LLM Answer Verification

The use of Large Language Models (LLMs) to support patients in addressing medical questions is becoming increasingly prevalent. However, most of the measures currently used to evaluate the performance of these models in this context only measure how closely a model's answers match semantically, and therefore do not provide a true indication of the model's medical accuracy or of the health equity risks associated with it. To address these shortcomings, we present a new evaluation framework for medical question answering called VB-Score (Verification-Based Score) that provides a separate evaluation of the four components of entity recognition, semantic similarity, factual consistency, and structured information completeness for medical question-answering models. We perform rigorous reviews of the performance of three well-known and widely used LLMs on 48 public health-related topics taken from high-quality, authoritative information sources. Based on our analyses, we discover a major discrepancy between the models' semantic and entity accuracy. Our assessments of the performance of all three models show that each of them has almost uniformly severe performance failures when evaluated against our criteria. Our findings indicate alarming performance disparities across various public health topics, with most of the models exhibiting 13.8% lower performance (compared to an overall average) for all the public health topics that relate to chronic conditions that occur in older and minority populations, which indicates the existence of what's known as condition-based algorithmic discrimination. Our findings also demonstrate that prompt engineering alone does not compensate for basic architectural limitations on how these models perform in extracting medical entities and raise the question of whether semantic evaluation alone is a sufficient measure of medical AI safety.

Verification of model outputs is rapidly emerging as a key primitive for both training and real-world deployment of large language models (LLMs). In practice, this often involves using imperfect LLM judges and reward models since ground truth acquisition can be time-consuming and expensive. We introduce Fully Unsupervised Score Ensembling (FUSE), a method for improving verification quality by ensembling verifiers without access to ground truth correctness labels. The key idea behind FUSE is to control conditional dependencies between verifiers in a manner that improves the unsupervised performance of a class of spectral algorithms from the ensembling literature. Despite requiring zero ground truth labels, FUSE typically matches or improves upon semi-supervised alternatives in test-time scaling experiments with diverse sets of generator models, verifiers, and benchmarks. In particular, we validate our method on both conventional academic benchmarks such as GPQA Diamond and on frontier, unsaturated benchmarks such as Humanity's Last Exam and IMO Shortlist questions.

Speculative decoding (SD) accelerates large language model inference by allowing a lightweight draft model to propose outputs that a stronger target model verifies. However, its token-centric nature allows erroneous steps to propagate. Prior approaches mitigate this using external reward models, but incur additional latency, computational overhead, and limit generalizability. We propose SpecGuard, a verification-aware speculative decoding framework that performs step-level verification using only model-internal signals. At each step, SpecGuard samples multiple draft candidates and selects the most consistent step, which is then validated using an ensemble of two lightweight model-internal signals: (i) an attention-based grounding score that measures attribution to the input and previously accepted steps, and (ii) a log-probability-based score that captures token-level confidence. These signals jointly determine whether a step is accepted or recomputed using the target, allocating compute selectively. Experiments across a range of reasoning benchmarks show that SpecGuard improves accuracy by 3.6% while reducing latency by ~11%, outperforming both SD and reward-guided SD.

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 $v_{\text{bal}}$, 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.

Quantized Large Language Models (LLMs) are used more often in qualitative analysis because they run fast and need fewer computing resources. This study examines how different lower bits quantization levels (8-bit, 4-bit, 3-bit, and 2-bit) and quantization types affect the performance of LLaMA-3.1 (8B) on qualitative analysis. The study uses expert and non-expert responses from 82 interview transcripts. Low-bit models often produce higher levels of hallucinations and unstable results, especially when reading non-expert language with unclear terms. To improve performance, we propose a quantization-aware multi-pass prompt verification method. This method guides the model through controlled steps that reduce hallucinations. It removes unreliable content and passes the results to the next transcript after verification, improving accuracy. To validate performance, human coders analyzed transcripts using NVivo and BF16 LLaMA. BF16 LLaMA-3.1 produced high-precision output but had semantic drift and hallucination. These errors were corrected manually. The corrected BF16 output and NVivo human coding were combined to create a gold-standard ground truth (GSGT) for thematic extraction and frequency analysis. The results show that 8-bit models stay closest to the GSGT. The 4-bit models lose accuracy but become stable when the proposed method is applied. The 3-bit and 2-bit models drop in performance because of heavy compression, but they improve with the proposed prompt design and verification. The study also finds that models at the same bit level behave differently depending on quantization type. Overall, the method helps low-resource LLMs become more stable, accurate, and suitable for qualitative research at lower cost.

Multi-hop QA benchmarks often reward Large Language Models (LLMs) for spurious correctness, where models reach correct answers through invalid intermediate reasoning. We propose SAFE, an LLM-as-verifier framework for evidence-grounded multi-hop QA. Rather than judging only the final answer after generation, SAFE verifies reasoning during generation by checking intermediate steps against the provided passages and previous reasoning trajectory. To make this process checkable, SAFE decomposes reasoning into atomic, evidence-grounded units represented with Knowledge Graph (KG) triples. At train-time, SAFE verifies benchmark supervision under KG-grounded constraints and constructs reliable verifier training data. At inference-time, an external verifier checks each generated step, identifies invalid reasoning, and provides correction feedback before errors propagate. Across three multi-hop QA benchmarks, SAFE improves accuracy by 8.8 pp on average. These results show that evidence-grounded multi-hop QA benefits from shifting LLM-based evaluation from post-hoc answer judgment to stepwise reasoning verification.

We present ThinknCheck, a 1B-parameter verifier for grounded claim verification that first produces a short, structured rationale and then a binary verdict. We construct LLMAggreFact-Think, a 24.1k reasoning-augmented training set derived from LLMAggreFact, and fine-tune a 4-bit Gemma3 model to follow this format. On LLMAggreFact, ThinknCheck attains 78.1 balanced accuracy (BAcc), surpassing MiniCheck-7B (77.4) with 7x fewer parameters; removing the reasoning step reduces BAcc to 57.5. On SciFact, ThinknCheck reaches 64.7 BAcc, a +14.7 absolute gain over MiniCheck-7B. By contrast, zero-shot chain-of-thought on the base Gemma3-1B harms accuracy relative to direct answers, and preference optimization with a simple format+accuracy reward underperforms supervised reasoning. To probe the latter, we introduce GSMClaims and a domain-specialized variant, ThinknCheck-Science, which improves across benchmarks, including 61.0% accuracy on GSMClaims. Overall, explicit, supervised reasoning enables compact verifiers that are competitive while remaining resource-efficient and interpretable.

When an LLM judge only has to assign a three-way support label to a candidate answer given a reference, does asking it to decompose the answer into atomic claims help, and at what cost? We compare four single-call designs that share the judge model, the inputs, and the level of instruction detail: candidate-side atomic decomposition, a matched holistic rubric, reference-side decomposition that checks whether each reference claim is covered, and a bidirectional combination. Support labels are constructed from TruthfulQA, ASQA, and QAMPARI references (200 questions and 400 rows per dataset). All four designs are run with Opus-4.6, GPT-4.1, and Gemini Flash Lite; Sonnet-4.6 is added for the candidate-side and holistic designs. Candidate-side decomposition is weak where the label depends on completeness: the holistic rubric is more accurate on ASQA and QAMPARI for every judge while using fewer tokens. Reference-side decomposition is 12.5-21.3 points more accurate than holistic on ASQA for 28-29% more tokens, and stays near the holistic ceiling on QAMPARI at 63-69% more. On TruthfulQA misconceptions, candidate-side decomposition is competitive and significantly better for two judges. On a 60-row single-author subset whose labels are looser than strict reference completeness, candidate-side decomposition edges ahead of holistic for every judge, reversing the construction-label order. What a judge decomposes should follow what the label measures.

We address LLM serving workloads where repeated requests share a common solution structure but differ in localized constraints, such as output schema, variable names, or numeric constants. Prior caching approaches typically reuse either full responses (semantic caching) or model-internal KV/prefix states, which are respectively brittle under partial changes or tightly coupled to specific backends. We present StepCache, a backend-agnostic step-level reuse layer that segments outputs into ordered steps, retrieves the best-matching cached request, verifies steps using lightweight task-aware checks, and regenerates only failing regions via selective patching. StepCache additionally supports strict structured-output enforcement for JSON, including single-step extraction, required-key constraints, and one-shot repair, as well as conservative skip-reuse fallbacks for semantic changes. For linear equations, StepCache promotes verification into correction via a bounded repair loop with a deterministic fallback that guarantees correctness when the backend model fails. In a CPU-only perturbation-heavy micro-benchmark on math and JSON variants, averaged over three seeds, StepCache reduces mean latency from 2.13 s to 0.67 s, median latency from 2.42 s to 0.01 s, and p95 latency from 3.38 s to 3.30 s. It also reduces total token usage from 36.1k to 27.3k and improves end-to-end correctness from 72.5% to 100% under task-specific checks and a stitched-output integrity check. Across requests, 79.7% take the reuse-only fast path, 5.4% require patching, and 14.9% trigger skip-reuse.

Retrieval-augmented language models can retrieve relevant evidence yet still commit to answers before explicitly checking whether the retrieved context supports the conclusion. We present PAVE (Premise-Grounded Answer Validation and Editing), an inference-time validation layer for evidence-grounded question answering. PAVE decomposes retrieved context into question-conditioned atomic facts, drafts an answer, scores how well that draft is supported by the extracted premises, and revises low-support outputs before finalization. The resulting trace makes answer commitment auditable at the level of explicit premises, support scores, and revision decisions. In controlled ablations with a fixed retriever and backbone, PAVE outperforms simpler post-retrieval baselines in two evidence-grounded QA settings, with the largest gain reaching 32.7 accuracy points on a span-grounded benchmark. We view these findings as proof-of-concept evidence that explicit premise extraction plus support-gated revision can strengthen evidence-grounded consistency in retrieval-augmented LLM systems.

Large Language Models (LLMs) have achieved unprecedented fluency but remain susceptible to "hallucinations" - the generation of factually incorrect or ungrounded content. This limitation is particularly critical in high-stakes domains where reliability is paramount. We propose a domain-grounded tiered retrieval and verification architecture designed to systematically intercept factual inaccuracies by shifting LLMs from stochastic pattern-matchers to verified truth-seekers. The proposed framework utilizes a four-phase, self-regulating pipeline implemented via LangGraph: (I) Intrinsic Verification with Early-Exit logic to optimize compute, (II) Adaptive Search Routing utilizing a Domain Detector to target subject-specific archives, (III) Refined Context Filtering (RCF) to eliminate non-essential or distracting information, and (IV) Extrinsic Regeneration followed by atomic claim-level verification. The system was evaluated across 650 queries from five diverse benchmarks: TimeQA v2, FreshQA v2, HaluEval General, MMLU Global Facts, and TruthfulQA. Empirical results demonstrate that the pipeline consistently outperforms zero-shot baselines across all environments. Win rates peaked at 83.7% in TimeQA v2 and 78.0% in MMLU Global Facts, confirming high efficacy in domains requiring granular temporal and numerical precision. Groundedness scores remained robustly stable between 78.8% and 86.4% across factual-answer rows. While the architecture provides a robust fail-safe for misinformation, a persistent failure mode of "False-Premise Overclaiming" was identified. These findings provide a detailed empirical characterization of multi-stage RAG behavior and suggest that future work should prioritize pre-retrieval "answerability" nodes to further bridge the reliability gap in conversational AI.

In factual question answering, many errors are not failures of access but failures of commitment: the system retrieves relevant evidence, yet still settles on the wrong answer. We present CounterRefine, a lightweight repair layer for short-form RAG that treats the first answer as a hypothesis to test. Given a draft, CounterRefine issues answer-conditioned expansion queries to retrieve candidate-specific evidence, then applies a constrained KEEP or REVISE refinement step whose proposed revisions are accepted only after deterministic validation. The design is intentionally narrow: it adds one evidence-gathering pass and one guarded refinement call rather than replacing the retriever or building a broad agentic system. On the full SimpleQA benchmark, CounterRefine improves a matched one-pass RAG baseline by up to 5.8 correct-rate points; in the full Claude trace, it changes only 5.6% of outputs, with 180 beneficial outcome changes and 8 harmful ones. These findings suggest a simple but important direction for knowledgeable foundation models: beyond accessing evidence, they should also be able to use that evidence to reconsider and, when necessary, repair their own answers.

Medication errors pose a significant threat to patient safety, making pharmacist verification (PV) a critical, yet heavily burdened, final safeguard. The direct application of Large Language Models (LLMs) to this zero-tolerance domain is untenable due to their inherent factual unreliability, lack of traceability, and weakness in complex reasoning. To address these challenges, we introduce PharmGraph-Auditor, a novel system designed for safe and evidence-grounded prescription auditing. The core of our system is a trustworthy Hybrid Pharmaceutical Knowledge Base (HPKB), implemented under the Virtual Knowledge Graph (VKG) paradigm. This architecture strategically unifies a relational component for set constraint satisfaction and a graph component for topological reasoning via a rigorous mapping layer. To construct this HPKB, we propose the Iterative Schema Refinement (ISR) algorithm, a framework that enables the co-evolution of both graph and relational schemas from medical texts. For auditing, we introduce the KB-grounded Chain of Verification (CoV), a new reasoning paradigm that transforms the LLM from an unreliable generator into a transparent reasoning engine. CoV decomposes the audit task into a sequence of verifiable queries against the HPKB, generating hybrid query plans to retrieve evidence from the most appropriate data store. Experimental results demonstrate robust knowledge extraction capabilities and show promises of using PharmGraph-Auditor to enable pharmacists to achieve safer and faster prescription verification.

Reinforcement Learning with Verifiable Rewards (RLVR) has proven effective in enhancing the reasoning capabilities of large language models, particularly in domains such as mathematics where reliable rule-based verifiers can be constructed. However, the reliance on handcrafted, domain-specific verification rules substantially limits the applicability of RLVR to general reasoning domains with free-form answers, where valid answers often exhibit significant variability, making it difficult to establish complete and accurate rules. To address this limitation, we propose Conditional Expectation Reward (CER), which leverages the large language model itself as an implicit verifier, and is therefore applicable to general domains and eliminates the need for external verifiers or auxiliary models. CER is defined as the expected likelihood of generating the reference answer conditioned on the generated answer. In contrast to rule-based verifiers that yield binary feedback, CER provides a soft, graded reward signal that reflects varying degrees of correctness, making it better suited to tasks where answers vary in correctness. Experimental results demonstrate that CER is effective across a wide range of reasoning tasks, spanning both mathematical and general domains, indicating that CER serves as a flexible and general verification mechanism. The code is available at https://github.com/changyi7231/CER.

Assessing whether an article supports an assertion is essential for hallucination detection and claim verification. While large language models (LLMs) have the potential to automate this task, achieving strong performance requires frontier models such as GPT-5 that are prohibitively expensive to deploy at scale. To efficiently perform biomedical evidence attribution, we present Med-V1, a family of small language models with only three billion parameters. Trained on high-quality synthetic data newly developed in this study, Med-V1 substantially outperforms (+27.0% to +71.3%) its base models on five biomedical benchmarks unified into a verification format. Despite its smaller size, Med-V1 performs comparably to frontier LLMs such as GPT-5, along with high-quality explanations for its predictions. We use Med-V1 to conduct a first-of-its-kind use case study that quantifies hallucinations in LLM-generated answers under different citation instructions. Results show that the format instruction strongly affects citation validity and hallucination, with GPT-5 generating more claims but exhibiting hallucination rates similar to GPT-4o. Additionally, we present a second use case showing that Med-V1 can automatically identify high-stakes evidence misattributions in clinical practice guidelines, revealing potentially negative public health impacts that are otherwise challenging to identify at scale. Overall, Med-V1 provides an efficient and accurate lightweight alternative to frontier LLMs for practical, real-world biomedical evidence attribution. Med-V1 is available at https://github.com/NLM-DIR/Med-V1.

Retrieval-augmented generation (RAG) is increasingly deployed in enterprise search and document-centric assistants, where responses must be grounded in long and complex source materials. In practice, verifying that generated answers faithfully reflect retrieved documents is difficult: large language models can check long contexts but are too slow and costly for interactive services, while lightweight classifiers operate within strict context limits and frequently miss evidence outside truncated passages. We present the design of a real-time verification component integrated into a production RAG pipeline that enables full-document grounding under latency constraints. The system processes documents up to 32K tokens and employs adaptive inference strategies to balance response time and verification coverage across workloads. We describe the architectural decisions, operational trade-offs, and evaluation methodology used to deploy the verifier, and show that full-context verification substantially improves detection of unsupported responses compared with truncated validation. Our experience highlights when long-context verification is necessary, why chunk-based checking often fails in real documents, and how latency budgets shape model design. These findings provide practical guidance for practitioners building reliable large-scale retrieval-augmented applications. (Model, benchmark, and code: https://huggingface.co/llm-semantic-router)

Large Language Models (LLMs) using chain-of-thought have demonstrated great potential for solving complex reasoning and planning tasks. Despite these advances, LLM-generated outputs remain susceptible to errors, making verification important for reliable reasoning systems. Learned verifiers can increase trust, enforce safety constraints, and ensure alignment with personal preferences, while also providing feedback to improve generation. This raises a central challenge: when learned verifiers are used to guide generation, the feedback loop between generator and verifier may induce a distribution shift. This is particularly salient for process reward models, a prominent class of learned verifiers that score or classify individual steps in a chain-of-thought reasoning trace. Motivated by this challenge, we propose a new online learning framework for chain-of-thought verifiers that, given a problem statement and a reasoning trace, check the correctness of each reasoning step given the preceding steps. Highlighting the asymmetric role of soundness errors (accepting an incorrect reasoning step) and completeness errors (flagging a correct step as wrong), we introduce novel notions of dimension that characterize their optimal tradeoff. We then show how our learned verifiers can boost the accuracy of a weak generator. Assuming that the generator can produce a correct next step with a small success probability, we show how to learn a strong generator with small error and abstention rates. Our results also allow learning from offline data when queries to an expert verifier can be simulated from a small set of correct reasoning traces. However, we establish a separation between our approach and learning from offline expert demonstrations: we show that learning from offline demonstrations cannot in general achieve the soundness-completeness guarantees produced by our interactive learning approach.

Parallel test-time scaling, which generates multiple candidate solutions for a single problem, is a powerful technique for improving large language model performance. However, it is hindered by two key bottlenecks: accurately selecting the correct solution from the candidate pool, and the high inference latency from generating many full solutions. We argue that both challenges are fundamentally linked to verifier calibration, as a well-calibrated verifier improves answer selection and enables early-stopping strategies to reduce latency. However, existing non-generative verifiers are limited as they score each candidate in isolation, overlooking rich contextual information across the set of candidates. To address this, we introduce the Multi-Sequence Verifier (MSV), a lightweight verifier that predicts each candidate's correctness conditioned on the full sampled set. MSV achieves improved calibration, which directly enhances best-of-N selection performance and empowers a novel early-stopping framework. Across challenging mathematical reasoning benchmarks, MSV improves best-of-64 accuracy by up to 6% relative to strong baselines, and in the early-stopping setting reaches the same accuracy as baselines with less than half the latency.

Despite the success of Retrieval-Augmented Generation (RAG) in grounding LLMs with external knowledge, its application over heterogeneous sources (e.g., private databases, global corpora, and APIs) remains a significant challenge. Existing approaches typically employ an LLM-as-a-Router to dispatch decomposed sub-queries to specific sources in a predictive manner. However, this "LLM-as-a-Router" strategy relies heavily on the semantic meaning of different data sources, often leading to routing errors when source boundaries are ambiguous. In this work, we introduce RealRoute System, a framework that shifts the paradigm from predictive routing to a robust Retrieve-then-Verify mechanism. RealRoute ensures evidence completeness through parallel, source-agnostic retrieval, followed by a dynamic verifier that cross-checks the results and synthesizes a factually grounded answer. Our demonstration allows users to visualize the real-time "re-routing" process and inspect the verification chain across multiple knowledge silos. Experiments show that RealRoute significantly outperforms predictive baselines in the multi-hop Rag reasoning task. The RealRoute system is released as an open-source toolkit with a user-friendly web interface. The code is available at the URL: https://github.com/Joseph1951210/RealRoute.

Large language models (LLMs) are increasingly used in scholarly question-answering (QA) systems to help researchers synthesize vast amounts of literature. However, these systems often produce subtle errors (e.g., unsupported claims, errors of omission), and current provenance mechanisms like source citations are not granular enough for the rigorous verification that scholarly domain requires. To address this, we introduce PaperTrail, a novel interface that decomposes both LLM answers and source documents into discrete claims and evidence, mapping them to reveal supported assertions, unsupported claims, and information omitted from the source texts. We evaluated PaperTrail in a within-subjects study with 26 researchers who performed two scholarly editing tasks using PaperTrail and a baseline interface. Our results show that PaperTrail significantly lowered participants' trust compared to the baseline. However, this increased caution did not translate to behavioral changes, as people continued to rely on LLM-generated scholarly edits to avoid a cognitively burdensome task. We discuss the value of claim-evidence matching for understanding LLM trustworthiness in scholarly settings, and present design implications for cognition-friendly communication of provenance information.

With the growing popularity of Large Reasoning Models and their results in solving mathematical problems, it becomes crucial to measure their capabilities. We introduce a pipeline for both automatic and interactive verification as a more accurate alternative to only checking the answer which is currently the most popular approach for benchmarks. The pipeline can also be used as a generator of correct solutions both in formal and informal languages. 3 AI agents, which can be chosen for the benchmark accordingly, are included in the structure. The key idea is the use of prompts to obtain the solution in the specific form which allows for easier verification using proof assistants and possible use of small models ($\le 8B$). Experiments on several datasets suggest low probability of False Positives. The open-source implementation with instructions on setting up a server is available at https://github.com/LogicEnj/lean4_verification_pipeline.

Structured Outputs from current LLMs exhibit sporadic errors, hindering enterprise AI deployment. We present CONSTRUCT, a real-time uncertainty estimator that scores the trustworthiness of LLM Structured Outputs. Lower-scoring outputs are more likely to contain errors, enabling automatic prioritization of limited human review bandwidth. CONSTRUCT additionally scores the trustworthiness of each field within a Structured Output, helping reviewers quickly identify which parts of the output are incorrect. Our method is suitable for any LLM (including black-box LLM APIs without logprobs), does not require labeled training data or custom model deployment, and supports complex Structured Outputs with heterogeneous fields and nested JSON schemas. We also introduce one of the first public LLM Structured Output benchmarks with reliable ground-truth values. Over this four-dataset benchmark, CONSTRUCT detects errors in outputs from various LLMs (including Gemini 3 and GPT-5) with significantly higher precision/recall than existing techniques.

Reasoning with LLMs increasingly unfolds inside a broader verification loop. Internally, systems use cheap checks, such as self-consistency or proxy rewards, which we call weak verification. Externally, users inspect outputs and steer the model through feedback until results are trustworthy, which we call strong verification. These signals differ sharply in cost and reliability: strong verification can establish trust but is resource-intensive, while weak verification is fast and scalable but noisy and imperfect. We formalize this tension through weak--strong verification policies, which decide when to accept or reject based on weak verification and when to defer to strong verification. We introduce metrics capturing incorrect acceptance, incorrect rejection, and strong-verification frequency. Over population, we show that optimal policies admit a two-threshold structure and that calibration and sharpness govern the value of weak verifiers. Building on this, we develop an online algorithm that provably controls acceptance and rejection errors without assumptions on the query stream, the language model, or the weak verifier.

In the past year, custom and unreleased math reasoning models reached gold medal performance on the International Mathematical Olympiad (IMO). Similar performance was then reported using large-scale inference on publicly available models but at prohibitive costs (e.g., 3000 USD per problem). In this work, we present an inference pipeline that attains best-in-class performance on IMO-style math problems at an average inference cost orders of magnitude below competing methods while using only general-purpose off-the-shelf models. Our method relies on insights about grader failure in solver-grader pipelines, which we call the Cognitive Well (iterative refinement converging to a wrong solution that the solver as well as the pipeline's internal grader consider to be basically correct). Our pipeline addresses these failure modes through conjecture extraction, wherein candidate lemmas are isolated from generated solutions and independently verified alongside their negations in a fresh environment (context detachment). On IMO-ProofBench Advanced (PB-Adv), our pipeline achieves 67.1 percent performance using Gemini 3.0 Pro with an average cost per question of approximately 31 USD. At the time of evaluation, this represented the state-of-the-art on PB-Adv among both public and unreleased models, and more than doubles the success rate of the next best publicly accessible pipeline, all at a fraction of the cost.

In agentic search, an LLM agent searches the web, reads the pages it finds, and decides what to look for next before returning an answer. But can we trust an answer simply because the agent returns it? Often not, and even a correct answer can be a lucky guess: on questions with several constraints, we find that agents conclude the task is complete while a constraint remains unverified in up to 48% of their correct answers. We call this illusory completion. To see how it arises, we introduce the Epistemic Ledger, which tracks at every turn what the retrieved pages establish about each constraint and what the agent claims. Across 13 agents, from 7B RL-trained models to frontier LLMs, training and scale raise accuracy but change the pattern of verification failures rather than eliminating them: constraints may be left unchecked, assumed without support, or retained despite refuting evidence. To measure what agents lose without tracking their constraints, we show them each constraint's state, approximated by LiveLedger, a lightweight 4B tracker. Agents then answer 4.4-16.1 points more questions correctly, suggesting that on their own, they may not track what they have verified and what remains.

Despite the syntactic fluency of Large Language Models (LLMs), ensuring their logical correctness in high-stakes domains remains a fundamental challenge. We present a neurosymbolic framework that combines LLMs with SMT solvers to produce verification-guided answers through iterative refinement. Our approach decomposes LLM outputs into atomic claims, autoformalizes them into first-order logic, and verifies their logical consistency using automated theorem proving. We introduce three key innovations: (1) multi-model consensus via formal semantic equivalence checking to ensure logic-level alignment between candidates, eliminating the syntactic bias of surface-form metrics, (2) semantic routing that directs different claim types to appropriate verification strategies: symbolic solvers for logical claims and LLM ensembles for commonsense reasoning, and (3) precise logical error localization via Minimal Correction Subsets (MCS), which pinpoint the exact subset of claims to revise, transforming binary failure signals into actionable feedback. Our framework classifies claims by their logical status and aggregates multiple verification signals into a unified score with variance-based penalty. The system iteratively refines answers using structured feedback until acceptance criteria are met or convergence is achieved. This hybrid approach delivers formal guarantees where possible and consensus verification elsewhere, advancing trustworthy AI. With the GPT-OSS-120B model, VERGE demonstrates an average performance uplift of 18.7% at convergence across a set of reasoning benchmarks compared to single-pass approaches.

LLMs have shown strong performance in data-rich domains such as programming, but their reliability in engineering tasks remains limited. Circuit analysis is particularly challenging because it requires both multimodal understanding and precise mathematical reasoning. This paper presents an enhanced end-to-end circuit problem-solving framework using Gemini 2.5 Pro as the backbone model for scalable engineering-education applications. We systematically evaluate Gemini 2.5 Pro on undergraduate circuit-analysis problems and identify two major failure modes: circuit-recognition hallucinations, especially source-polarity errors, and reasoning-process hallucinations, such as incorrect current-direction assumptions. To reduce recognition errors, we integrate a fine-tuned YOLO detector with OpenCV-based processing to isolate voltage and current sources for polarity re-identification. To mitigate reasoning errors, we introduce an ngspice-driven verification loop that supports iterative refinement with optional human feedback. On 83 problems, the proposed pipeline achieves 97.59% accuracy, compared with 79.52% for baseline Gemini. Across four hand-drawn diagram variations, accuracy improves from 60.61%--71.21% to 89.39%--92.42%, with statistically significant gains (p<0.005). On 43 problems from a different textbook, accuracy increases from 58.14% to 83.72%, further supporting cross-textbook generalizability. Error analysis shows that circuit recognition remains the dominant source of residual failures, particularly under varying diagram representations. Overall, the framework substantially improves the robustness, scalability, and generalizability of LLM-based circuit problem solving for engineering education and practical circuit analysis.

The integration of Formal Verification tools with Large Language Models (LLMs) offers a path to scale software verification beyond manual workflows. However, current methods remain unreliable: without a solid theoretical footing, the refinement process acts as a black box that may oscillate, loop, or diverge. This work bridges this critical gap by developing an LLM-Verifier Convergence Theorem, providing the first formal framework with provable guarantees for termination in multi-stage verification pipelines. We model the interaction not as a generic loop, but as a sequential absorbing Markov Chain comprising four essential engineering stages: CodeGen, Compilation, InvariantSynth, and SMTSolving. We prove that for any non-zero stage success probability ($δ> 0$), the system reaches the Verified state almost surely. Furthermore, because of the sequential nature of the pipeline, we derive a precise latency bound of $\mathbb{E}[n] \leq 4/δ$. We stress-tested this prediction in an extensive empirical campaign comprising over 90,000 trials. The results match the theory with striking consistency: every run reached verification, and the empirical convergence factor clustered tightly around $C_f\approx 1.0$, confirming that the $4/δ$ bound accurately mirrors system behavior rather than serving as a loose buffer. Based on this data, we identify three distinct operating zones -- marginal, practical, and high-performance -- and propose a dynamic calibration strategy to handle parameter drift in real-world environments. Together, these contributions replace heuristic guesswork with a rigorous architectural foundation, enabling predictable resource planning and performance budgeting for safety-critical software.

Automatic verification is a critical component in building math-solving agents and reinforcement learning, yet it often falls short in generalizability, performance, and cost-efficiency. Identifying that the primary bottleneck of verification lies in error detection capability, we propose pessimistic verification, a paradigm of agentic workflows that rejects a solution if any of multiple parallel verifiers identifies a flaw. We further introduce progressive pessimistic verification, which employs fine-grained proof decomposition to significantly enhance verification accuracy and efficiency. Our approach surpasses the performance and token efficiency of extended long chain-of-thought (long CoT) and mainstream verification workflows, crucially, our analysis reveals that existing benchmarks underestimate its effectiveness on stronger models due to inherent annotation errors. To further validate the effectiveness of our method, we applied a verification-based solving workflow on the IMO 2025 and MathArena Apex 2025 datasets, where the workflow with progressive pessimistic verification exhibits remarkable improvements in both efficiency and accuracy on highly challenging contest-level math problems with state-of-the-art models. Code is available at https://github.com/THUNLP-MT/pverify.

Although large language models (LLMs) have achieved remarkable performance, the inherent stochasticity of their reasoning processes and varying conclusions present significant challenges. Majority voting or Best-of-N with external verifiers has been explored to mitigate this, but these approaches are limited in applicability or require additional training. To address this problem, we propose a novel framework that Recycles Few-shot examples to verify LLM outputs (ReFeri). Our key idea is to utilize the given few-shot examples not only to generate outputs, but also to evaluate the candidate outputs. Specifically, ReFeri combines a forward confidence score with a backward reconstruction penalty to select candidates that follow few-shot guidance while avoiding demonstration-specific overfitting. Experiments with three different LLMs across seven diverse tasks demonstrate that our framework significantly improves the accuracy of LLMs---achieving an average relative gain of 8.2%---through effective response selection.