaps) and widens as more people take part in the judgment, consistent with Collins' collective tacit knowledge. We show two consequences: (1) on human generations, the full model more closely matches human preferences, often in disagreement with articulated criteria, and (2) in an analogy to Goodhart's law, articulating preference shifts it away from the tacit dimension. Articulability and verifiability gaps are consequential; we give recommendations on when tasks can be prompted; how learning mechanisms might improve; and when to leave judgments with humans.

Large language models (LLMs) struggle with multi-step reasoning, where inference-time scaling has emerged as a promising strategy for performance improvement. Verifier-guided search outperforms repeated sampling when sample size is limited by selecting and prioritizing valid reasoning paths. However, we identify a critical limitation: scaling flaws, prevalent across different models (Mistral 7B and DeepSeekMath 7B), benchmarks (GSM8K and MATH), and verifiers (outcome value models and process reward models). As sample size increases, verifier-guided search exhibits diminishing advantages and eventually underperforms repeated sampling. Our analysis attributes this to verifier failures, where imperfect verifiers misrank candidates and erroneously prune all valid paths. These issues are further exacerbated in challenging and out-of-distribution problems, restricting search effectiveness. To mitigate verifier failures, we explore reducing reliance on verifiers and conduct preliminary investigations using two simple methods. Our findings reveal fundamental limitations in verifier-guided search and suggest future directions.

We describe a verification pipeline that takes production Rust cryptographic code and produces machine-checked correctness proofs in Lean 4. The pipeline combines three components: symbolic extraction tools (Charon and Aeneas, or Hax) that lift Rust into Lean 4; formal cryptographic specification libraries (ArkLib and CompPoly, from the Verified zkEVM project) that provide the mathematical targets; and AI provers (Aristotle from Harmonic AI and Aleph from Logical Intelligence) that close the resulting proof obligations. Every proof is checked by the Lean kernel, so AI output cannot compromise soundness. Within the scope of the Ethereum Foundation's zkEVM Verification Project, we applied the pipeline to cryptographic primitives in Plonky3 (FRI folding, Mersenne31 and KoalaBear field arithmetic, Horner polynomial evaluation) and RISC Zero (Merkle inclusion verification). In addition, Aleph authored proofs of two bounds-style theorems in Plonky3's compute_log_arity_for_round that previously stood as sorry. The paper describes the architecture, walks through a running example based on Aleph's two proofs, reports which classes of proof obligations AI closed and which required manual work, and discusses the engineering gaps we encountered: Lean 4 toolchain drift across tools and specific Aeneas/Hax extraction limits. We also document concrete missing lemmas, tactic gaps, and code-generation friction points discovered during proof development. We hope this contribution lowers the barrier to adoption of formal verification and facilitates more effective use of AI in this pipeline. The result is a working pipeline for formal verification of Rust, with kernel-checked proofs and reproducible artefacts.

When should an AI system's answer be trusted? Formal proof assistants offer certainty but cannot reach most of the problem distribution; scalar LLM judges offer coverage but produce opaque scores that cannot be audited after the fact and are subject to the same coherence issues as any LLM. We present Theoria, a verification architecture that closes this gap. A candidate solution is rewritten into a sequence of typed state transitions, each licensed by an explicit justification, whether that be a citation, computation, or problem-given fact, and every transition is independently auditable. The foundational invariant is completeness of change: every difference between consecutive proof states must be accounted for, so hidden premises surface as unlicensed mutations rather than passing silently. On HLE-Verified Gold (185 text-only expert problems), Theoria certifies 105 at 91.4% strict precision (Wilson 95% CI [84.5%, 95.4%]). Every certification produces a human readable proof trace in which each step can be independently challenged. Holistic LLM judges achieve comparable precision at matched coverage but fail on different problems (Jaccard 0.14-0.36), making the approaches complementary. On 95 adversarial poisoned proofs across 15 domains, structured judges catch 94.7% versus 83.2% for holistic judging (p= 0.0017). The overall 11.5 pp gap concentrates in hidden premises (90.6% vs. 62.5%, a 28 pp difference) and fabricated citations (100% vs. 90%), the error classes where the formal analysis predicts an advantage; performance is identical on arithmetic and theorem-misapplication errors, where no advantage is predicted. On GPQA Diamond (n= 65), certified precision is 97.1% (Wilson CI [85.1%, 99.5%]).

Large language models (LLMs) are increasingly integrated in software development, but ensuring correctness in LLM-generated code remains challenging and often requires costly manual review. Verifiable code generation -- jointly generating code, specifications, and proofs of code-specification alignment -- offers a promising path to address this limitation and further unleash LLMs' benefits in coding. Yet, there exists a significant gap in evaluation: current benchmarks often lack support for end-to-end verifiable code generation. In this paper, we introduce Verina (Verifiable Code Generation Arena), a high-quality benchmark enabling a comprehensive and modular evaluation of code, specification, and proof generation as well as their compositions. Verina consists of 189 manually curated coding tasks in Lean, with detailed problem descriptions, reference implementations, formal specifications, and extensive test suites. Our extensive evaluation of state-of-the-art LLMs reveals significant challenges in verifiable code generation, especially in proof generation, underscoring the need for improving LLM-based theorem provers in verification domains. The best model, OpenAI o4-mini, generates only 61.4% correct code, 51.0% sound and complete specifications, and 3.6% successful proofs, with one trial per task. We hope Verina will catalyze progress in verifiable code generation by providing a rigorous and comprehensive benchmark. We release our dataset on https://huggingface.co/datasets/sunblaze-ucb/verina and our evaluation code on https://github.com/sunblaze-ucb/verina.

From a first-principles perspective, it may seem odd that the strongest results in foundation model fine-tuning (FT) are achieved via a relatively complex, two-stage training procedure. Specifically, one first trains a reward model (RM) on some dataset (e.g. human preferences) before using it to provide online feedback as part of a downstream reinforcement learning (RL) procedure, rather than directly optimizing the policy parameters on the dataset via offline maximum likelihood estimation. In fact, from an information-theoretic perspective, we can only lose information via passing through a reward model and cannot create any new information via on-policy sampling. To explain this discrepancy, we scrutinize several hypotheses on the value of RL in FT through both theoretical and empirical lenses. Of the hypotheses considered, we find the most support for the explanation that on problems with a generation-verification gap, the combination of the ease of learning the relatively simple RM (verifier) from the preference data, coupled with the ability of the downstream RL procedure to then filter its search space to the subset of policies (generators) that are optimal for relatively simple verifiers is what leads to the superior performance of online FT.

Visible tests are a common gate for LLM-generated code, but passing them does not certify specification correctness. We study a deployment-like monitoring problem: after code has passed public tests, can a weaker LLM verifier identify the residual hidden bugs? We introduce Code Monitor Red Teaming, a monitor-red-teaming protocol that fixes a public-check information boundary while varying generator pressure, verifier scaffolding, and weak-to-strong capability. We instantiate it as CodeMonitorBench, spanning function-level, data-science, and workflow code. Across 71,000 generated candidates, 43,677 pass public tests and 23,081 of those fail hidden tests. Weak verifiers improve with scaffolding and model family, but still miss most hidden bugs at 5% false-positive rate. As a robustness stress test, adversarial public-test-overfit pressure lowers verifier AUROC and raises low-FPR miss rates in most cells. A GLM-5.1 verifier recovers part of the gap under the same evidence boundary; an inferability audit shows that remaining misses mix verifier failures with M1 evidence limits.

Agents need reliable ways to validate their outputs, both when completing tasks and when filtering synthetic training data. We study verifier generation: from the instruction and environment available for solving a task, an agent writes an executable verifier that must accept every valid solution and reject invalid ones. We introduce VerifierBench, a benchmark of 222 tasks from six agent benchmarks spanning tool-call, state-based, and code verification, with 4,993 hidden test cases that include alternative valid solutions and attacks; Verifier4Verifier (V4V), a six-check audit, screens its source verifiers and test suites. Across 19 models in a common agent harness, normalized balanced accuracy ranges from 10.6 to 62.1, with generated verifiers consistently rejecting invalid solutions more reliably than they accept valid ones. This apparent strength can mask overly restrictive verification: most false rejections in the analyzed records come from verifiers that accept no tested solution to a task. Case studies reveal unsupported assumptions about solution formats, tool use, and code changes that exclude valid solutions. These errors survive self-testing because agents test solutions that share the verifier's mistaken assumptions, or submit verifiers without confirming that any correct solution passes. These findings suggest testing both whether a verifier accepts a correct solution and whether it accepts other solutions permitted by the task. The project page and dataset are publicly available.

It is tempting to assume any task solvable by a short program can be taught to a model as its chain-of-thought: write the steps out, fine-tune, and the model follows. This paper shows the assumption fails for an identifiable class of procedures. The testbed is nine reasoning tasks, each from a deterministic generator; public and hidden splits share generators, so held-out data proxies test accuracy. I reverse-engineer the generators into Python solvers, render them as chain-of-thought, and distill into a rank-<= 32 LoRA over a 30B (3.5B-active) Nemotron model. Forward-computable tasks install readily: lookup/arithmetic and an 8-bit boolean task transfer (>= 0.99 and 0.68). Cryptarithm does not: distilling its backtracking search holds at 0.01-0.07 across eleven chain-of-thought designs, RL from verifiable rewards, and self-training, even though a search solver answers 71% of instances. This is not a capability gap. The model does the arithmetic on 97-100% of lines and ranks the correct cipher in its top eight on 71%; it cannot carry the search forward as a left-to-right derivation. Fine-tuning learns the shape of a verifiable elimination step while its verdicts become unconditional templates, correct only 16-57% of the time ("verdict-as-token"). The ceiling holds across backbones from 3B to 671B and across fine-tuning and prompting; a controlled intervention isolates the cause: revealing the cipher key, which turns the derivation forward, lifts the same instances from 0.03 to 0.57. When a procedure's only solution is search over information-free structure, no faithful forward chain-of-thought exists to imitate. The task becomes learnable only by removing the search, precomputing its combinatorial core into a catalog and reducing the trace to recall plus verification; the 1st-place solution reaches Private LB 0.92 this way. What distills is memorization and verification, not search.

Large language models often produce plausible numerical claims from financial filings while using the wrong reporting period, unit, line item, or formula. Verifying such claims requires more than rechecking arithmetic: a verifier must ground the relevant facts, establish the authorized calculation, and determine whether the candidate value follows from both. We propose VeriFin, a neurosymbolic verification framework for numerical financial question answering that grounds operands in filed XBRL facts, derives calculations from the question, filing linkbases, or documented metric definitions, and checks claims using Z3. When a claim is inconsistent, solver-derived unsatisfiable cores identify the conflicting facts, formula, and candidate value, enabling targeted repair. We evaluate VeriFin on XBRLFiling, a new 600-question benchmark constructed from 10-K filings of 28 U.S. companies, and on FinanceBench. On fixed candidate pools shared by all verification methods, VeriFin accepts none of the incorrect claims, whereas the baselines accept 6 to 92 incorrect claims on XBRLFiling's 600 claims and 4 to 21 incorrect claims on FinanceBench's 67 claims. The zero false-accept result persists across multiple answer-generator models. Solver-derived feedback further improves repair, achieving up to 69.9% recovery among true catches. These results show that source-grounded symbolic verification can provide a reliable and auditable acceptance control for LLM-generated financial claims.

Automated formal verification remains challenging for large language models because data for proof assistants and verification-aware languages is scarce, and correctness depends on satisfying precise machine-checkable specifications rather than producing plausible code. This thesis studies how verifier environments can improve LLM generation of verified programs and proofs through reinforcement learning from verifiable rewards (RLVR) and verifier-guided inference-time search. First, we train open-source models in Dafny with RLVR using Group Relative Policy Optimization (GRPO) and related variants, assembling generated candidates into complete programs and scoring them with compiler and verifier outcomes. Initial experiments on an APPS-derived Dafny dataset increased verified reward from 2.2% to 58.1%, but revealed specification hacking, where models exploit weak formal specifications instead of implementing the intended solutions. After filtering underspecified and vulnerable tasks, multi-turn RLVR on the refined benchmark improves the verified pass rate from 9.7% to 31.1%. Second, we develop a verifier-guided inference scaffold in Lean that treats proof generation as structured search over decomposed subgoals, verifier feedback, diagnostics, and repair. With a fixed base model, the full scaffold with proof reviser improves pass rate on an initial VeriCoding pilot set from 46.2% under direct repair to 69.2%. On the larger VERINA dataset, whole-task decomposition plus proof reviser solves 7 of 42 previously unsolved tasks. We also introduce Dalek-Bench, a repository-scale Lean benchmark derived from the Rust curve25519-dalek verification project; preliminary results remain weak, indicating that stronger progress evaluation and task-specific tool-use policies are still needed.

Vericoding refers to the generation of formally verified code from rigorous specifications. Recent AI models show promise in vericoding, but a unified methodology for cross-paradigm evaluation is lacking. Existing benchmarks test only individual languages/tools (e.g., Dafny, Verus, and Lean) and each covers very different tasks, so the performance numbers are not directly comparable. We address this gap with AlgoVeri, a benchmark that evaluates vericoding of 77 classical algorithms in Dafny, Verus, and Lean. By enforcing identical functional contracts, AlgoVeri reveals critical capability gaps in verification systems. While frontier models achieve tractable success in Dafny (40.3% for Gemini-3 Flash), where high-level abstractions and SMT automation simplify the workflow, performance collapses under the systems-level memory constraints of Verus (24.7%) and the explicit proof construction required by Lean (7.8%). Beyond aggregate metrics, we uncover a sharp divergence in test-time compute dynamics: Gemini-3 effectively utilizes iterative repair to boost performance (e.g., tripling pass rates in Dafny), whereas GPT-OSS saturates early. Finally, our error analysis shows that language design affects the refinement trajectory: while Dafny allows models to focus on logical correctness, Verus and Lean trap models in persistent syntactic and semantic barriers. All data and evaluation code can be found at https://github.com/haoyuzhao123/algoveri.

Speculative decoding typically characterizes speedup by draft tokens accepted per verification round. Lossless verification preserves the target model's output distribution and thus expected completion length, making this metric a meaningful efficiency proxy. Relaxed (lossy) verification accepts tokens that lossless verification would reject, potentially altering that distribution. We identify a length inflation phenomenon: relaxed verification lengthens completions, primarily by extending reasoning traces, largely offsetting apparent gains in accepted tokens per round. We evaluate existing training-free relaxed verification rules on twelve target-drafter pairs against lossless verification on identical prompts, measuring completion length, verifier rounds, wall-clock time, and task accuracy. At each rule's most relaxed evaluated setting, relaxed verification on average accepts 1.34times as many tokens per round but produces 1.54times longer completions. For rules with higher acceptance lengths, tokens accepted only under relaxed verification tend to have low target-model probability, contributing to length inflation. For rules that reduce verifier rounds, accuracy typically drops substantially. Further analysis traces some failures to continued reasoning that exhausts the budget needed to deliver a final answer. Overall, accepted tokens per round alone is an unreliable measure of decoding speed under relaxed verification. Completion length and verifier rounds must also be reported to assess actual efficiency.

Agentic theorem provers combine a reasoning model, retrieval, search, and a proof assistant verifier, yet it remains unclear which components actually improve finite-budget proof success and why they help on real mathematical workloads. We study this question through statistical provability: the probability of reaching a verified proof within a budget on a specified stream of theorem instances. We model formal proof search as a finite-horizon reachability MDP with deterministic verifier dynamics, and show that under a faithful state abstraction the optimal success probability coincides with ordinary syntactic provability. We then analyze a simple but practically important pipeline: depth-wise offline action-value regression followed by greedy test-time proving. Our main theorem bounds the provability gap between the learned prover and the optimal prover by an occupancy-weighted sum of uniform action-value errors; in the common uniform-error reading, the leading complexity multiplier is the learned prover's average truncated proof length. The error decomposes into approximation error, geometric coverage of the training distribution, and Monte Carlo label noise, and improves to a fast rate under an action-gap margin condition. The result gives a component-sensitive account of why verifier feedback, retrieval, representation geometry, and proof-shortening mechanisms help on biased theorem workloads, without contradicting classical worst-case hardness.

Answer verification is crucial not only for evaluating large language models (LLMs) by matching their unstructured outputs against standard answers, but also serves as the reward model to guide LLM optimization. Most evaluation frameworks rely on regularized matching or employ general LLMs for answer verification, which demands extensive, repetitive customization for regex rules or evaluation prompts. Two fundamental limitations persist in current methodologies: 1) the absence of comprehensive benchmarks that systematically evaluate verification capabilities across different LLMs; and 2) the nascent stage of verifier development, where existing approaches lack both the robustness to handle complex edge cases and the generalizability across different domains. In this work, we develop CompassVerifier, an accurate and robust lightweight verifier model for evaluation and outcome reward. It demonstrates multi-domain competency spanning math, knowledge, and diverse reasoning tasks, with the capability to process various answer types, including multi-subproblems, formulas, and sequence answers, while effectively identifying abnormal/invalid responses. We introduce VerifierBench benchmark comprising model outputs collected from multiple data sources, augmented through manual analysis of metaerror patterns to enhance CompassVerifier. We anticipate that CompassVerifier and VerifierBench will facilitate answer verification, evaluation protocols, and reinforcement learning research. Code and dataset are available at https://github.com/open-compass/CompassVerifier.

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.

Reasoning in Large Language Models (LLMs) has recently shown strong potential in enhancing generative recommendation through deep understanding of complex user preference. Existing approaches follow a {reason-then-recommend} paradigm, where LLMs perform step-by-step reasoning before item generation. However, this paradigm inevitably suffers from reasoning degradation (i.e., homogeneous or error-accumulated reasoning) due to the lack of intermediate verification, thus undermining the recommendation. To bridge this gap, we propose a novel \textit{reason-verify-recommend} paradigm, which interleaves reasoning with verification to provide reliable feedback, guiding the reasoning process toward more faithful user preference understanding. To enable effective verification, we establish two key principles for verifier design: 1) reliability ensures accurate evaluation of reasoning correctness and informative guidance generation; and 2) multi-dimensionality emphasizes comprehensive verification across multi-dimensional user preferences. Accordingly, we propose an effective implementation called VRec. It employs a mixture of verifiers to ensure multi-dimensionality, while leveraging a proxy prediction objective to pursue reliability. Experiments on four real-world datasets demonstrate that VRec substantially enhances recommendation effectiveness and scalability without compromising efficiency. The codes can be found at https://github.com/Linxyhaha/Verifiable-Rec.

Large language models are often adapted through parameter efficient fine tuning, but current release practices provide weak assurances about what data were used and how updates were computed. We present Verifiable Fine Tuning, a protocol and system that produces succinct zero knowledge proofs that a released model was obtained from a public initialization under a declared training program and an auditable dataset commitment. The approach combines five elements. First, commitments that bind data sources, preprocessing, licenses, and per epoch quota counters to a manifest. Second, a verifiable sampler that supports public replayable and private index hiding batch selection. Third, update circuits restricted to parameter efficient fine tuning that enforce AdamW style optimizer semantics and proof friendly approximations with explicit error budgets. Fourth, recursive aggregation that folds per step proofs into per epoch and end to end certificates with millisecond verification. Fifth, provenance binding and optional trusted execution property cards that attest code identity and constants. On English and bilingual instruction mixtures, the method maintains utility within tight budgets while achieving practical proof performance. Policy quotas are enforced with zero violations, and private sampling windows show no measurable index leakage. Federated experiments demonstrate that the system composes with probabilistic audits and bandwidth constraints. These results indicate that end to end verifiable fine tuning is feasible today for real parameter efficient pipelines, closing a critical trust gap for regulated and decentralized deployments.

As LLMs are deployed in high-stakes settings, users must judge the correctness of individual responses, often relying on model-generated justifications such as reasoning chains or explanations. Yet, no standard measure exists for whether these justifications help users distinguish correct answers from incorrect ones. We formalize this idea as error verifiability and propose v_{bal}, a balanced metric that measures whether justifications enable raters to accurately assess answer correctness, validated against human raters who show high agreement. We find that neither common approaches, such as post-training and model scaling, nor more targeted interventions recommended improve verifiability. We introduce two methods that succeed at improving verifiability: reflect-and-rephrase (RR) for mathematical reasoning and oracle-rephrase (OR) for factual QA, both of which improve verifiability by incorporating domain-appropriate external information. Together, our results establish error verifiability as a distinct dimension of response quality that does not emerge from accuracy improvements alone and requires dedicated, domain-aware methods to address.

Organizations deploying AI face two fundamental governance challenges: managing AI risk and sustaining AI value. Both depend on evidence whose sufficiency cannot be taken for granted. We call the shared underlying challenge the AI Evaluability Gap: the condition in which organizations lack sufficient evidence to support high-confidence governance decisions regarding either risk or value. We argue that this gap reflects a category error in current practice. Existing governance approaches focus primarily on properties of systems, such as safety, fairness, reliability, compliance, and value, while paying comparatively little attention to the evidentiary foundations required to justify decisions about those properties. We further argue that AI governance encompasses both operational decisions regarding whether a system may operate and investment decisions regarding whether it merits continued organizational resources. To address this problem, we introduce Evaluability, defined as the capability of a system to generate, maintain, and renew evidence sufficient to support high-confidence governance decisions over time. We formalize governance decisions as functions of calibrated confidence Conf(D|E) and identify six properties of evaluable evidence: observability, attributability, intervenability, verifiability, calibration, and temporal validity. The framework distinguishes Operational Certification, which relies primarily on structural evidence to justify deployment decisions, from Investment Certification, which relies primarily on causal evidence to justify continued resource allocation. We argue that evidence sufficiency is a missing layer of AI governance and that closing the AI Evaluability Gap is a prerequisite for both managing risk and sustaining value in AI-enabled organizations.

Zero-Knowledge Proofs (ZKPs) are a cryptographic primitive that allows a prover to demonstrate knowledge of a secret value to a verifier without revealing anything about the secret itself. ZKPs have shown to be an extremely powerful tool, as evidenced in both industry and academic settings. In recent years, the utilization of user data in practical applications has necessitated the rapid development of privacy-preserving techniques, including ZKPs. This has led to the creation of several robust open-source ZKP frameworks. However, there remains a significant gap in understanding the capabilities and real-world applications of these frameworks. Furthermore, identifying the most suitable frameworks for the developers' specific applications and settings is a challenge, given the variety of options available. The primary goal of our work is to lower the barrier to entry for understanding and building applications with open-source ZKP frameworks. In this work, we survey and evaluate 25 general-purpose, prominent ZKP frameworks. Recognizing that ZKPs have various constructions and underlying arithmetic schemes, our survey aims to provide a comprehensive overview of the ZKP landscape. These systems are assessed based on their usability and performance in SHA-256 and matrix multiplication experiments. Acknowledging that setting up a functional development environment can be challenging for these frameworks, we offer a fully open-source collection of Docker containers. These containers include a working development environment and are accompanied by documented code from our experiments. We conclude our work with a thorough analysis of the practical applications of ZKPs, recommendations for ZKP settings in different application scenarios, and a discussion on the future development of ZKP frameworks.

Commercial LLM services often conceal internal reasoning traces while still charging users for every generated token, including those from hidden intermediate steps, raising concerns of token inflation and potential overbilling. This gap underscores the urgent need for reliable token auditing, yet achieving it is far from straightforward: cryptographic verification (e.g., hash-based signature) offers little assurance when providers control the entire execution pipeline, while user-side prediction struggles with the inherent variance of reasoning LLMs, where token usage fluctuates across domains and prompt styles. To bridge this gap, we present PALACE (Predictive Auditing of LLM APIs via Reasoning Token Count Estimation), a user-side framework that estimates hidden reasoning token counts from prompt-answer pairs without access to internal traces. PALACE introduces a GRPO-augmented adaptation module with a lightweight domain router, enabling dynamic calibration across diverse reasoning tasks and mitigating variance in token usage patterns. Experiments on math, coding, medical, and general reasoning benchmarks show that PALACE achieves low relative error and strong prediction accuracy, supporting both fine-grained cost auditing and inflation detection. Taken together, PALACE represents an important first step toward standardized predictive auditing, offering a practical path to greater transparency, accountability, and user trust.

We propose a framework for robust evaluation of reasoning capabilities of language models, using functional variants of benchmarks. Models that solve a reasoning test should exhibit no difference in performance over the static version of a problem compared to a snapshot of the functional variant. We have rewritten the relevant fragment of the MATH benchmark into its functional variant MATH(), with functionalization of other benchmarks to follow. When evaluating current state-of-the-art models over snapshots of MATH(), we find a reasoning gap -- the percentage difference between the static and functional accuracies. We find reasoning gaps from 58.35% to 80.31% among the state-of-the-art closed and open weights models that perform well on static benchmarks, with the caveat that the gaps are likely to be smaller with more sophisticated prompting strategies. Here we show that models which anecdotally have good reasoning performance over real-world tasks, have quantifiable lower gaps, motivating the open problem of building "gap 0" models. Code for evaluation and new evaluation datasets, three MATH() snapshots, are publicly available at https://github.com/consequentai/fneval/.

As LLM agents undertake increasingly complex, long-horizon tasks, verifying their outputs becomes increasingly challenging. We study how verification capability can be strengthened with a fixed base model, without access to reference answers or grading rubrics at test time. Repeated sampling yields multiple rollouts that can contain complementary correct claims, but we need a reliable verification mechanism to determine which claims to trust. We first find that disagreement often exposes correct alternatives, while consensus can conceal errors. These observations motivate VeriHarness, which turns the underlying LLM a generator uses into an agentic verifier by giving it a workspace, evidence tools, and reusable verification skills. A disagreement resolver checks competing claims against environmental evidence, while a consensus challenger tests shared claims and searches for omitted requirements. Their findings guide the selection and revision of the final artifact. Across five long-horizon workspace benchmarks and two frontier models, VeriHarness achieves the highest selection scores among the evaluated baselines. Evidence-backed revision further improves average performance, bringing gains over a single rollout to 6.2 points with Gemini 3.5 Flash and 6.4 points with Claude Opus 4.8. We further show that verification skills can self-improve from failure feedback, demonstrating VeriHarness as a novel and critical approach for scaling long-horizon agentic verification. We release the full pool of approximately 26,000 rollouts across all five benchmarks and both models, produced at a cost of over $100,000, to support future research on agentic verification.

Large language models often solve a theorem forward yet fail to disprove a closely related false one: a falsification gap that supervised fine-tuning does not close and can actively worsen. We frame counterexample generation as constrained witness emission against a deterministic per-theorem Python verifier, and release SymCE, a corpus of 4,707 false undergraduate-algebra and real-analysis conjectures, each paired with executable verifiers. The verifier also serves as the reward function, making SymCE a training environment. Training Qwen3-4B with SFT followed by GRPO under this oracle reveals an imitation trap: counterexample-only SFT collapses true-theorem recognition from 0.27 to 0.00, while RLVR with a sparse outcome-only reward repairs this and exceeds the base, to 0.66. The collapse replicates across four seeds and on Gemma-3-4B. Sparse and dense rewards yield statistically indistinguishable in-domain success yet diverge by 33 points on a held-out calibration probe, a dissociation we trace to the partial-credit term. Our 4B model outperforms every evaluated 7B open-weights math specialist, remains competitive with six frontier commercial APIs, and transfers under unchanged prompting to GSM8K, MATH-500 and MMLU-college-math. A human audit of 177 verifier decisions finds 97.7% accuracy. Code, data, verifier modules and annotations: https://github.com/ce-rlvr/SymCE.

Large language models have made significant progress in mathematical reasoning, which serves as an important testbed for AI and could impact scientific research if further advanced. By scaling reasoning with reinforcement learning that rewards correct final answers, LLMs have improved from poor performance to saturating quantitative reasoning competitions like AIME and HMMT in one year. However, this approach faces fundamental limitations. Pursuing higher final answer accuracy doesn't address a key issue: correct answers don't guarantee correct reasoning. Moreover, many mathematical tasks like theorem proving require rigorous step-by-step derivation rather than numerical answers, making final answer rewards inapplicable. To push the limits of deep reasoning, we believe it is necessary to verify the comprehensiveness and rigor of mathematical reasoning. Self-verification is particularly important for scaling test-time compute, especially for open problems without known solutions. Towards self-verifiable mathematical reasoning, we investigate how to train an accurate and faithful LLM-based verifier for theorem proving. We then train a proof generator using the verifier as the reward model, and incentivize the generator to identify and resolve as many issues as possible in their own proofs before finalizing them. To maintain the generation-verification gap as the generator becomes stronger, we propose to scale verification compute to automatically label new hard-to-verify proofs, creating training data to further improve the verifier. Our resulting model, DeepSeekMath-V2, demonstrates strong theorem-proving capabilities, achieving gold-level scores on IMO 2025 and CMO 2024 and a near-perfect 118/120 on Putnam 2024 with scaled test-time compute.

Reinforcement learning with verifiable rewards has enabled strong post-training gains in domains such as math and coding, though many open-ended settings rely on rubric-based rewards. We study reward hacking in rubric-based RL, where a policy is optimized against a training verifier but evaluated against a cross-family panel of three frontier judges, reducing dependence on any single evaluator. Our framework separates two sources of divergence: verifier failure, where the training verifier credits rubric criteria that reference verifiers reject, and rubric-design limitations, where even strong rubric-based verifiers favor responses that rubric-free judges rate worse overall. Across medical and science domains, weak verifiers produce large proxy-reward gains that do not transfer to the reference verifiers; exploitation grows over training and concentrates in recurring failures such as partial satisfaction of compound criteria, treating implicit content as explicit, and imprecise topical matching. Stronger verifiers substantially reduce, but do not eliminate, verifier exploitation. We also introduce a self-internalization gap, a verifier-free diagnostic based on policy log-probabilities, which tracks reference-verifier quality, detecting when the policy trained using the weak verifier stops improving. Finally, in our setting, stronger verification does not prevent reward hacking when the rubric leaves important failure modes unspecified: rubric-based verifiers prefer the RL checkpoint, while rubric-free judges prefer the base model. These disagreements coincide with gains concentrated in completeness and presence-based criteria, alongside declines in factual correctness, conciseness, relevance, and overall quality. Together, these results suggest that stronger verification reduces reward hacking, but does not by itself ensure that rubric gains correspond to broader quality gains.

Large language models (LLMs) can act as both problem solvers and solution verifiers, where the latter select high-quality answers from a pool of solver-generated candidates. This raises the question of under what conditions verification pays off in solver-verifier systems. Prior work has conducted only limited studies of the factors influencing verification performance, focusing primarily on self-verification and examining neither the relationship between solver and verifier model families nor the effects of reasoning post-training. To rectify this, we present a systematic study across 37 models spanning multiple families, sizes, and base vs. post-trained variants, evaluated on 9 benchmarks covering logical reasoning, structured puzzles, symbolic computation, mathematics, commonsense, factual recall, and domain knowledge. In order to support our analysis, we introduce and empirically validate verifier gain, a metric that predicts the performance improvements from test-time verifier-based rejection sampling. Our experiments find that 1) verification across model families is more effective than either self-verification or verification within the same family, and more generally that the benefits of verification decrease as the solver and verifier become more similar, 2) reasoning post-training weakens self-improvement abilities but strengthens cross-family improvement, and 3) some tasks are inherently more amenable to improvement through verification, particularly mathematical and logical tasks.

Verifier-based selection improves LLM performance by generating multiple candidate solutions and using a verifier to select the most promising one. However, existing methods typically treat verification only as a ranking step and discard its feedback once a fixed candidate pool has been evaluated. In this paper, we ask whether verification can also improve the candidate set itself. To this end, we introduce LLM-as-an-Improver and propose Verify--Repair--Reselect (VRR), which uses verification feedback to generate and reselect improved candidates. VRR retains the initial winner while conditionally generating three complementary alternatives: repaired versions of the winner and runner-up, and a solution based on a new approach. It filters invalid and duplicate candidates using only inference-time information and then reselects the final answer under the original evaluation criteria. Across diverse models and code-generation and reasoning benchmarks, VRR improves over fixed-pool verifier-based selection in many settings and can recover correct solutions even when all candidates in the initial pool are incorrect. These results highlight a broader role for LLMs as improvers: verification feedback can not only select among existing solutions but also construct stronger candidates beyond the initial pool.

Recent advances have shown that scaling test-time computation enables large language models (LLMs) to solve increasingly complex problems across diverse domains. One effective paradigm for test-time scaling (TTS) involves LLM generators producing multiple solution candidates, with LLM verifiers assessing the correctness of these candidates without reference answers. In this paper, we study generative verifiers, which perform verification by generating chain-of-thought (CoT) reasoning followed by a binary verdict. We systematically analyze verification dynamics across three dimensions - problem difficulty, generator capability, and verifier generation capability - with empirical studies on 12 benchmarks across mathematical reasoning, knowledge, and natural language reasoning tasks using 14 open-source models (2B to 72B parameter range) and GPT-4o. Our experiments reveal three key findings about verification effectiveness: (1) Easy problems allow verifiers to more reliably certify correct responses; (2) Weak generators produce errors that are easier to detect than strong generators; (3) Verification ability is generally correlated with the verifier's own problem-solving capability, but this relationship varies with problem difficulty. These findings reveal opportunities to optimize basic verification strategies in TTS applications. First, given the same verifier, some weak generators can nearly match stronger ones in post-verification TTS performance (e.g., the Gemma2-9B to Gemma2-27B performance gap shrinks by 75.5%). Second, we identify cases where strong verifiers offer limited advantage over weak ones, as both fail to provide meaningful verification gains, suggesting that verifier scaling alone cannot overcome fundamental verification challenges.

We introduce the Formally Verified Automated Programming Progress Standards, or FVAPPS, a benchmark of 4715 samples for writing programs and proving their correctness, the largest formal verification benchmark, including 1083 curated and quality controlled samples. Previously, APPS provided a benchmark and dataset for programming puzzles to be completed in Python and checked against unit tests, of the kind seen in technical assessments in the software engineering industry. Building upon recent approaches for benchmarks in interactive theorem proving, we generalize the unit tests to Lean 4 theorems given without proof (i.e., using Lean's "sorry" keyword). On the 406 theorems of 100 randomly selected samples, Sonnet correctly proves 30% and Gemini correctly proves 18%. We challenge the machine learning and program synthesis communities to solve both each general purpose programming problem and its associated correctness specifications. The benchmark is available at https://huggingface.co/datasets/quinn-dougherty/fvapps.

Autonomous research agents produce competitive solutions and professional-looking manuscripts, yet their outputs contain verifiability failures undetectable by surface-level evaluation: fabricated citations, unreproducible scores, and method descriptions that diverge from the implementation. We address this through three contributions. First, Chain-of-Evidence (CoE), a verifiability framework requiring every claim to be traceable to its evidence source. Second, ScientistOne, an end-to-end autonomous research system that maintains evidence chains by construction throughout literature review, solution discovery, and paper writing. Third, CoE Audit, a post-hoc audit whose four integrity checks -- score verification, specification violation, reference verification, and method-code alignment -- apply uniformly to all systems. Across 75 papers spanning five systems and five frontier research tasks, every baseline exhibits at least one systematic failure mode: hallucinated reference rates reach 21%, score verification passes in as few as 42% of papers, and method-code alignment ranges from 20% to 80%. ScientistOne achieves zero hallucinated references (0/337), perfect score verification (12/12), and the highest method-code alignment (14/15), while matching or exceeding human expert performance on all five tasks. ScientistOne further generalizes to six additional tasks spanning medical imaging, fine-grained recognition, 3D perception, and language modeling, achieving state-of-the-art on Parameter Golf and gold medals on MLE-Bench tasks where baselines fail entirely.

Large language models for code generation increasingly rely on synthetic data, where both problem solutions and verification tests are generated by models. While this enables scalable data creation, it introduces a previously unexplored bottleneck: the verification ceiling, in which the quality and diversity of training data are fundamentally constrained by the capabilities of synthetic verifiers. In this work, we systematically study how verification design and strategies influence model performance. We investigate (i) what we verify by analyzing the impact of test complexity and quantity: richer test suites improve code generation capabilities (on average +3 pass@1), while quantity alone yields diminishing returns, (ii) how we verify by exploring relaxed pass thresholds: rigid 100% pass criteria can be overly restrictive. By allowing for relaxed thresholds or incorporating LLM-based soft verification, we can recover valuable training data, leading to a 2-4 point improvement in pass@1 performance. However, this benefit is contingent upon the strength and diversity of the test cases used, and (iii) why verification remains necessary through controlled comparisons of formally correct versus incorrect solutions and human evaluation: retaining diverse correct solutions per problem yields consistent generalization gains. Our results show that Verification as currently practiced is too rigid, filtering out valuable diversity. But it cannot be discarded, only recalibrated. By combining calibrated verification with diverse, challenging problem-solution pairs, we outline a path to break the verification ceiling and unlock stronger code generation models.

The widespread adoption of deep neural networks (DNNs) requires efficient techniques for verifying their safety. DNN verifiers are complex tools, which might contain bugs that could compromise their soundness and undermine the reliability of the verification process. This concern can be mitigated using proofs: artifacts that are checkable by an external and reliable proof checker, and which attest to the correctness of the verification process. However, such proofs tend to be extremely large, limiting their use in many scenarios. In this work, we address this problem by minimizing proofs of unsatisfiability produced by DNN verifiers. We present algorithms that remove facts which were learned during the verification process, but which are unnecessary for the proof itself. Conceptually, our method analyzes the dependencies among facts used to deduce UNSAT, and removes facts that did not contribute. We then further minimize the proof by eliminating remaining unnecessary dependencies, using two alternative procedures. We implemented our algorithms on top of a proof producing DNN verifier, and evaluated them across several benchmarks. Our results show that our best-performing algorithm reduces proof size by 37%-82% and proof checking time by 30%-88%, while introducing a runtime overhead of 7%-20% to the verification process itself.

Reinforcement learning with verifiable rewards improves mathematical reasoning when a final answer can be checked, but a correct answer does not certify the derivation behind it, and proof problems offer no answer to check. Replacing the check with a reasoning model as an in-loop verifier faces a fundamental obstacle: the policy must already produce occasional correct attempts for any reward to exist, which it may never do at olympiad level. We instead move verification and repair out of the training loop and into offline data generation with SPLICE, a workflow in which a second model, given only a reference solution, edits a reasoning model's traces as they are generated: it verifies progress, truncates a failed trace at its last verified step, and splices in an adaptively strengthened hint until progress resumes. The resulting dataset includes repaired examples whose sampled unaided trajectory failed, with failed continuations removed and progress repeatedly checked against the reference solution. A model finetuned on SPLICE traces outperforms the same model distilled from the reasoning model's unedited traces on AIME and, under a grading protocol validated against human grades, on olympiad proofs. The trained model's own traces show a corresponding behavioral signature: at difficult points it announces a branch in the style of the spliced hints and takes riskier steps, with predictive entropy rising there and just after, a pattern absent from the model trained on unedited traces.

AI agents are increasingly used for programming, but do not provide any guarantee on the correctness of generated code. Verified code generation, in which an agent produces both an implementation and a machine-checked proof of its specification, offers a stronger path toward trustworthy AI-generated software. Existing benchmarks in this direction either focus on individual functions or only evaluate proof generation with provided implementations. It is still an open question whether agents can make coherent implementation and proof choices across real multi-module codebases. To bridge this gap, we introduce Vero, the first benchmark to evaluate joint implementation and proof synthesis at the repository level. Vero contains 43 multi-module instances sourced from real-world repositories spanning Python, Dafny, Verus, and Coq, and covering diverse domains from cryptographic protocols to distributed systems. Each instance consists of a multi-module Lean 4 repository with predetermined API interfaces, manually curated formal specifications, and reference implementations, supporting both proof-only and code-and-proof evaluation modes. To improve benchmark reliability, Vero also includes an audit mechanism where agents are allowed to formally prove unsatisfiability of provided specification or incorrectness of reference code, which surfaces and corrects latent code and specification errors during curation. We evaluate frontier coding-agent configurations with Lean toolchain access. The strongest agent fully solves only 27 of 43 instances and closes no specifications on the hardest repositories. Vero provides a concrete testbed for measuring progress toward repository-scale verified software synthesis, where current agents still fall short. We release the benchmark, curation pipeline, and evaluation harness at https://github.com/sunblaze-ucb/vero.

Advances in training, post-training, and inference-time methods have enabled frontier reasoning models to win gold medals in math competitions and settle challenging open problems. Gaining trust in the responses of these models requires that natural language proofs be checked for errors. LLM judges are increasingly being adopted to meet the growing demand for evaluating such proofs. While verification is considered easier than generation, what model capability does reliable verification actually require? We systematically evaluate four open-source and two frontier LLMs on datasets of human-graded natural language proofs of competition-level problems. We consider two key metrics: verifier accuracy and self-consistency (the rate of agreement across repeated judgments on the same proof). We observe that smaller open-source models are only up to ~10% behind frontier models in accuracy but they are up to ~25% more inconsistent. Furthermore, we see that verifier accuracy is sensitive to prompt choice across all models. We then demonstrate that the smaller models, in fact, do possess the mathematical capabilities to verify proofs at the level of frontier models, but they struggle to reliably elicit these capabilities with general judging prompts. Through an LLM-guided prompt search, we synthesize an ensemble of specialized prompts that overcome the specific failure modes of smaller models, boosting their performance by up to 9.1% in accuracy and 15.9% in self-consistency. These gains are realized across models and datasets, allowing models like Qwen3.5-35B to perform on par with frontier models such as Gemini 3.1 Pro for proof verification.

Recent research has generated hope that inference scaling could allow weaker language models to match or exceed the accuracy of stronger models, such as by repeatedly sampling solutions to a coding problem until it passes unit tests. The central thesis of this paper is that there is no free lunch for inference scaling: indefinite accuracy improvement through resampling can only be realized if the "verifier" (in this case, a set of unit tests) is perfect. When the verifier is imperfect, as it almost always is in domains such as reasoning or coding (for example, unit tests have imperfect coverage), there is a nonzero probability of false positives: incorrect solutions that pass the verifier. Resampling cannot decrease this probability, so it imposes an upper bound to the accuracy of resampling-based inference scaling even with an infinite compute budget. We find that there is a very strong correlation between the model's single-sample accuracy (i.e. accuracy without unit tests) and its false positive rate on coding benchmarks HumanEval and MBPP, whose unit tests have limited coverage. Therefore, no amount of inference scaling of weaker models can enable them to match the single-sample accuracy of a sufficiently strong model (Fig. 1a). When we consider that false positives have a negative utility compared to abstaining from producing a solution, it bends the inference scaling curve further downward. Empirically, we find that the optimal number of samples can be less than 10 under realistic assumptions (Fig. 1b). Finally, we show that beyond accuracy, false positives may have other undesirable qualities, such as poor adherence to coding style conventions.

Public-service recommendations require evidence that matches the requested service, scope, and date. Yet treating every missing detail as decisive can withhold useful recommendations. We introduce EviGraph, which distinguishes critical decision requirements from information that can remain unresolved. A language agent links these requirements to evidence in a temporal knowledge graph, while a deterministic checker establishes whether a recommendation is supported. Evaluation on a bilingual Hong Kong public-service benchmark with executable policy references shows that this distinction reduces unnecessary abstention. Additional verification, however, can withdraw supported recommendations without improving decision quality. These findings suggest that reliable evidence-based navigation depends on specifying what must be established for a decision, rather than simply adding more verification.

AI coding agents produce code faster than humans can review it. In our approach, the prover is the judge of whether the code is correct. Under a verifier-driven loop, AI agents wrote and verified bare-metal security software in Ada/SPARK spanning classical and post-quantum cryptography, TLS 1.3, IKEv2, X.509, and a Matrix client. GNATprove discharged 49,280 proof obligations, established functional correctness for selected primitives, and proved the absence of run-time errors for the rest, at roughly 20-40 times lower supervision cost than comparable hand verification. GNATprove alone was insufficient: some defects could not be detected and were resolved using known-answer tests, interoperability, or human review of specifications. Given weak checks, the agent tried to bypass them and reported success. We report where each layer caught faults and draw the central lesson: what an agent can be trusted to establish is bounded by the strength of its feedback.

Large language models can generate useful code from natural language, but their outputs come without correctness guarantees. Verifiable code generation offers a path beyond testing by requiring models to produce not only executable code, but also formal specifications and machine-checkable proofs. Progress in this direction, however, is difficult to measure: existing benchmarks are often small, focus on only one part of the pipeline, lack ground-truth proofs or rigorous specification validation, or target verification settings far from mainstream software development. We present VeriContest, a benchmark of 946 competitive-programming problems from LeetCode and Codeforces for verifiable code generation in Rust with Verus. Each problem pairs a natural language description with expert-validated formal specifications, judge-accepted Rust code, Verus-checked proofs, and positive and negative test suites. VeriContest is constructed through a three-phase pipeline that scales from manually verified seed problems to semi-automated expansion with human-in-the-loop review. To further strengthen benchmark quality, we use testing as an additional quality-assurance layer for validating postcondition completeness. VeriContest supports isolated and compositional evaluation of specification generation, code generation, proof generation, and end-to-end verified program synthesis. Evaluating ten state-of-the-art models reveals a sharp gap between coding ability and verifiable code generation: the strongest model reaches 92.18% on natural-language-to-code generation, but only 48.31% on specification generation, 13.95% on proof generation, and 5.29% end-to-end. These results identify proof and specification generation as the central bottlenecks for models and establish VeriContest as a rigorous platform for measuring and training future systems that generate code with machine-checkable correctness.

As post-training techniques evolve, large language models (LLMs) are increasingly augmented with structured multi-step reasoning abilities, often optimized through reinforcement learning. These reasoning-enhanced models outperform standard LLMs on complex tasks and now underpin many commercial LLM APIs. However, to protect proprietary behavior and reduce verbosity, providers typically conceal the reasoning traces while returning only the final answer. This opacity introduces a critical transparency gap: users are billed for invisible reasoning tokens, which often account for the majority of the cost, yet have no means to verify their authenticity. This opens the door to token count inflation, where providers may overreport token usage or inject synthetic, low-effort tokens to inflate charges. To address this issue, we propose CoIn, a verification framework that audits both the quantity and semantic validity of hidden tokens. CoIn constructs a verifiable hash tree from token embedding fingerprints to check token counts, and uses embedding-based relevance matching to detect fabricated reasoning content. Experiments demonstrate that CoIn, when deployed as a trusted third-party auditor, can effectively detect token count inflation with a success rate reaching up to 94.7%, showing the strong ability to restore billing transparency in opaque LLM services. The dataset and code are available at https://github.com/CASE-Lab-UMD/LLM-Auditing-CoIn.

Large language model (LLM)-based reasoning systems have recently achieved gold medal-level performance in the IMO 2025 competition, writing mathematical proofs where, to receive full credit, each step must be not only correct but also sufficiently supported. To train LLM-based reasoners in such challenging, open-ended settings, strong verifiers capable of catching step-level mistakes are necessary prerequisites. We introduce Hard2Verify, a human-annotated, step-level verification benchmark produced with over 500 hours of human labor. Hard2Verify is designed to rigorously assess step-level verifiers at the frontier: Verifiers must provide step-level annotations or identify the first error in responses generated by frontier LLMs for very recent, challenging, and open-ended math questions. We evaluate 29 generative critics and process reward models, demonstrating that, beyond a few standouts, open-source verifiers lag closed source models. We subsequently analyze what drives poor performance in step-level verification, the impacts of scaling verifier compute, as well as fundamental questions such as self-verification and verification-generation dynamics.

Test-time scaling via solution sampling and aggregation has become a key paradigm for improving the reasoning performance of Large Language Models (LLMs). While reward model selection is commonly employed in this approach, it often fails to identify minority-yet-correct answers, which limits its effectiveness beyond that of simple majority voting. We argue that this limitation stems from a lack of informative critique signals during verifier training. To bridge this gap, we introduce Mirror-Critique, a framework that trains a verifier with informative critiques. Our key insight is to leverage the rich critique signal by contrasting model-generated solutions with ground-truth solutions. We deploy a small instruction-tuned model to synthesize high-quality critique data with rejection sampling that teaches the verifier not only what is wrong, but also why. The synthetic data is used to cold-start the LLMs in the RLVR process to further improve the verification ability. The resulting Mirror-Verifier is deployed to evaluate candidate solutions by generating multiple critiques per solution, aggregating them into a verify score used for weighted voting or selective abstention. The experimental results show that our Mirror-Verifier significantly outperforms majority voting in terms of solution accuracy and also improves the solver's honesty to recognize and abstain from answering beyond its capability boundaries.

A semantic decision engine such as Jev can return a valid answer and still miss a network deadline, select an infeasible action or leave the service unverified. We systematize 139 paper families by decision interface, execution path and check ownership. Fifty families claim that their engine fits a control loop or time budget, but only four support the claim with matched measurement. Across all 139, four report deadline attainment. The gap concentrates where the decision has no deterministic computation step. Those 72 families make 22 of the claims, none supported, and name a coverage owner in only two. Bounded tests under one event model show that each gap can reverse an admission verdict. A decision that meets a 10 s budget for every isolated request meets it for none once decisions queue ahead of replayed execution times. The same engine passes one coverage check and fails another. We derive a minimum reporting record, design rules and a research agenda for admitting decision engines to control loops.

Visual fluency in generated video does not imply physical reliability, and a scalar quality score alone is incapable of indicating the obligation a clip violates or the moment it fails. We present VeriPhy, an auditable physical-verification system in which a text-only planner compiles the prompt into typed physical obligations and a statically validated execution plan before any frame is observed. During execution, observations gate and scope only declared calls to frozen low-level experts (e.g., segmentation and tracking, counting, eleven typed physical measurements over the resulting tracks, depth, OCR, and audio-event detection). Each action returns a provenance-carrying evidence record whose payload, when usable, is either a typed measurement or an explicitly tagged learned state. Typed resolvers and fixed composition map usable records to a three-valued state (supported, contradicted, or unknown, surfaced as plausible, implausible, or abstain) with full provenance, so that every verdict is traceable to the evidence that produced it. We anchor evaluation in a 1,500-clip corpus of human-annotated flaw records that localize real generation failures in prompt reference, space, and time. On a 149-clip core carrying 304 such records, VeriPhy accounts for 228, against 164 for a published question-decomposition evaluator given the same clips and the same claims. Recall alone does not separate it from prompting the same backbone monolithically, which reaches 222; what separates them is that each decision retains its evidence record and provenance, making the traces auditable one verdict at a time and usable as the interface through which a critic verdict could be written back into generation.

Benchmarks like MMLU suggest flagship language models approach factuality saturation above 90\%. LLMpedia shows this picture is incomplete. We materialize {sim}1.3M encyclopedia articles entirely from parametric memory across three model families, then audit every claim against Wikipedia and curated web evidence. For gpt-5-mini, the verifiable true rate is 68.4\% on Wikipedia-covered subjects - more than 21\,pp below MMLU - and the gap is driven by unverifiability (30.5\%), not refutation (1.2\%). Beyond Wikipedia, frontier articles audited against curated web evidence reach 57.6\%; Wikipedia covers only 56.7\% of model-surfaced subjects, and three model families overlap in just 7.3\% of subject choices. In a retrieval-trap benchmark inspired by prior analysis of Grokipedia, LLMpedia is more factual at roughly half the textual similarity to Wikipedia. Every prompt, article, and verdict is released. Data, code, interface: https://llmpedia.net.

Digital signatures prove key possession, not authorship. An author who generates text with AI, constructs intermediate document states post-hoc, and signs each hash produces a signature chain indistinguishable from genuine composition. We address this gap between cryptographic integrity and process provenance. We introduce proof-of-process, a primitive category for evidence that a physical process, not merely a signing key, produced a digital artifact. Our construction, the jitter seal, injects imperceptible microsecond delays derived via HMAC from a session secret, keystroke ordinal, and cumulative document hash. Valid evidence requires that real keystrokes produced the document through those intermediate states. We propose the Adversarial Collapse Principle as an evaluation criterion: evidence systems should be judged by whether disputing them requires a conjunction of specific, testable allegations against components with independent trust assumptions. We present Witnessd, an architecture combining jitter seals with Verifiable Delay Functions, external timestamp anchors, dual-source keystroke validation, and optional hardware attestation. Each layer forces allegations at different capability levels; disputing authentic evidence requires coordinated claims across independent trust boundaries. The system does not prevent forgery: a kernel-level adversary can defeat it, and typing AI-generated content produces valid evidence. The contribution is converting vague doubt into falsifiable allegations. We evaluate across 31,000 verification trials with deterministic rejection of invalid proofs.

Large Language Models (LLMs) demonstrate impressive mathematical reasoning abilities, but their solutions frequently contain errors that cannot be automatically verified. Formal theorem proving systems such as Lean 4 offer automated verification with complete accuracy, motivating recent efforts to build specialized prover LLMs that generate verifiable proofs in formal languages. However, a significant gap remains: current prover LLMs solve substantially fewer problems than general-purpose LLMs operating in natural language. We introduce Hilbert, an agentic framework that bridges this gap by combining the complementary strengths of informal reasoning and formal verification. Our system orchestrates four components: an informal LLM that excels at mathematical reasoning, a specialized prover LLM optimized for Lean 4 tactics, a formal verifier, and a semantic theorem retriever. Given a problem that the prover is unable to solve, Hilbert employs recursive decomposition to split the problem into subgoals that it solves with the prover or reasoner LLM. It leverages verifier feedback to refine incorrect proofs as necessary. Experimental results demonstrate that Hilbert substantially outperforms existing approaches on key benchmarks, achieving 99.2% on miniF2F, 6.6% points above the best publicly available method. Hilbert achieves the best known result on PutnamBench. It solves 462/660 problems (70.0%), outperforming proprietary approaches like SeedProver (50.4%) and achieving a 422% improvement over the best publicly available baseline. Thus, Hilbert effectively narrows the gap between informal reasoning and formal proof generation.

We present EasyProver, a Lean proving system built from recursive agents with isolated contexts, asynchronous search, parallel tactic probing, and an elastic elaboration service. Using an off-the-shelf frontier model with no additional training, it proves 669/672 PutnamBench problems and all six IMO 2026 formalizations and resolves 597/600 NTP4VC verification conditions and all 189 VERINA tasks by proof or refutation. Because these benchmarks are close to saturation, we also introduce Aegis, which converts competitive programming problems into much harder verification tasks over real implementations. The pipeline writes safe Rust solutions, translates them into Lean with Aeneas, generates specifications, and labels each frozen statement with a kernel-checked proof or refutation. It produces 2,045 differentially validated Rust-Lean program pairs and 11,533 candidate specifications. Unlike hand-written benchmarks, these tasks carry implementation-level obligations from fixed-width arithmetic, explicit failure, generated loops, casts, and functionalized mutation. On a stratified sample of 2,000 statements, EasyProver proves 1,054 (52.7%) within a 30-minute budget per statement. The same tasks can also be used to benchmark other provers and as rewards for reinforcement learning.

Formal theorem proving (FTP) has emerged as a critical foundation for evaluating the reasoning capabilities of large language models, enabling automated verification of mathematical proofs at scale. However, progress has been constrained by limited datasets due to the high cost of manual curation and the scarcity of challenging problems with verified formal-informal correspondences. We propose leveraging theoretical computer science (TCS) as a scalable source of rigorous proof problems, where algorithmic definitions enable automated generation of arbitrarily many challenging theorem-proof pairs. We demonstrate this approach on two TCS domains: Busy Beaver problems, which involve proving bounds on Turing machine halting behavior, and Mixed Boolean Arithmetic problems, which combine logical and arithmetic reasoning. Our framework automatically synthesizes problems with parallel formal (Lean4) and informal (Markdown) specifications, creating a scalable pipeline for generating verified proof challenges. Evaluation on frontier models reveals substantial gaps in automated theorem proving: while DeepSeekProver-V2-671B achieves 57.5\% success on Busy Beaver problems, it manages only 12\% on Mixed Boolean Arithmetic problems. These results highlight the difficulty of long-form proof generation even for problems that are computationally easy to verify, demonstrating the value of TCS domains for advancing automated reasoning research.

Large language models (LLMs) have made misinformation inexpensive to produce but not to verify, creating a growing asymmetry in the information ecosystem. Under tight time, labor, and budget constraints, media organizations, platforms, and fact-checkers rely on screening to prioritize which content to verify. We introduce VEX-Bench, a unified benchmark for evaluating the verification complexity of LLM-generated misinformation, as perceived during screening, across models and generation methods. Verification complexity is assessed along multiple dimensions derived from journalistic and fact-checking practices, capturing checkability, harm potential, source credibility signals, imposter legitimacy, and expected verification effort. We define the VEX score as an integrated measure combining elicitation yield and verification complexity to quantify how generated content consumes limited verification capacity. We construct a benchmark spanning two misinformation categories, 6 high-stakes domains, and 60 real-world topics, and evaluate 7 frontier LLMs and 7 generation methods, yielding 5{,}880 articles. We employ an LLM-as-judge for scalable evaluation and validate it using content-analysis methodology, including ordinal Krippendorff α for inter-annotator reliability, complemented by fact-checking agents for verification. Our findings show that no single method dominates all dimensions, underscoring the need for multi-dimensional evaluation. LLMs can generate high-VEX misinformation at 3times to 169times lower cost than agent-based verification. Such content is often prioritized during screening, consuming scarce verification resources and introducing a systematic risk of misallocation in resource-constrained verification systems. The code is publicly available in our https://github.com/HanxunH/VEX-Bench{GitHub repository}.

Partial differential equations (PDEs) are foundational to modeling in science and engineering, but constructing reliable numerical solvers remains labor-intensive, demanding expert knowledge of discretization schemes, stability conditions, and boundary treatments. Recent work has begun to frame PDE solving as a code-generation task for large language models (LLMs), yet existing approaches operate primarily at inference time: relying on prompting, debugging, self-refinement, and test-time scaling rather than adapting the model itself. In parallel, reinforcement learning with verifiable rewards has emerged as a post-training paradigm for code and math reasoning, but its verifiers are typically binary: a compiler runs, or a test passes. Such signals discard the graded structure of scientific correctness, where two solvers may both execute and yet differ in solution accuracy by orders of magnitude. In this work, we introduce RLVP: Reinforcement Learning with Verifiable Physics, an RL post-training framework for multi-PDE solver code generation. RLVP addresses this verifiability gap with a hybrid verifier: hard program-validity checks ensure executability, while continuous physics rewards score function-space accuracy and PDE-residual consistency. A single policy is post-trained across diverse PDE families spanning hyperbolic, parabolic, elliptic, and incompressible-flow systems. RLVP improves over both pre-trained and supervised-only baselines on PDE benchmarks, and shows zero-shot improvement transfer to held-out PDEs. We show that a smaller LLM post-trained with RLVP can outperform prompting a frontier model on in-distribution PDE solver generation. The trained policy shows evidence of compositionality in numerical motifs: it recombines stencils, time-stepping schemes, and boundary-handling primitives learned from the PDEs used in training into generated solvers for unseen PDE problems.

Natural-language software requirements are often ambiguous, inconsistent, and underspecified; in safety-critical domains, these defects propagate into formal models that verify the wrong specification and into implementations that ship unsafe behavior. We show that large language models, equipped with an SMT solver, can audit such requirements: translating them into formal logic, detecting ambiguity through stochastic variation in the generated formalization, and exposing inconsistency, vacuousness, and safety violations through solver queries on the resulting specification. We present VERIMED, a neurosymbolic pipeline that operationalizes this idea for medical-device software requirements, and report two findings. First, stochastic variation across independent formalizations is a signal of ambiguity: requirements that admit multiple plausible interpretations produce SMT-inequivalent formalizations, and bidirectional SMT equivalence checking turns this disagreement into a solver-checkable test. Second, the usefulness of symbolic feedback depends on its granularity: in counterexample-guided repair on a hemodialysis question-answering benchmark, concrete SMT counterexamples raise verified accuracy from 55.4% to 98.5%. Over an extensive experimental evaluation on open-source hemodialysis safety requirements, we show that the LLM-based approach in VERIMED successfully reduces ambiguity-sensitive requirements and enables rigorous auditing of software requirements through SMT-based queries.

Verifiability is a core content policy of Wikipedia: claims that are likely to be challenged need to be backed by citations. There are millions of articles available online and thousands of new articles are released each month. For this reason, finding relevant sources is a difficult task: many claims do not have any references that support them. Furthermore, even existing citations might not support a given claim or become obsolete once the original source is updated or deleted. Hence, maintaining and improving the quality of Wikipedia references is an important challenge and there is a pressing need for better tools to assist humans in this effort. Here, we show that the process of improving references can be tackled with the help of artificial intelligence (AI). We develop a neural network based system, called Side, to identify Wikipedia citations that are unlikely to support their claims, and subsequently recommend better ones from the web. We train this model on existing Wikipedia references, therefore learning from the contributions and combined wisdom of thousands of Wikipedia editors. Using crowd-sourcing, we observe that for the top 10% most likely citations to be tagged as unverifiable by our system, humans prefer our system's suggested alternatives compared to the originally cited reference 70% of the time. To validate the applicability of our system, we built a demo to engage with the English-speaking Wikipedia community and find that Side's first citation recommendation collects over 60% more preferences than existing Wikipedia citations for the same top 10% most likely unverifiable claims according to Side. Our results indicate that an AI-based system could be used, in tandem with humans, to improve the verifiability of Wikipedia. More generally, we hope that our work can be used to assist fact checking efforts and increase the general trustworthiness of information online.

Large language models (LLMs) have achieved remarkable performance on high-school and competition-level mathematics, yet their capabilities on advanced mathematics remain poorly understood. Existing benchmarks, however, fall short in both scope and evaluation granularity: they provide limited disciplinary coverage and often rely on final-answer correctness or coarse judgments, leaving the validity of the reasoning process inadequately assessed. To bridge this gap, we introduce AdvancedMathBench, a benchmark suite designed to evaluate the reasoning capabilities of LLMs on advanced mathematical proofs. Its core generation benchmark, ProverBench, contains 245 problems spanning undergraduate (UG) and doctoral qualifying-exam (QE) levels. To reliably evaluate these proofs, we develop a dedicated automatic verification pipeline that is trained on large-scale expert annotations, produces both correctness verdicts and fine-grained analyses, and exhibits strong agreement with human experts on held-out proof trajectories. We further introduce VerifierBench, consisting of 888 model-generated proof trajectories paired with expert ground truth, to evaluate whether models can correctly judge proof validity and provide sound verification rationales. Experiments show that AdvancedMathBench remains challenging for frontier models. On proof generation, the best-performing model, GPT-5.5-xhigh, achieves only 64.5 and 48.9 on the UG and QE splits, respectively. On proof verification, the best model only attains a Balanced F1 of 65.1. Further analysis reveals a notable mismatch between proof generation and verification capabilities across models.

A paraphrase-quality audit of MathCheck (ICLR 2025) detected 4 semantically incorrect paraphrases in 129 groups (3.1%); removing them drops GPT-4o from rank 2 to rank 4 and elevates Claude Haiku and DeepSeek V3 above it; these ranking changes are invisible to any single-model evaluation. Cross-model unanimity found these errors automatically (>= 3/4 models for MathCheck; >= 6/9 for our primary evaluation) for under $10; in our own dataset the same protocol found that 47% of auto-generated connective-variation paraphrases were semantically incorrect. That flaw compounds a deeper measurement gap: Claude Haiku 4.5 achieves 86% accuracy yet SCR=50%, meaning half its theorems are answered differently under semantically equivalent restatements, while aggregate accuracy across 9 models spans only 86-96% yet Semantic Consistency Rates (SCR) span 50-82% -- a 32-point gap invisible to standard benchmarks. Formally, for any target ranking over 9 frontier models there exists a weighting over paraphrase families that realizes it (No-Free-Benchmark corollary), because no model Pareto-dominates all families -- so benchmark designers who select families are implicitly choosing which model wins. FormInv supplies the audit protocol (replicated on external benchmarks at 100% recall), SCR and per-theorem Cochran's Q as primary invariance measures evaluated on 9 models across 366-811 items (on Lean4-verified theorems), and FormInvSelector for regime-aware model selection.

This paper presents a quantitative program verification infrastructure for discrete probabilistic programs. Our infrastructure can be viewed as the probabilistic analogue of Boogie: its central components are an intermediate verification language (IVL) together with a real-valued logic. Our IVL provides a programming-language-style for expressing verification conditions whose validity implies the correctness of a program under investigation. As our focus is on verifying quantitative properties such as bounds on expected outcomes, expected run-times, or termination probabilities, off-the-shelf IVLs based on Boolean first-order logic do not suffice. Instead, a paradigm shift from the standard Boolean to a real-valued domain is required. Our IVL features quantitative generalizations of standard verification constructs such as assume- and assert-statements. Verification conditions are generated by a weakest-precondition-style semantics, based on our real-valued logic. We show that our verification infrastructure supports natural encodings of numerous verification techniques from the literature. With our SMT-based implementation, we automatically verify a variety of benchmarks. To the best of our knowledge, this establishes the first deductive verification infrastructure for expectation-based reasoning about probabilistic programs.

Large Language Model (LLM) reasoning for complex tasks inherently involves a trade-off between solution accuracy and computational efficiency. The subsequent step of verification, while intended to improve performance, further complicates this landscape by introducing its own challenging trade-off: sophisticated Generative Reward Models (GenRMs) can be computationally prohibitive if naively integrated with LLMs at test-time, while simpler, faster methods may lack reliability. To overcome these challenges, we introduce FlexiVe, a novel generative verifier that flexibly balances computational resources between rapid, reliable fast thinking and meticulous slow thinking using a Flexible Allocation of Verification Budget strategy. We further propose the Solve-Detect-Verify pipeline, an efficient inference-time scaling framework that intelligently integrates FlexiVe, proactively identifying solution completion points to trigger targeted verification and provide focused solver feedback. Experiments show FlexiVe achieves superior accuracy in pinpointing errors within reasoning traces on ProcessBench. Furthermore, on challenging mathematical reasoning benchmarks (AIME 2024, AIME 2025, and CNMO), our full approach outperforms baselines like self-consistency in reasoning accuracy and inference efficiency. Our system offers a scalable and effective solution to enhance LLM reasoning at test time.

Large language models demonstrate remarkable reasoning capabilities but often produce unreliable or incorrect responses. Existing verification methods are typically model-specific or domain-restricted, requiring significant computational resources and lacking scalability across diverse reasoning tasks. To address these limitations, we propose VerifiAgent, a unified verification agent that integrates two levels of verification: meta-verification, which assesses completeness and consistency in model responses, and tool-based adaptive verification, where VerifiAgent autonomously selects appropriate verification tools based on the reasoning type, including mathematical, logical, or commonsense reasoning. This adaptive approach ensures both efficiency and robustness across different verification scenarios. Experimental results show that VerifiAgent outperforms baseline verification methods (e.g., deductive verifier, backward verifier) among all reasoning tasks. Additionally, it can further enhance reasoning accuracy by leveraging feedback from verification results. VerifiAgent can also be effectively applied to inference scaling, achieving better results with fewer generated samples and costs compared to existing process reward models in the mathematical reasoning domain. Code is available at https://github.com/Jiuzhouh/VerifiAgent

While tool-augmented Large Language Models have significantly improved multi-step reasoning in quantitative STEM tasks, a critical residual failure mode remains: intermediate reasoning steps that are syntactically well-formed, mathematically executable, and unit-consistent, yet contextually ungrounded. Current approaches either rely on formal verifiers that cannot assess semantic intent, or burden Process Reward Models (PRMs) with the dual task of checking both arithmetic and logic. In this paper, we propose a neuro-symbolic framework that cleanly decouples reasoning into two formal dimensions: Symbolic Validity (V) and Semantic Groundedness (G). We guarantee V by construction using a deterministic symbolic verifier acting as a hard filter. To assess G, we train a PRM conditionally on the verifier-accepted manifold. To train this PRM efficiently, we introduce Counterfactual Symbolic Perturbation (CSP), a novel data synthesis strategy that algorithmically generates constraint-preserving hard negatives (steps that perfectly pass the verifier but are logically flawed). At inference, we deploy a verifier-first constrained search that guarantees execution consistency for verifier-covered operations while relying on the PRM solely to rank semantic grounding. By targeting the exact residual error class of strong tool-using LLMs, our method significantly improves reasoning reliability without the sprawling heuristics of prior frameworks.

Verifying the final answer of a long reasoning trace may show that the trace is wrong but it does not distinguish three cases. The problem may lie beyond the model's capability, another attempt would succeed, or one step in an otherwise recoverable trace made the trace fail. Standard hidden-state probes are trained on final-answer correctness, a label that conflates these cases. Monte Carlo rollouts distinguish the cases and locate the failing step at additional inference cost. We test the hidden states of 4B and 8B reasoning models for the failure type and the failing step and compare each probe with the strongest baseline that uses no hidden states. On incorrect 8B traces, the confidence probe, a calibrated correctness probe at a midpoint cutoff, misses 81% of cases in which a single step makes the trace unrecoverable and 47% of cases in which another attempt would succeed. A probe with a different training design over the same states reverses the ordering of these two kinds of failure, so either ordering can be read from those states. At both scales, no single tested probe locates the failing step more accurately than a rule based only on trace length, and at 4B a probe trained to predict that step fails to outperform this rule in any of 10 runs on identical traces. The confidence probe therefore indicates that an answer is wrong, but its failure-type ordering depends on its training design.

Large language models (LLMs) increasingly rely on reinforcement learning (RL) to enhance their reasoning capabilities through feedback. A critical challenge is verifying the consistency of model-generated responses and reference answers, since these responses are often lengthy, diverse, and nuanced. Rule-based verifiers struggle with complexity, prompting the use of model-based verifiers. However, specialized verifiers lack flexibility, while general LLM judges can be inconsistent. Existing research primarily focuses on building better verifiers, yet a systematic evaluation of different types of verifiers' performance across domains remains lacking, severely constraining the reliable development of Reinforcement Learning with Verifiable Reward (RLVR). To address this, we propose VerifyBench--a cross-domain comprehensive benchmark for systematically evaluating verifiers. We construct 4,000 expert-level questions covering mathematics, physics, chemistry, and biology. Each question is equipped with reference answers and diverse responses. The reliability of the evaluation is ensured through a rigorous annotation process conducted by a multidisciplinary expert team. We design a four-dimensional experimental framework to comprehensively compare the performance boundaries of specialized verifiers and general LLMs under combined conditions of extracted answers vs. complete responses, and short vs. long outputs. Our evaluation uncovers fundamental trade-offs in verifiers: while specialized verifiers achieve leading accuracy, they exhibit deficiencies in recall; general models show stronger inclusivity but unstable precision. More importantly, we discover verifiers' high sensitivity to input structure and inherent limitations in cross-domain generalization, providing critical insights into the bottlenecks of current verifier technology.

Retrieval-augmented generation (RAG) grounds answers in retrieved passages, yet relevance does not guarantee sufficiency: a topical passage may still fail to justify the answer. We study evidence sufficiency verification for selective RAG answering, in which a verifier receives a question, a candidate answer, and retrieved evidence and decides whether the evidence supports, refutes, or is insufficient for the answer, answering only when support is established. We present SURE-RAG, an aggregation protocol that treats evidence sufficiency as a set-level property: missing hops and unresolved conflicts cannot be detected by scoring passages independently. A shared claim-evidence verifier produces a local relation distribution for each (claim, passage) pair, which SURE-RAG aggregates into four interpretable answer-level feature blocks (coverage, relation strength, uncertainty, and retrieval), producing a three-way decision and an auditable selective score. We evaluate on HotpotQA-RAG v3, a controlled multi-hop benchmark, under an artifact-aware protocol (shortcut baselines, counterfactual swaps, no-oracle checks, and GPT-4o audits). Calibrated SURE-RAG attains 0.9075 Macro-F1 (raw 0.8951 +/- 0.0069), well above DeBERTa mean-pooling (0.6516) and a GPT-4o judge (0.7284), and on par with a strong concat cross-encoder (0.8888 +/- 0.0109) while remaining fully auditable. At 30% coverage, risk falls from 0.2588 to 0.1642, a 37% relative reduction. As a boundary-mapping experiment, we contrast SURE-RAG with GPT-4o on HaluBench unsafe detection: the ranking reverses (0.3343 vs. 0.7389 unsafe-F1), indicating that controlled sufficiency verification and natural hallucination detection are distinct problems.

For humans to trust the fluent generations of large language models (LLMs), they must be able to verify their correctness against trusted, external sources. Recent efforts aim to increase verifiability through citations of retrieved documents or post-hoc provenance. However, such citations are prone to mistakes that further complicate their verifiability. To address these limitations, we tackle the verifiability goal with a different philosophy: we trivialize the verification process by developing models that quote verbatim statements from trusted sources in pre-training data. We propose Quote-Tuning, which demonstrates the feasibility of aligning LLMs to leverage memorized information and quote from pre-training data. Quote-Tuning quantifies quoting against large corpora with efficient membership inference tools, and uses the amount of quotes as an implicit reward signal to construct a synthetic preference dataset for quoting, without any human annotation. Next, the target model is aligned to quote using preference optimization algorithms. Experimental results show that Quote-Tuning significantly increases the percentage of LLM generation quoted verbatim from high-quality pre-training documents by 55% to 130% relative to untuned models while maintaining response quality. Further experiments demonstrate that Quote-Tuning generalizes quoting to out-of-domain data, is applicable in different tasks, and provides additional benefits to truthfulness. Quote-Tuning not only serves as a hassle-free method to increase quoting but also opens up avenues for improving LLM trustworthiness through better verifiability.

Multi-step symbolic reasoning is essential for robust financial analysis, yet most benchmarks neglect intermediate reasoning steps. FINCHAIN introduced verifiable Chain-of-Thought (CoT) evaluation but is limited to English. FINESSE-Bench includes a Russian block but relies on multiple-choice questions without step-level supervision. We present RusFinChain, the first Russian-language symbolic benchmark for verifiable CoT reasoning in finance. It spans 17 domains, 172 topics, and comprises 5,280 parameterized examples from executable Python templates, ensuring contamination-free evaluation. Each example includes a gold-standard reasoning chain with intermediate numeric values for automatic verification. We also introduce enhanced metrics: Fuzzy Numeric Alignment and Soft-Attention Alignment. We evaluate 8 open-weight LLMs on a stratified sample, generating 8,100 responses. Results reveal a substantial reasoning gap: models achieve Hard F1 of ~0.65 for step alignment, but only ~29% of final answers are correct. Our fuzzy and soft metrics show stronger correlation with final-answer correctness (Spearman rho approx 0.48) than the original ChainEval (rho approx 0.38-0.46), demonstrating superior diagnostic power. We release dataset, code, and evaluation framework to foster verifiable financial AI for the Russian-speaking community.

Large language model (LLM) agents are increasingly used across the scientific research lifecycle: ideation, literature search, experiment design and execution, analysis, manuscript drafting, and review. End-to-end AI scientist systems can now produce paper-like manuscripts, but their claims are often harder to verify than their code is to run. This survey studies that gap in computational AI/ML research, where code, benchmarks, experiments, and write-ups are most visible. We screen 125 candidate works and include 35, with full-text coding of 26 entries: 24 runnable systems and two study or position papers. We code seven audit dimensions: lifecycle stage, autonomy level, evaluation method, released artifacts, human-in-the-loop points, novelty verification, and result-selection disclosure. The main pattern is that code release is now common, but reproducibility-grade and claim-verification artifacts remain much less common. In the 24 runnable systems, 83 percent release code, while 38 percent release seeds or execution traces and 38 percent report any novelty-verification method. Among nine closed-loop L4 systems, seven are mechanical reruns and one is author-claimed without an external check; no LLM-era system in the corpus demonstrates an externally validated in-loop oracle under our coding rule. We contribute a coded corpus, a lifecycle-by-autonomy map, an auditability-gap analysis, and a reviewer-facing reporting checklist. The survey argues that the field's central bottleneck is no longer only whether agents can complete research tasks, but whether reviewers can verify the claims those agents produce.

Verification is crucial for effective mathematical reasoning. We present a new temporal consistency method where verifiers iteratively refine their judgments based on the previous assessment. Unlike one-round verification or multi-model debate approaches, our method leverages consistency in a sequence of self-reflection actions to improve verification accuracy. Empirical evaluations across diverse mathematical process error identification benchmarks (Mathcheck, ProcessBench, and PRM800K) show consistent performance improvements over baseline methods. When applied to the recent DeepSeek R1 distilled models, our method demonstrates strong performance, enabling 7B/8B distilled models to outperform all 70B/72B models and GPT-4o on ProcessBench. Notably, the distilled 14B model with our method achieves performance comparable to Deepseek-R1. Our codes are available at https://github.com/jcguo123/Temporal-Consistency

AI systems now write long and difficult mathematical proofs, and checking their step-by-step validity is becoming a bottleneck. A learned verifier could check proofs cheaply, but training one requires a correctness label on every step, and such labels are hard to obtain for open-ended proofs. Existing methods take them from human annotators, which is costly, or derive them from final answers, which proofs do not have, and formalizing each proof in Lean requires expert effort. We introduce LeanPRM, a framework that uses the Lean proof assistant to create step-level labels for natural-language proofs. A model writes a natural-language proof together with a paired formal proof, without feedback from Lean. Lean then labels every step, and a second model discards failures caused only by the formal syntax, so that the negative labels mark errors in the mathematics. LeanPRM contains 10,634 proofs with 152K labeled steps and no human annotation. A Qwen3-4B verifier trained on LeanPRM raises exact step-label accuracy on a held-out set of 200 proofs (LeanPRM-Bench) from 48.2 to 74.7, as opposed to 60.3 when the same base model is trained on PRM800K. It also selects better proofs in best-of-N selection on IMO-level problems, and outperforms the PRM800K-trained verifier on three of four external verification benchmarks. Training on LeanPRM improves verifiers at every size from 0.6B to 8B parameters on LeanPRM-Bench, Hard2Verify and IMO-GradingBench. Given the labels need no human annotation, LeanPRM can grow with the supply of formal statements, offering a path to step-level feedback on AI-generated proofs that are hard for humans to check.

As Large Language Models (LLMs) become more pervasive across various users and scenarios, identifying potential issues when using these models becomes essential. Examples of such issues include: bias, inconsistencies, and hallucination. Although auditing the LLM for these problems is often warranted, such a process is neither easy nor accessible for most. An effective method is to probe the LLM using different versions of the same question. This could expose inconsistencies in its knowledge or operation, indicating potential for bias or hallucination. However, to operationalize this auditing method at scale, we need an approach to create those probes reliably and automatically. In this paper we propose the LLMAuditor framework which is an automatic, and scalable solution, where one uses a different LLM along with human-in-the-loop (HIL). This approach offers verifiability and transparency, while avoiding circular reliance on the same LLM, and increasing scientific rigor and generalizability. Specifically, LLMAuditor includes two phases of verification using humans: standardized evaluation criteria to verify responses, and a structured prompt template to generate desired probes. A case study using questions from the TruthfulQA dataset demonstrates that we can generate a reliable set of probes from one LLM that can be used to audit inconsistencies in a different LLM. This process is enhanced by our structured prompt template with HIL, which not only boosts the reliability of our approach in auditing but also yields the delivery of less hallucinated results. The novelty of our research stems from the development of a comprehensive, general-purpose framework that includes a HIL verified prompt template for auditing responses generated by LLMs.

Reinforcement learning from verifiable rewards (RLVR) has improved the reasoning abilities of large language models (LLMs), but most existing approaches rely on sparse outcome-level feedback. This sparsity creates a credit assignment challenge in long-horizon agentic reasoning: a trajectory may fail despite containing many correct intermediate decisions, or succeed despite containing flawed ones. In this work, we study a class of densely-verifiable agentic reasoning problems, where intermediate actions can be objectively checked by symbolic or algorithmic oracles. We propose Verifiable Process Rewards (VPR), a framework that converts such oracles into dense turn-level supervision for reinforcement learning, and instantiate it in three representative settings: search-based verification for dynamic deduction, constraint-based verification for logical reasoning, and posterior-based verification for probabilistic inference. We further provide a theoretical analysis showing that dense verifier-grounded rewards can improve long-horizon credit assignment by providing more localized learning signals, with the benefit depending on the reliability of the verifier. Empirically, VPR outperforms outcome-level reward and rollout-based process reward baselines across controlled environments, and more importantly, transfers to both general and agentic reasoning benchmarks, suggesting that verifiable process supervision can foster general reasoning skills applicable beyond the training environments. Our results indicate that VPR is a promising approach for enhancing LLM agents whenever reliable intermediate verification is available, while also highlighting its dependence on oracle quality and the open challenge of extending VPR to less structured, open-ended environments.

This position paper argues that computer science conferences should require tamper-evident, nonrepudiable attestations of experimental results. We name the underlying problem experiment nonrepudiation: a compliant protocol must bind the numbers in a paper to an actual executed computation in a way the author cannot later alter or deny. The current system relies on self-reported checklists, optional code sharing, and author-controlled logging. None of these mechanisms answer the question a reviewer cannot check: did the code the paper describes produce the numbers the paper reports? We define the problem formally, state the security properties any compliant protocol must satisfy, and describe a threat model that includes attacks current approaches do not prevent. To show that the problem is solvable, we built K-Veritas, a reference implementation in Go that produces signed reports without accessing training data. K-Veritas is a testbed, not a finished answer. We call on conferences and the community to treat nonrepudiation as a first-class requirement and to help build an open, independent standard for it.

This position paper provides a critical but constructive discussion of current practices in benchmarking and evaluative practices in the field of formal reasoning and automated theorem proving. We take the position that open code, open data, and benchmarks that are complete and error-free will accelerate progress in this field. We identify practices that create barriers to contributing to this field and suggest ways to remove them. We also discuss some of the practices that might produce misleading evaluative information. We aim to create discussions that bring together people from various groups contributing to automated theorem proving, autoformalization, and informal reasoning.

Large language models (LLMs) have demonstrated significant potential in formal theorem proving, yet state-of-the-art performance often necessitates prohibitive test-time compute via massive roll-outs or extended context windows. In this work, we address this scalability bottleneck by exploiting an informative structure in formal verification: the observation that compilers map a vast space of diverse proof attempts to a compact set of structured failure modes. We introduce a learning-to-refine framework that leverages this compression to perform efficient learning and proof exploration. We perform tree search that corrects errors locally conditioned on explicit verifier feedback, thereby circumventing the costs associated with accumulating a long history of proof attempts. Extensive evaluations show that our method consistently amplifies the reasoning capabilities of base provers across varying scales. Notably, our approach achieves state-of-the-art performance on PutnamBench among publicly reported sim8B and sim32B parameter models under comparable test-time budgets, offering a scalable paradigm for next-generation verifier-guided reasoning.

Recent Language Models (LMs) have shown impressive capabilities in generating texts with the knowledge internalized in parameters. Yet, LMs often generate the factually incorrect responses to the given queries, since their knowledge may be inaccurate, incomplete, and outdated. To address this problem, previous works propose to augment LMs with the knowledge retrieved from an external knowledge source. However, such approaches often show suboptimal text generation performance due to two reasons: 1) the model may fail to retrieve the knowledge relevant to the given query, or 2) the model may not faithfully reflect the retrieved knowledge in the generated text. To overcome these, we propose to verify the output and the knowledge of the knowledge-augmented LMs with a separate verifier, which is a small LM that is trained to detect those two types of errors through instruction-finetuning. Then, when the verifier recognizes an error, we can rectify it by either retrieving new knowledge or generating new text. Further, we use an ensemble of the outputs from different instructions with a single verifier to enhance the reliability of the verification processes. We validate the effectiveness of the proposed verification steps on multiple question answering benchmarks, whose results show that the proposed verifier effectively identifies retrieval and generation errors, allowing LMs to provide more factually correct outputs. Our code is available at https://github.com/JinheonBaek/KALMV.

In this paper we present a new "external checker" for the Lean theorem prover, written in Lean itself. This is the first complete typechecker for Lean 4 other than the reference implementation in C++ used by Lean itself, and our new checker is competitive with the original, running between 20% and 50% slower and usable to verify all of Lean's mathlib library, forming an additional step in Lean's aim to self-host the full elaborator and compiler. Moreover, because the checker is written in a language which admits formal verification, it is possible to state and prove properties about the kernel itself, and we report on progress to formalize the Lean type theory abstractly and prove some theorems about it. Finally, we combine these to get a proof of correctness of parts of the kernel. We plan to use this project to help justify any future changes to the kernel and type theory and ensure unsoundness does not sneak in through either the abstract theory or implementation bugs. The verification is already paying off, as one soundness bug has been spotted and fixed as a result of this work.

One way to increase confidence in the outputs of Large Language Models (LLMs) is to support them with reasoning that is clear and easy to check -- a property we call legibility. We study legibility in the context of solving grade-school math problems and show that optimizing chain-of-thought solutions only for answer correctness can make them less legible. To mitigate the loss in legibility, we propose a training algorithm inspired by Prover-Verifier Game from Anil et al. (2021). Our algorithm iteratively trains small verifiers to predict solution correctness, "helpful" provers to produce correct solutions that the verifier accepts, and "sneaky" provers to produce incorrect solutions that fool the verifier. We find that the helpful prover's accuracy and the verifier's robustness to adversarial attacks increase over the course of training. Furthermore, we show that legibility training transfers to time-constrained humans tasked with verifying solution correctness. Over course of LLM training human accuracy increases when checking the helpful prover's solutions, and decreases when checking the sneaky prover's solutions. Hence, training for checkability by small verifiers is a plausible technique for increasing output legibility. Our results suggest legibility training against small verifiers as a practical avenue for increasing legibility of large LLMs to humans, and thus could help with alignment of superhuman models.

We introduce our Leanabell-Prover-V2, a 7B large language models (LLMs) that can produce formal theorem proofs in Lean 4, with verifier-integrated Long Chain-of-Thoughts (CoT). Following our previous work Leanabell-Prover-V1, we continual to choose to posttrain existing strong prover models for further performance improvement. In our V2 version, we mainly upgrade the Reinforcement Learning (RL) with feedback provided by the Lean 4 verifier. Crucially, verifier feedback, such as indicating success or detailing specific errors, allows the LLM to become ``self-aware'' of the correctness of its own reasoning process and learn to reflexively correct errors. Leanabell-Prover-V2 directly optimizes LLM reasoning trajectories with multi-turn verifier interactions, together with feedback token masking for stable RL training and a simple reward strategy. Experiments show that Leanabell-Prover-V2 improves performance by 3.2% (pass@128) with Kimina-Prover-Preview-Distill-7B and 2.0% (pass@128) with DeepSeek-Prover-V2-7B on the MiniF2F test set. The source codes, curated data and models are available at: https://github.com/Leanabell-LM/Leanabell-Prover-V2.

Large Language Models (LLMs) have demonstrated remarkable potential in handling complex reasoning tasks by generating step-by-step rationales.Some methods have proven effective in boosting accuracy by introducing extra verifiers to assess these paths. However, existing verifiers, typically trained on binary-labeled reasoning paths, fail to fully utilize the relative merits of intermediate steps, thereby limiting the effectiveness of the feedback provided. To overcome this limitation, we propose Tree-based Preference Learning Verifier (Tree-PLV), a novel approach that constructs reasoning trees via a best-first search algorithm and collects step-level paired data for preference training. Compared to traditional binary classification, step-level preferences more finely capture the nuances between reasoning steps, allowing for a more precise evaluation of the complete reasoning path. We empirically evaluate Tree-PLV across a range of arithmetic and commonsense reasoning tasks, where it significantly outperforms existing benchmarks. For instance, Tree-PLV achieved substantial performance gains over the Mistral-7B self-consistency baseline on GSM8K (67.55% to 82.79%), MATH (17.00% to 26.80%), CSQA (68.14% to 72.97%), and StrategyQA (82.86% to 83.25%).Additionally, our study explores the appropriate granularity for applying preference learning, revealing that step-level guidance provides feedback that better aligns with the evaluation of the reasoning process.

Retrieval-enhanced methods have become a primary approach in fact verification (FV); it requires reasoning over multiple retrieved pieces of evidence to verify the integrity of a claim. To retrieve evidence, existing work often employs off-the-shelf retrieval models whose design is based on the probability ranking principle. We argue that, rather than relevance, for FV we need to focus on the utility that a claim verifier derives from the retrieved evidence. We introduce the feedback-based evidence retriever(FER) that optimizes the evidence retrieval process by incorporating feedback from the claim verifier. As a feedback signal we use the divergence in utility between how effectively the verifier utilizes the retrieved evidence and the ground-truth evidence to produce the final claim label. Empirical studies demonstrate the superiority of FER over prevailing baselines.

Recent progress in reasoning models suggests that generating plausible attempts for research-level mathematics may be within reach, but verification remains a bottleneck, consuming scarce expert time. We hypothesize that a meaningful solution should contain enough method-level information that, when applied to a neighborhood of related questions, it should yield better downstream performance than incorrect solutions. Building on this idea, we propose Consequence-Based Utility, an oracle-free evaluator that scores each candidate by testing its value as an in-context exemplar in solving related yet verifiable questions. Our approach is evaluated on an original set of research-level math problems, each paired with one expert-written solution and nine LLM-generated solutions. Notably, Consequence-Based Utility consistently outperforms reward models, generative reward models, and LLM judges on ranking quality. Specifically, for GPT-OSS-120B, it improves Acc@1 from 67.2 to 76.3 and AUC from 71.4 to 79.6, with similarly large AUC gains on GPT-OSS-20B (69.0 to 79.2). Furthermore, compared to LLM-Judges, it also exhibits a larger solver-evaluator gap, maintaining a stronger correct-wrong separation even on instances where the underlying solver often fails to solve.

The scarcity of high-quality, logically sound data is a critical bottleneck for advancing the mathematical reasoning of Large Language Models (LLMs). Our work confronts this challenge by turning decades of automated theorem proving research into a scalable data engine. Rather than relying on error-prone LLMs or complex proof-assistant syntax like Lean and Isabelle, our framework leverages E-prover's saturation capabilities on the vast TPTP axiom library to derive a massive, guaranteed-valid corpus of theorems. Our pipeline is principled and simple: saturate axioms, filter for "interesting" theorems, and generate tasks. With no LLMs in the loop, we eliminate factual errors by construction. This purely symbolic data is then transformed into three difficulty-controlled challenges: entailment verification, premise selection, and proof reconstruction. Our zero-shot experiments on frontier models reveal a clear weakness: performance collapses on tasks requiring deep, structural reasoning. Our framework provides both the diagnostic tool to measure this gap and a scalable source of symbolic training data to address it. We make the code and data publicly available. https://github.com/sileod/reasoning_core https://hf.co/datasets/reasoning-core/rc1

Mathematical reasoning remains a significant challenge for Large Language Models (LLMs) due to hallucinations. When combined with formal proof assistants like Lean, these hallucinations can be eliminated through rigorous verification, making theorem proving reliable. However, even with formal verification, LLMs still struggle with long proofs and complex mathematical formalizations. While Lean with LLMs offers valuable assistance with retrieving lemmas, generating tactics, or even complete proofs, it lacks a crucial capability: providing a sense of proof progress. This limitation particularly impacts the overall development efficiency in large formalization projects. We introduce LeanProgress, a method that predicts the progress in the proof. Training and evaluating our models made on a large corpus of Lean proofs from Lean Workbook Plus and Mathlib4 and how many steps remain to complete it, we employ data preprocessing and balancing techniques to handle the skewed distribution of proof lengths. Our experiments show that LeanProgress achieves an overall prediction accuracy of 75.1\% in predicting the amount of progress and, hence, the remaining number of steps. When integrated into a best-first search framework using Reprover, our method shows a 3.8\% improvement on Mathlib4 compared to baseline performances of 41.2\%, particularly for longer proofs. These results demonstrate how proof progress prediction can enhance both automated and interactive theorem proving, enabling users to make more informed decisions about proof strategies.

We study the accuracy of differentially private mechanisms in the continual release model. A continual release mechanism receives a sensitive dataset as a stream of T inputs and produces, after receiving each input, an accurate output on the obtained inputs. In contrast, a batch algorithm receives the data as one batch and produces a single output. We provide the first strong lower bounds on the error of continual release mechanisms. In particular, for two fundamental problems that are widely studied and used in the batch model, we show that the worst case error of every continual release algorithm is tilde Omega(T^{1/3}) times larger than that of the best batch algorithm. Previous work shows only a polylogarithimic (in T) gap between the worst case error achievable in these two models; further, for many problems, including the summation of binary attributes, the polylogarithmic gap is tight (Dwork et al., 2010; Chan et al., 2010). Our results show that problems closely related to summation -- specifically, those that require selecting the largest of a set of sums -- are fundamentally harder in the continual release model than in the batch model. Our lower bounds assume only that privacy holds for streams fixed in advance (the "nonadaptive" setting). However, we provide matching upper bounds that hold in a model where privacy is required even for adaptively selected streams. This model may be of independent interest.

Lean verifies that a generated declaration is well typed, but cannot check whether it states what the user intended. We measure how far compilation overstates faithfulness in statement autoformalization without canonical Lean targets. Our criterion requires Lean compilation and agreement of two LLM judges from different model families. On an independent random audit, it agrees with the human majority on 91.5% of cases; it rarely accepts an unfaithful output but rejects some faithful ones. Across eight systems on 227 university-level statements, the gap between compiling and accepted outputs ranges from 1.3 to 29.5 percentage points: our tool-augmented agent compiles 87.2% of the statements, but only 57.7% are accepted. Human review of all the agent's rejections confirms that 23.8 of its 29.5 points are real translation errors. For the gaps of the seven one-shot systems, a two-stage review by Claude Opus 5.5 with Lean and GPT-6.1 Sol, validated on these human labels, finds that 78-100% are real translation errors for six of them. When GPT-5.2, one of the two judges, grades the translations of two models released after it, GPT-6 Astra and Claude Opus 5.5, it accepts 83% of the statements for each, against 21% for one-shot GPT-5.2. For these two models, the same review classifies only 4 of their 54 rejections (7%) as translation errors, so most of their measured gap is judge errors rather than translation errors. This is consistent with these systems erring less while the judge does not: the fewer errors a system makes, the smaller the share of its rejections that are translation errors. Finally, a 2^3 tool ablation of the agent shows that Lean feedback drives most of its compilation gain, and that nearly half of this gain fails the semantic criterion. Code and data are available at https://github.com/grenadeComing/beyond-compilation.

Verifiable instruction-following benchmarks emphasize output-constraint satisfaction but can overlook whether responses meaningfully answer the underlying request, creating an evaluation blind spot. We show that frontier models can receive full credit for responses with little substantive answer content. Using the same verifier-based criterion as a training reward can induce reward hacking, as publicly released checkpoints exhibit higher constraint satisfaction but lower response quality after reinforcement learning with verifiable rewards. To address this reward hacking, we introduce CALIF, a lightweight method that augments verifier rewards with a calibrated likelihood-based bonus to preserve response quality. CALIF evaluates response likelihood under the request alone using the existing frozen reference policy, requiring no additional judge or learned reward model. Across models ranging from 1.7B to 8B in reasoning and non-reasoning modes, CALIF improves constraint satisfaction on IFBench while maintaining response quality near or above pre-RL levels. It also outperforms corresponding base policies in 40 of 45 model-metric comparisons across three rubric-based instruction-following benchmarks, demonstrating gains beyond training verifiers.

Polymarket has emerged as a prominent prediction market platform and one of the fastest-growing applications in DeFi. To achieve low-latency trading, it adopts a hybrid architecture that matches orders off-chain but settles them on-chain for final execution. This design creates a consistency gap we call Ghost Fills: an order that is successfully matched off-chain may later fail during on-chain settlement. To understand the security implications of this gap, we investigate such failed settlements by building GHOSTHUNTER, which reconstructs them from on-chain traces and attributes to concrete attack patterns. Across 1,952,440 reverted match-order transactions, we find that attackers exploit the time gap between matching and settlement to invalidate already matched orders before they are finalized on-chain. We then identify four attack vectors from these incidents: nonce bump, balance drain, allowance revoke, and proxy trap, realized via 35 evolving variants. These vectors allow attackers to selectively revert 980,133 filled orders, enabling risk-free prediction, arbitrage-bot hunting, and liquidity reward manipulation, realizing at least \1.49M in profit, which places 1.78 B USD at risk and 2.17 M POL (about \212 K) paid by operator. During peak hours, more than 24.3% of all filled orders reverted, causing de facto DoS attacks. We also find that code derived from the flawed contract still appears in 167 independent contracts across 10 chains holding at least 23 M in user funds, extending the impact beyond Polymarket. We have disclosed our evidence to affected parties, and the issue has been partially mitigated.

Current research on the Decompose-Then-Verify paradigm for evaluating the factuality of long-form text typically treats decomposition and verification in isolation, overlooking their interactions and potential misalignment. We find that existing decomposition policies, typically hand-crafted demonstrations, do not align well with downstream verifiers in terms of atomicity -- a novel metric quantifying information density -- leading to suboptimal verification results. We formulate finding the optimal decomposition policy for optimal verification as a bilevel optimization problem. To approximate a solution for this strongly NP-hard problem, we propose dynamic decomposition, a reinforcement learning framework that leverages verifier feedback to learn a policy for dynamically decomposing claims to verifier-preferred atomicity. Experimental results show that dynamic decomposition outperforms existing decomposition policies, improving verification confidence by 0.07 and accuracy by 0.12 (on a 0-1 scale) on average across varying verifiers, datasets, and atomcities of input claims.

The recent paradigm shift towards training large language models (LLMs) using DeepSeek-R1-Zero-style reinforcement learning (RL) on verifiable rewards has led to impressive advancements in code and mathematical reasoning. However, this methodology is limited to tasks where rule-based answer verification is possible and does not naturally extend to real-world domains such as chemistry, healthcare, engineering, law, biology, business, and economics. Current practical workarounds use an additional LLM as a model-based verifier; however, this introduces issues such as reliance on a strong verifier LLM, susceptibility to reward hacking, and the practical burden of maintaining the verifier model in memory during training. To address this and extend DeepSeek-R1-Zero-style training to general reasoning domains, we propose a verifier-free method (VeriFree) that bypasses answer verification and instead uses RL to directly maximize the probability of generating the reference answer. We compare VeriFree with verifier-based methods and demonstrate that, in addition to its significant practical benefits and reduced compute requirements, VeriFree matches and even surpasses verifier-based methods on extensive evaluations across MMLU-Pro, GPQA, SuperGPQA, and math-related benchmarks. Moreover, we provide insights into this method from multiple perspectives: as an elegant integration of training both the policy and implicit verifier in a unified model, and as a variational optimization approach. Code is available at https://github.com/sail-sg/VeriFree.

Current mathematical reasoning benchmarks for large language models (LLMs) are approaching saturation, with some achieving > 90% accuracy, and are increasingly compromised by training-set contamination. We introduce Putnam-AXIOM, a benchmark of 522 university-level competition problems drawn from the prestigious William Lowell Putnam Mathematical Competition, and Putnam-AXIOM Variation, an unseen companion set of 100 functional variants generated by programmatically perturbing variables and constants. The variation protocol produces an unlimited stream of equally difficult, unseen instances -- yielding a contamination-resilient test bed. On the Original set, OpenAI's o1-preview -- the strongest evaluated model -- scores 41.9%, but its accuracy drops by 19.6% (46.8% relative decrease) on the paired Variations. The remaining eighteen models show the same downward trend, ten of them with non-overlapping 95% confidence intervals. These gaps suggest memorization and highlight the necessity of dynamic benchmarks. We complement "boxed" accuracy with Teacher-Forced Accuracy (TFA), a lightweight metric that directly scores reasoning traces and automates natural language proof evaluations. Putnam-AXIOM therefore provides a rigorous, contamination-resilient evaluation framework for assessing advanced mathematical reasoning of LLMs. Data and evaluation code are publicly available at https://github.com/brando90/putnam-axiom.

We present and evaluate LeanFlow, an LLM agent system specialized for translating mathematical papers into buildable Lean projects. Recent verifier-in-the-loop systems show that large formal artifacts can be produced, but it remains unclear which runtime mechanisms affect completion, auditability, or efficiency in document-to-project formalization. We study this question through case studies on two previously unformalized mathematical papers in number theory and measure theory, using model, proof-workflow, and toolset ablations with Kimi2.6 and GPT5.5; we report task outcome, API calls, input tokens, and output tokens. With Kimi2.6, the full workflow completes both document-level projects within the 2000-call budget, while no-queue variants reach the budget limit; with GPT5.5, all document-level variants complete, and the full workflow has the lowest or tied-lowest input-token cost on both sources. As complementary calibration, LeanFlow reaches 75.7% BEq+ on the PFR slice of RLM25 and solves all five ICML 2026 AI for Math TCS challenge projects in our GPT5.5 runs.

A verifier for robot policies reads a candidate behavior and returns a score for how well it did, used both to evaluate vision-language-action policies and to train them. Verifiers range from success detectors and reward models to runtime monitors, safety filters, and temporal-logic specifications. We survey roughly 150 verifiers and compare them along two properties. Availability is how much a verdict costs, how early in a rollout the verdict arrives, and how often a verdict can be asked for. Availability rises as verdicts get cheaper, earlier, and denser. Credibility is how much a high score tells us about the task. Credibility falls as the judgment becomes gameable and self-serving. We group the verifiers by who supplies the judgment: human verifiers, rule-based and formal verifiers, learned and pretrained verifiers, and model-intrinsic verifiers. Across the four families, we find that credibility falls as availability rises. Regardless of who supplies the judgment, there is no free checker. We then examine what validates a verifier itself, and how much a high score tells us. Three measures appear in the literature: agreement with human labels, the performance of the policy it trains, and behavior under reward hacking. We close with nine metrics that make a verifier claim checkable, and coordinates for the verifiers still to be built.

Deep Research agents extend LLM-based assistants into long-horizon workflows involving planning, retrieval, evidence synthesis, and report generation, yet their reliability in open information environments remains underexplored. A key concern is whether apparently credible but factually misleading knowledge encountered in such environments can propagate through these workflows and be adopted as false conclusions in final reports. To study this failure mode, we introduce MisKnow-Agent, a framework for constructing and validating misleading knowledge for Deep Research tasks. MisKnow-Agent generates misleading instances with controllable authority levels and styles, yielding 5,933 quality-controlled instances built on DeepResearch Benchmark tasks. Extensive experiments across open-source and closed-source Deep Research agents show that even limited exposure to misleading knowledge can induce false-conclusion adoption in final reports, revealing a broad reliability vulnerability in current Deep Research agents. Although search-enabled verifier models consistently identify the retained instances as misleading during focused corpus validation, the same instances can still be adopted during long-horizon research, revealing a disconnect between focused verification and workflow-level evidence use. Finally, we evaluate pre- and post-research defenses, both individually and in combination, finding that all three configurations mitigate but do not fully prevent false-conclusion adoption. Our findings suggest that reliable Deep Research requires evidence verification and correction capabilities at both the model and framework levels, beyond improvements in planning, retrieval, evidence integration, or report-generation abilities.

Artificial Intelligence (AI) is starting to transform the research process as we know it by automating the discovery of new solutions. Given a task, the typical AI-driven approach is (i) to generate a set of diverse solutions, and then (ii) to verify these solutions and select one that solves the problem. Crucially, this approach assumes the existence of a reliable verifier, i.e., one that can accurately determine whether a solution solves the given problem. We argue that systems research, long focused on designing and evaluating new performance-oriented algorithms, is particularly well-suited for AI-driven solution discovery. This is because system performance problems naturally admit reliable verifiers: solutions are typically implemented in real systems or simulators, and verification reduces to running these software artifacts against predefined workloads and measuring performance. We term this approach as AI-Driven Research for Systems (ADRS), which iteratively generates, evaluates, and refines solutions. Using penEvolve, an existing open-source ADRS instance, we present case studies across diverse domains, including load balancing for multi-region cloud scheduling, Mixture-of-Experts inference, LLM-based SQL queries, and transaction scheduling. In multiple instances, ADRS discovers algorithms that outperform state-of-the-art human designs (e.g., achieving up to 5.0x runtime improvements or 50% cost reductions). We distill best practices for guiding algorithm evolution, from prompt design to evaluator construction, for existing frameworks. We then discuss the broader implications for the systems community: as AI assumes a central role in algorithm design, we argue that human researchers will increasingly focus on problem formulation and strategic guidance. Our results highlight both the disruptive potential and the urgent need to adapt systems research practices in the age of AI.

A growing number of agentic pipelines search for governing equations: an agent proposes laws and a verifier checks them against simulated trajectories. The verifier sees only a numerical summary: integrals against test functions on a time grid. That summary carries its own discretization error, making the verifier part of the simulator. We freeze the trajectory, candidate library, verifier and thresholds, leaving the agent one move: which permitted summary to admit. Without scores or language models, finer time grids reduce numerical discrepancies and false library rejections. On PDEBench, the verifier instead accepts equations missing diffusion on score-selected summaries. Maximizing a public score rewarding decisive verdicts and larger mismatches when rejecting the library wrongly rejects all 1,500 in-library cases; retrospectively retuning score constants on a separate bank reduces errors to 13/180 at best. We call this evidence-view gaming: reward hacking by choice of permitted summary. In single-turn forced-choice tests, Claude models rarely choose summaries giving wrong verdicts when asked to be accurate, but often do so when asked to maximize the score or report missing physics. In separate trust prompts, they often rank those harmful summaries least trustworthy. Requiring several summaries to agree reduces errors but often withholds a verdict and performs worse than a fixed fine view on PDEBench: a safeguard, not a certificate. Evidence chosen after its verdict is known cannot alone authorize a claim.

The formalization of existing mathematical proofs is a notoriously difficult process. Despite decades of research on automation and proof assistants, writing formal proofs remains arduous and only accessible to a few experts. While previous studies to automate formalization focused on powerful search algorithms, no attempts were made to take advantage of available informal proofs. In this work, we introduce Draft, Sketch, and Prove (DSP), a method that maps informal proofs to formal proof sketches, and uses the sketches to guide an automated prover by directing its search to easier sub-problems. We investigate two relevant setups where informal proofs are either written by humans or generated by a language model. Our experiments and ablation studies show that large language models are able to produce well-structured formal sketches that follow the same reasoning steps as the informal proofs. Guiding an automated prover with these sketches enhances its performance from 20.9% to 39.3% on a collection of mathematical competition problems.

Process attestation verifies human authorship by collecting behavioral biometric evidence, including keystroke dynamics, typing patterns, and editing behavior, during the creative process. However, the very data needed to prove authenticity can reveal intimate details about an author's cognitive state, health conditions, and identity, constituting sensitive biometric data under GDPR Article 9. We resolve this privacy-attestation paradox using zero-knowledge proofs. We present ZK-PoP, a construction that allows a verifier to confirm that (a) sequential work function chains were computed correctly, (b) behavioral feature vectors fall within human population distributions, and (c) content evolution is consistent with incremental human editing, all without learning the underlying behavioral data, exact timing, or intermediate content. Our construction uses Groth16 proofs over arithmetic circuits with Pedersen commitments and Bulletproof range proofs. We prove that ZK-PoP is computationally zero-knowledge, computationally sound, and achieves unlinkability across sessions. Evaluation shows proof generation in under 30 seconds for a 1-hour writing session, with 192-byte proofs verifiable in 8.2 ms, while incurring less than 5% accuracy loss in simulation at practical privacy levels (epsilon >= 1.0) compared to non-private baselines.

Test-time scaling often uses an external verifier, such as compilers and test cases in coding or trained value functions in robotics applications, to obtain high-quality rollouts. Verifier-free test-time scaling (or VF-TTS) is gaining extensive attention as a mechanism to enhance Large Language Model (LLM) reasoning, primarily because we do not have access to such high-quality verifiers in many real-world applications. Among existing VF-TTS methods, confidence-based VF-TTS methods, which compute and rank rollouts solely by confidence, are particularly promising. Such methods introduce near-zero overhead for sample evaluation and require minimal access to internal model states, making the methods highly flexible across models and tasks. In this paper, we demonstrate a critical limitation of existing confidence-based VF-TTS methods by showing that such methods catastrophically break down on complex tasks. We observe a very interesting phenomenon: uniformly high confidence frequently indicates a failure to explore, favoring confidently wrong answers. To address this, our core insight is that robust cognitive search requires a specific confidence trajectory pattern: such methods perform exploratory branching at the beginning, as manifested by low initial confidence, and converge to a high final confidence solution. To implement this insight, we introduce consilience, a novel selection framework that explicitly evaluates the temporal asymmetry of confidence in reasoning. We operationalize this via a combinatorial metric that actively penalizes high initial confidence while strictly demanding final certainty. Extensive experiments covering both graduate-level mathematics problems and free-form code generation demonstrate that consilience effectively outperforms existing baselines, validating our novel perspective on completion confidence.

Large language models (LLMs) are increasingly used to generate Verus contracts for Rust programs together with the proofs that establish them. Yet verifier acceptance, the common measure of success, shows only that the implementation satisfies the generated contract, so it leaves open whether the contract captures the intended behavior. Assessing specification generation therefore requires an evaluation methodology, not a single metric. We present VerusEval, a framework that evaluates generated contracts along six dimensions, including Verus-native checks for semantic triviality, validated input-output (I/O) suites, and contract equivalence, and that relates them so that the strength and blind spots of each become measurable. Evaluating 13,716 artifacts from four workflows, five LLMs, and two prompting settings on 762 tasks, we find that acceptance overstates quality: only 42.3\% of the accepted artifacts complete all three semantic checks without a flag, and across workflows and models, higher acceptance does not consistently come with more correct contracts. I/O evaluation rejects 23.6\% of the accepted artifacts, yet only 63.2\% of those that pass it are proved equivalent. Of the accepted artifacts flagged by contract comparison, 84.1\% have a rejected completeness obligation, and the direction of a failure prioritizes inspection of the generated contract or the reference; it does not by itself establish which artifact is defective. Screening and review further identify 36 candidate defects among 61 selected reference contracts, and repairing one of them makes 10 of its 11 failing artifacts equivalent. Based on these findings, we derive four actionable lessons for evaluation, generation, and benchmark maintenance.

Reliable verifiable data has become a key driver of capability gains in modern language models, enabling stable reinforcement learning with verifiable rewards and effective distillation that transfers competence across math, coding, and agentic tasks. Yet constructing generalizable synthetic verifiable data remains difficult due to hallucination-prone generation, and weak or trivial verification artifacts that fail to separate strong from weak solutions. Existing approaches often rely on task-specific heuristics or post-hoc filters that do not transfer across domains and lack a principled, universal evaluator of verifiability. In this work, we introduce an evolutionary, task-agnostic, strategy-guided, executably-checkable data synthesis framework that, from minimal seed supervision, jointly synthesizes problems, diverse candidate solutions, and verification artifacts, and iteratively discovers strategies via a consistency-based evaluator that enforces agreement between human-annotated and strategy-induced checks. This pipeline upgrades filtering into principled synthesis: it reliably assembles coherent, verifiable training instances and generalizes without domain-specific rules. Our experiments demonstrate the effectiveness of the proposed approach under both RLVR and model distillation training paradigms. The results show that training with our synthesized data yields significant improvements on both the LiveCodeBench and AgentBench-OS tasks, highlighting the robust generalization of our framework.

We propose claim-level falsification as a principle for test-time scaling and instantiate it through Claim-Level Reliability Assessment (CLR), a training-free framework that reallocates test-time compute from additional solution sampling to targeted verification. Since whole-trace evaluation often obscures decisive errors due to signal dilution from routine tokens, CLR condenses each reasoning trace into a compact set of decision-critical claims, thereby isolating its logical anchors. Furthermore, recognizing the inherent difficulty of generating entirely correct solutions under fixed model capabilities, CLR shifts the focus to semantic falsification. This approach exploits a fundamental asymmetry between solution construction and claim refutation. Constructing a valid solution requires a flawless reasoning path, whereas refuting an incorrect claim requires identifying only a single decisive flaw. This targeted search for negative evidence systematically compresses the survival space of high-confidence incorrect traces, effectively suppressing erroneous consensus via nonlinear reliability scoring. Across four LLMs and four reasoning benchmarks under matched budgets, CLR generally improves upon pass@1 and self-consistency. On GPT-OSS-20B/CMIMC25, for instance, CLR exceeds pass@1 by 27.15 percentage-points and raises self-consistency accuracy from 77.50\% to 82.19\% with 37.0\% fewer tokens.

We present gap-lean4-port (Release v0.2.0), a machine-checked formalization of foundational algorithms in computational discrete algebra and permutation group theory within the Lean 4 interactive theorem prover and Mathlib4. While modern proof assistants feature extensive abstract algebraic hierarchies, constructive and operational permutation group algorithms, such as Charles Sims' 1970 Schreier-Sims algorithm, transversal tree lookups, and backtrack ordered partition refinement, have remained largely unformalized in dependent type theory. Here, we formalize the operational algorithmic core of the Groups, Algorithms, Programming (GAP) system library across four foundational modules, proving 35 machine-checked theorems with zero unproven conjectures (sorry) and zero custom axioms under standard Lean 4 foundations (propext, Classical.choice, Quot.sound). We machine-check: (1) Schreier-Sims stabilizer chain hierarchies (StabLevel, StabChain) and the invariance of incremental transversal tree extensions (extendSchreierPoint_invariant); (2) single-level and multi-level Schreier sifting reductions (siftOneLevel, siftFull), proving that full sifting strictly fixes all base points (siftFull_fixes_all_basePoints); (3) constructive soundness and completeness of Base and Strong Generating Set (BSGS) membership testing (membershipTestKnownBase_iff_mem); (4) backtrack ordered partition cell refinement (splitCellByPred), proving mutual cell disjointness, union conservation, and exact cardinality preservation; (5) cyclotomic extension rings Z/nZ(eps_m) and machine-check GAP's exact size theorem |Z/nZ(eps_m)| = n^m; and (6) executable Bezout inverses via Extended Euclidean GCD (Nat.gcdA) over residue rings Z/nZ. The entire codebase compiles deterministically under 'lake build RequestProject' and is openly available at https://github.com/pCwOrM/gap-lean4-port.

Software testing plays a critical role in ensuring that systems behave as intended. However, existing automated testing approaches struggle to match the capabilities of human engineers due to key limitations such as test locality, lack of general reliability, and business logic blindness. In this work, we propose a novel framework that leverages functional programming and type systems to translate Scala backend code into formal Lean representations. Our pipeline automatically generates theorems that specify the intended behavior of APIs and database operations, and uses LLM-based provers to verify them. When a theorem is proved, the corresponding logic is guaranteed to be correct and no further testing is needed. If the negation of a theorem is proved instead, it confirms a bug. In cases where neither can be proved, human intervention is required. We evaluate our method on realistic backend systems and find that it can formally verify over 50% of the test requirements, which suggests that half of a testing engineer's workload can be automated. Additionally, with an average cost of only $2.19 per API, LLM-based verification is significantly more cost-effective than manual testing and can be scaled easily through parallel execution. Our results indicate a promising direction for scalable, AI-powered software testing, with the potential to greatly improve engineering productivity as models continue to advance.

Verifiers play a crucial role in large language model (LLM) reasoning, needed by post-training techniques such as reinforcement learning. However, reliable verifiers are hard to get for difficult coding problems, because a well-disguised wrong solution may only be detected by carefully human-written edge cases that are difficult to synthesize. To address this issue, we propose HARDTESTGEN, a pipeline for high-quality test synthesis using LLMs. With this pipeline, we curate a comprehensive competitive programming dataset HARDTESTS with 47k problems and synthetic high-quality tests. Compared with existing tests, HARDTESTGEN tests demonstrate precision that is 11.3 percentage points higher and recall that is 17.5 percentage points higher when evaluating LLM-generated code. For harder problems, the improvement in precision can be as large as 40 points. HARDTESTS also proves to be more effective for model training, measured by downstream code generation performance. We will open-source our dataset and synthesis pipeline at https://leililab.github.io/HardTests/.

The global economy is entering the era of Agentic Commerce, where autonomous agents can discover services, negotiate prices, and transact value. However adoption towards agentic commerce faces a foundational trust gap: current systems are built for direct human interactions rather than agent-driven operations. It lacks core primitives across three critical stages of agentic transactions. First, Task Delegation lacks means to translate user intent into defined scopes, discover appropriate agents, and securely authorize actions. Second, Payment Settlement for tasks is processed before execution, lacking verifiable evidence to validate the agent's work. Third, Audit Mechanisms fail to capture the full transaction lifecycle, preventing clear accountability for disputes. While emerging standards address fragments of this trust gap, there still remains a critical need for a unified infrastructure that binds the entire transaction lifecycle. To resolve this gap, we introduce TessPay, a unified infrastructure that replaces implicit trust with a 'Verify-then-Pay' architecture. It is a two plane architecture separating control and verification from settlement. TessPay operationalizes trust across four distinct stages: Before execution, agents are anchored in a canonical registry and user intent is captured as verifiable mandates, enabling stakeholder accountability. During execution, funds are locked in escrow while the agent executes the task and generates cryptographic evidence (TLS Notary, TEE etc.) to support Proof of Task Execution (PoTE). At settlement, the system verifies this evidence and releases funds only when the PoTE satisfies verification predicates; modular rail adapters ensure this PoTE-gated escrow remains chain-agnostic across heterogeneous payment rails. After settlement, TessPay preserves a tamper-evident audit trail to enable clear accountability for dispute resolution.

Modern Lean theorem provers achieve strong performance only with substantial training and inference compute, driven in part by scarce verified proof data and the long reasoning traces of formal proof search, making both supervised fine-tuning (SFT) and sampling expensive. We introduce Pythagoras-Prover, a compute-efficient open-source family of Lean theorem provers built for practical compute budgets. The family spans two generation paradigms: autoregressive models at 4B and 32B parameters, and a first proof-of-concept diffusion-based prover (4B) that iteratively refines Lean proofs at inference time. For training efficiency, we build a Lean-verified corpus stratified into easy, medium, and hard problems for curriculum SFT, so models acquire proof skills progressively from shorter, simpler proofs to longer, harder ones. During SFT, a dynamic proof-reasoning filtering scheme preserves informative proof traces while keeping each instance within an 8k-token context budget. We also introduce Augmented Lean Formalisation (ALF), which expands scarce verified corpora into variants of formal statements, populated via self-distillation for extra training signal without formally verifying every mutated instance. By perturbing known problems while preserving their formal character, ALF reduces reliance on any statement's surface form. Empirically, Pythagoras-Prover-4B surpasses DeepSeek-Prover-V2-671B at pass@32 on MiniF2F-Test (86.1% vs 82.4%) with ~167x fewer parameters, while Pythagoras-Prover-32B sets the open-source state of the art at 93.0% on MiniF2F-Test and solves 93 of 672 PutnamBench problems. We release MiniF2F-ALF, an ALF-mutated contamination-sensitive benchmark on which every evaluated model loses accuracy; here our 32B remains strongest and our 4B matches the prior state of the art, Goedel-Prover-V2-32B.

Large language models (LLMs) can generate plausible code but offer limited guarantees of correctness. Formally verifying that implementations satisfy specifications requires constructing machine-checkable proofs, a task that remains beyond current automation. We propose a hierarchical proof search framework for automated code verification in Lean~4 that decomposes complex verification goals into structurally simpler subgoals before attempting tactic-level proving. Central to our approach is a principled decomposition score that combines constructive justification with structural effectiveness. Crucially, this score serves as both the training reward and the inference-time ranking criterion, ensuring strict alignment between optimization and deployment. We train Goedel-Code-Prover-8B, a single unified policy for both decomposition and completion, via supervised initialization followed by hybrid reinforcement learning, where a continuous decomposition reward drives planning exploration while supervised replay stabilizes proof generation. On three Lean-based code verification benchmarks comprising 427 tasks, our 8B-parameter model achieves a 62.0\% prove success rate, a 2.6times improvement over the strongest baseline, surpassing neural provers up to 84times larger. We further observe consistent inference-time scaling: success rates improve monotonically with search iterations and sampling budget, with our trained model achieving greater efficiency than frontier off-the-shelf models of comparable scale.

Large language models (LLMs) often struggle with maintaining accuracy across a sequence of intermediate reasoning steps in mathematical reasoning, leading to error propagation that undermines the final result. The current methodology to mitigate this issue primarily involves using a verifier model to assess the correctness of generated solution candidates, focusing either on the overall reasoning path or on an incomplete reasoning path. By rethinking this approach, we argue that assessing potentials of incomplete reasoning paths could be more advantageous as it guides towards correct final answers, transforming the task into a planning problem. Our proposed verifier, the Outcome-supervision Value Model (OVM), employs outcome supervision for training, offering an efficient and intuitive method for planning by prioritizing steps that lead to accurate conclusions over mere per-step correctness. Furthermore, the OVM eschews the need for labor-intensive annotations on step-level correctness, enhancing its scalability. Our experiments on two multi-step mathematical reasoning datasets, GSM8K and Game of 24, demonstrate the superior performance of the OVM model. Notably, in GSM8K, our OVM-7B model achieves state-of-the-art results among LLMs up to 13B parameters; especially it does not utilize GPT-4 or code execution. These findings offer a novel perspective on the role of outcome supervision in training verifiers for multi-step reasoning tasks and provide theoretical justification for its advantage in value estimation for planning.

Fact verification is essential for ensuring the reliability of LLM applications. In this study, we evaluate 12 pre-trained LLMs and one specialized fact-verifier, including frontier LLMs and open-weight reasoning LLMs, using a collection of examples from 14 fact-checking benchmarks. We share three findings intended to guide future development of more robust fact verifiers. First, we highlight the importance of addressing annotation errors and ambiguity in datasets, demonstrating that approximately 16\% of ambiguous or incorrectly labeled data substantially influences model rankings. Neglecting this issue may result in misleading conclusions during comparative evaluations, and we suggest using a systematic pipeline utilizing LLM-as-a-judge to help identify these issues at scale. Second, we discover that frontier LLMs with few-shot in-context examples, often overlooked in previous works, achieve top-tier performance. We therefore recommend future studies include comparisons with these simple yet highly effective baselines. Lastly, despite their effectiveness, frontier LLMs incur substantial costs, motivating the development of small, fine-tuned fact verifiers. We show that these small models still have room for improvement, particularly on instances that require complex reasoning. Encouragingly, we demonstrate that augmenting training with synthetic multi-hop reasoning data significantly enhances their capabilities in such instances. We release our code, model, and dataset at https://github.com/just1nseo/verifying-the-verifiers

Large language models are increasingly used to generate code from natural language, but ensuring correctness remains challenging. Formal verification offers a principled way to obtain such guarantees by proving that a program satisfies a formal specification. However, specifications are frequently missing in real-world codebases, and writing high-quality specifications remains expensive and expertise-intensive. We present VeriSpecGen, a traceable refinement framework that synthesizes intent-aligned specifications in Lean through requirement-level attribution and localized repair. VeriSpecGen decomposes natural language into atomic requirements and generates requirement-targeted tests with explicit traceability maps to validate generated specifications. When validation fails, traceability maps attribute failures to specific requirements, enabling targeted clause-level repairs. VeriSpecGen achieve 86.6% on VERINA SpecGen task using Claude Opus 4.5, improving over baselines by up to 31.8 points across different model families and scales. Beyond inference-time gains, we generate 343K training examples from VeriSpecGen refinement trajectories and demonstrate that training on these trajectories substantially improves specification synthesis by 62-106% relative and transfers gains to general reasoning abilities.

A classical intuition holds that verifying a solution is easier than producing one. For today's coding agents, this intuition is being inverted: as foundation models develop stronger reasoning capabilities and engineering harnesses grow more sophisticated, generating complex candidate solutions is no longer difficult -- reliably verifying them has become the harder problem. Every verifier we can build is only a proxy for human intent, never the intent itself. This makes verification subject to a twofold difficulty: first, intent is underspecified by nature, making it inherently hard to faithfully check whether it has been fulfilled; second, during model training, optimization widens the gap between proxy and intent -- manifesting as reward hacking or signal saturation. To address this, we characterize the quality of verification signals along three dimensions -- scalability, faithfulness, and robustness -- and argue that achieving all three simultaneously is the central challenge. We further study four reward constructions: a test verifier for general coding tasks, a rubric verifier for frontend tasks, the user as verifier for real-world agent tasks, and an automated agent verifier for long-horizon tasks. Across different task types and policy capability levels, we conduct in-depth analysis and experiments on the core challenges of reward design and how to more effectively leverage reward signals. Experiments show that targeted verification design can effectively suppress reward hacking, improve task completion quality, and achieve significant gains across multiple internal and public benchmarks. These experiences collectively point to a core observation: no fixed reward function can remain effective as policy capability continues to grow; and verification must co-evolve with the generator.

Serial verification gates are a core reliability primitive in LLM harnesses: a candidate answer is returned only if k verifier calls all accept it. Under conditionally independent gates, the recent Odds Law (arXiv:2606.15712) shows that posterior log-odds grow linearly in k, so failure decays exponentially, and states that "a tight theory of partially correlated verifier cascades remains open." This note gives a minimal such theory. Modeling the per-instance false-accept rate on the generator's own errors as a latent variable αsim G (de Finetti), the exact cascade posterior is ell_k = ell_0 - ln m_k, with m_k the k-th moment of G. Then: (i) ell_k is concave in k for every non-degenerate G -- the Odds Law is its tangent at the first gate and an upper bound; (ii) for Beta(a,b) latents, failure decays polynomially, 1-r_k asymp k^{-b}, with correlation parameter ρ_v = 1/(a+b+1); (iii) a blind-spot atom of mass 1-π at α=1 caps the evidence extractable from any number of gates at -ln(1-π) nats, so reliability saturates below 1; (iv) letting the true-accept rate also vary (βsim H) yields a trichotomy -- gates eventually always help, plateau, or actively harm -- decided by the upper-tail exponents of G and H, with closed-form crossover k^dagger. The mechanism is survivorship: errors surviving gates are the high-α ones. The theory is measurable: R repeated verdicts per instance identify the first R moments of G, so two verdicts identify ρ_v; beta-binomial likelihood and NPMLE recover the reliability curve and the ill-posed ceiling. In synthetic tests, independence-based extrapolation underestimates failure by 20x at k=5 and ~3000x at k=10; the correlated fit at R=8 tracks held-out depths. The practical lever is decorrelation -- changing model family, modality, or evidence source -- not adding gates.

Recent work on reinforcement learning with verifiable rewards (RLVR) has shown that large language models (LLMs) can be substantially improved using outcome-level verification signals, such as unit tests for code or exact-match checks for mathematics. In parallel, process supervision has long been explored as a way to shape the intermediate reasoning behaviour of LLMs, but existing approaches rely on neural judges to score chain-of-thought steps, leaving them vulnerable to opacity, bias, and reward hacking. To address this gap, we introduce Verifiable Process Reward Models (VPRMs), a reinforcement-learning framework in which intermediate reasoning steps are checked by deterministic, rule-based verifiers. We apply VPRMs to risk-of-bias assessment for medical evidence synthesis, a domain where guideline-defined criteria and rule-based decision paths enable programmatic verification of reasoning traces. Across multiple datasets, we find that VPRMs generate reasoning that adheres closely to domain rules and achieve substantially higher coherence between step-level decisions and final labels. Results show that VPRMs achieve up to 20% higher F1 than state-of-the-art models and 6.5% higher than verifiable outcome rewards, with substantial gains in evidence grounding and logical coherence.

Verifiers have been demonstrated to enhance LLM reasoning via test-time scaling (TTS). Yet, they face significant challenges in complex domains. Error propagation from incorrect intermediate reasoning can lead to false positives for seemingly plausible solutions, while lacking external grounding makes verifiers unreliable on computation or knowledge-intensive tasks. To address these challenges, we propose Agentic Verifier, a framework that transforms reward modeling into a multi-turn, tool-augmented deliberative process. We introduce complementary forward and backward agents: one traces solutions from premises to conclusions, while the other re-checks conclusions against their underlying premises. This bidirectional process enables a comprehensive, reliable, and interpretable assessment of solutions. To facilitate practical deployment, we propose AgentV-RL. Through proactive exploration and reinforcement learning, the verifier autonomously interleaves tool-use with internal reasoning. Extensive experiments show that Agentic Verifier yields consistent performance gains under both parallel and sequential TTS. Notably, our 4B variant surpasses state-of-the-art ORMs by 25.2%, positioning it as a promising paradigm for agentic reward modeling.

A fundamental challenge in formal theorem proving by LLMs is the lack of high-quality training data. Although reinforcement learning or expert iteration partially mitigates this issue by alternating between LLM generating proofs and finetuning them on correctly generated ones, performance quickly plateaus due to the scarcity of correct proofs (sparse rewards). To keep improving the models with limited data, we draw inspiration from mathematicians, who continuously develop new results, partly by proposing novel conjectures or exercises (which are often variants of known results) and attempting to solve them. We design the Self-play Theorem Prover (STP) that simultaneously takes on two roles, conjecturer and prover, each providing training signals to the other. The conjecturer is trained iteratively on previously generated conjectures that are barely provable by the current prover, which incentivizes it to generate increasingly challenging conjectures over time. The prover attempts to prove the conjectures with standard expert iteration. We evaluate STP with both Lean and Isabelle formal versifiers. With 19.8 billion tokens generated during the training in Lean, STP proves 26.3% of the statements in the LeanWorkbook dataset, doubling the previous best result of 13.2% achieved through expert iteration. The final model achieves state-of-the-art performance among whole-proof generation methods on miniF2F-test (61.7%, pass@3200), Proofnet-test (23.1%, pass@3200) and PutnamBench (8/644, pass@3200).

Inference-time planning with large language models frequently breaks under partial observability: when task-critical preconditions are not specified at query time, models tend to hallucinate missing facts or produce plans that violate hard constraints. We introduce Self-Querying Bidirectional Categorical Planning (SQ-BCP), which explicitly represents precondition status (Sat/Viol/Unk) and resolves unknowns via (i) targeted self-queries to an oracle/user or (ii) bridging hypotheses that establish the missing condition through an additional action. SQ-BCP performs bidirectional search and invokes a pullback-based verifier as a categorical certificate of goal compatibility, while using distance-based scores only for ranking and pruning. We prove that when the verifier succeeds and hard constraints pass deterministic checks, accepted plans are compatible with goal requirements; under bounded branching and finite resolution depth, SQ-BCP finds an accepting plan when one exists. Across WikiHow and RecipeNLG tasks with withheld preconditions, SQ-BCP reduces resource-violation rates to 14.9\% and 5.8\% (vs.\ 26.0\% and 15.7\% for the best baseline), while maintaining competitive reference quality.

Large reasoning models (e.g., R1, o3) have demonstrated remarkable mathematical problem-solving abilities. However, the high reported accuracy of these advanced models on popular datasets, reliance on purely numerical evaluation and potential benchmark leakage, often masks their true reasoning shortcomings. To address this, we propose leveraging the inherent rigor and methodological complexity of mathematical proofs as a diagnostic tool to expose these hidden failures. Specifically, we introduce the RFMDataset (Reveal Failure Modes), a collection of 200 diverse mathematical proof problems, and thoroughly evaluate advanced models' performance on it. Our in-depth analysis of their failures uncovers 10 fine-grained error types, which shows fundamental limitations in current large reasoning models: 1) large reasoning models grapple profoundly with mathematical proofs, with some generating entirely correct proofs for less than 20% of problems and failing even on basic ones; 2) models exhibit a diverse spectrum of reasoning failures, prominently demonstrating the lack of guarantees for the correctness and rigor of single-step reasoning; and 3) models show hallucination and incompleteness during the reasoning process. Our findings reveal that models' self-reflection is insufficient to resolve the current logical dilemmas, necessitating formalized and fine-grained logical training.

Improving a language model today means retraining it: enormous compute, a new opaque model each cycle, non-deterministic output. We take the opposite path: the model stays frozen, and a persistent memory of verified solutions grows beside it. Once a problem family is solved and has passed an independent verification step that never consults the answer key, every new instance of that family is answered at zero generation tokens, bit-exact, deterministically. Across 180 fresh instances spanning nine problem families, four architectures from four vendors - dense and mixture-of-experts - each score 180/180 at zero generation tokens per answer: execution-bound capability decoupled from parameter scaling. A negative control attributes the capability fully to the memory: emptied, it solves nothing. The same verify-before-store contract holds for open-ended reasoning: 88/88 consistency-gated acceptances across all four models, machine-checked formal proof, and reasoning-method transfer at 77/80. Memory selection takes 1.4 microseconds; a full reuse completes in 6-23 ms at 36 mWh. Approximate similarity retrieval selects the wrong item 94.3% of the time on a 4,500-item verified store where exact addressing makes zero errors. The store also serves as working context at a scale no shipped engine matches: a 6,000,000-token movable window on a single 46 GB GPU at flat memory, where vLLM stops at 30,399 tokens and SGLang silently truncates past 32,000. On published benchmarks, frontier models remain far ahead of any 12B at raw from-scratch reasoning; on everything this system has solved and verified, the comparison inverts: a frontier API call pays a fresh generation pass on every query, forever, while verified reuse costs zero tokens and returns the identical bits every time. A public testbench with free, rate-limited access accompanies this report: https://corbenic-galahad-bench.hf.space

The standard paradigm for solving coding tasks via large language models (LLMs) is to generate-then-rank programs, where the latter step uses a verifier in the ranking process. The growing consensus is that a comprehensive verifier (e.g., a full test suite) should be prioritized over an outcome reward model (ORM) whenever possible, with little consideration given to the trade-offs involved. We aim to challenge this assumption by systematically exploring the tradeoff between speed and accuracy. We find that ORMs play a crucial role in scaling verification through trading accuracy for speed, even when a comprehensive verifier is available. Their value becomes especially apparent when used in a generate-prune-then-rank approach, where a faster but less accurate verifier removes incorrect solutions prior to ranking -- leading to a system that is 11.65x faster while only being 8.33% less accurate than the full test suite. We analyze the generate-prune-then-rank approach and show that it works by filtering out incorrect but highly ranked solutions. These findings enable the design of scalable and accurate program ranking systems.

Trustworthy verifiers are essential for the success of reinforcement learning with verifiable reward (RLVR), which is the core methodology behind various large reasoning models such as DeepSeek-R1. In complex domains like mathematical reasoning, rule-based verifiers have been widely adopted in previous works to train strong reasoning models. However, the reliability of these verifiers and their impact on the RL training process remain poorly understood. In this work, we take mathematical reasoning as a case study and conduct a comprehensive analysis of various verifiers in both static evaluation and RL training scenarios. First, we find that current open-source rule-based verifiers often fail to recognize equivalent answers presented in different formats across multiple commonly used mathematical datasets, resulting in non-negligible false negative rates. This limitation adversely affects RL training performance and becomes more pronounced as the policy model gets stronger. Subsequently, we investigate model-based verifiers as a potential solution to address these limitations. While the static evaluation shows that model-based verifiers achieve significantly higher verification accuracy, further analysis and RL training results imply that they are highly susceptible to hacking, where they misclassify certain patterns in responses as correct (i.e., false positives). This vulnerability is exploited during policy model optimization, leading to artificially inflated rewards. Our findings underscore the unique risks inherent to both rule-based and model-based verifiers, aiming to offer valuable insights to develop more robust reward systems in reinforcement learning.