FaithformBench: Benchmarking Faithfulness of Mathematical Chain-of-Thought Autoformalisation

Autoformalisation (AF) systems map natural language reasoning steps into formal statements in a proof assistant such as Lean. We consider how to assess the faithfulness of these systems. Existing approaches require expensive human-annotated ground truth, or rely on LLM judges or embedding models, which come with limited guarantees of accuracy. In addition, these methods typically only consider inputs that are known to be correct, and therefore do not assess whether the AF translates incorrect inputs faithfully. To address these limitations, we propose a new benchmark for AF faithfulness that is cheap to apply, sound under weak assumptions, and assesses both positive and negative examples. Our method is based on automatically generating perturbed reasoning steps that are designed to be invalid, and then measuring validity preservation on unperturbed steps and invalidity preservation on perturbed steps. We apply our method to eight AF systems across four mathematical datasets, and observe pervasive sycophancy: many AFs "silently correct" invalid inputs into provable statements. The most validity-preserving fine-tuned AFs are also the most sycophantic, suggesting a tension between validity and invalidity preservation in current AF systems.

A Reference-Free Score for Detecting Silent Reasoning Failures in Large Language Models

Mathematical chain of thought (CoT) evaluation is commonly reduced to whether the final answer matches a reference. This conflates producing a correct conclusion with producing a valid derivation an invalid chain can accidentally reach the right answer, while a valid calculation can be followed by a transcription error. We call this mismatch the reasoning answer consistency gap. This framework paper introduces the Reasoning Answer Faithfulness Score (RAFS), a reference free, instance level diagnostic of whether an emitted mathematical trace is locally credible, supports its answer, and is stable under resampling and targeted counterfactual interventions. RAFS combines step validity, reasoning to answer entailment and counterfactual sensitivity, answer consensus, and conditional reasoning stability. It evaluates transcript level agreement, not a models private computation and not factual correctness outside the tested mathematical setting. We retain a preregistered, results blind confirmatory study on GSM8K and MATH, with hypotheses, admissibility rules, calibration, and tests fixed before confirmatory outcomes are inspected. A separate feasibility pilot is specified to verify end to end execution and estimate interven tion coverage before that freeze numerical pilot claims are re ported only when trace level artifacts are available. We formalize four reasoning answer outcomes, justify the non compensatory aggregator, instantiate semantic trace distance, quantify compute and abstention tradeoffs, and define verifier independence and power analyses. RAFS is intended to complement mathematical answer accuracy with an auditable warning signal for silent reasoning failures and answer extraction errors

First-Order Modal Logic in HOL: Deep and Shallow Embeddings with Automated Faithfulness (Extended Preprint)

We extend, in Isabelle/HOL, the deep-and-shallow embedding methodology of our prior work from propositional to first-order modal logic (FML) with constant-domain Kripke semantics. Three embeddings of FML into classical higher-order logic (HOL) are provided side by side: a deep embedding, a heavyweight maximal-shallow embedding, and a lightweight minimal-shallow embedding. The minimal-shallow embedding is presented as an Isabelle/HOL locale, parametrised by an accessibility relation, a world-indexed interpretation, a universe of worlds, and a variable assignment; the locale form admits a global faithfulness theorem, stating that quantifying over all minimal-shallow interpretations recovers exactly deep validity. A central technical contribution is a mechanisation, for FML under constant-domain Kripke semantics, of the (countable) downward Löwenheim-Skolem theorem, which underpins the automation of our faithfulness proof between the deep and minimal-shallow embeddings. Deploying it inside an extension of the minimal-shallow locale resolves the surjectivity problem that arises against an uncountable domain of individuals -- where the locale's variable assignment, having countable domain V = nat, cannot be surjective onto the domain -- and thereby yields faithfulness over the full domain. Since prior work treats only the propositional fragment, we develop here the substitution machinery (free/bound-variable predicates, the fresh-variable function, capture-avoiding substitution, alphabetic renaming, the substitutability predicate, the substitution lemma, and size-based induction principles) needed for the first-order quantifiers.

Validating Causal Abstraction Metrics on Simulated Complex Systems

A central goal of science is to produce valid explanations of complex systems: high-level causal accounts that faithfully reflect the behavior of lower-level mechanisms. Yet no consensus exists on how to measure whether a proposed high-level explanation is actually valid. We introduce a benchmark of ten complex systems spanning both discrete and continuous state spaces, as well as static and dynamical regimes, each equipped with consensual ground-truth causal explanations and invalid contrastive conditions. Within a unified causal abstraction framework, we systematically evaluate over thirty candidate metrics drawn from observational, functional, information-theoretic, and causal families. Our results show that only the latter reliably discriminates valid from invalid abstractions, and only when incorporating faithfulness testing over unmapped variables. Building on these findings, we introduce the Causal Abstraction Error (CAE), a continuous validity metric with an explicit faithfulness test, which passes all discrimination tests across every system and can converge with as few as 30 sampled interventions. We offer it as a general-purpose metric for the discovery and validation of high-level explanations.

Decodable but Not Faithful: Coupling Natural-Language Rationales to Programmatic Verifiers

Language models can generate plausible rationales for their predictions, but these explanations may not faithfully represent the model's internal reasoning. We propose verifier-coupled reasoning, a framework that inserts inline claims into reasoning traces and trains an auxiliary consistency head to predict programmatic verifier outputs from rationale-span hidden states. The central finding is a gap between decodability and faithfulness: consistency training reliably makes verifier information decodable from rationale representations, but decodability does not guarantee faithful generation. In LeanCheck (formal theorem proving), rationale-only and proof-only pooling achieve perfect directional separation under counterfactual conflict. In KataGo (Go engine), commentary spans encode 10-way win-rate buckets at 81% accuracy. Yet in a code setting, the model achieves 98.6% coupling while its generated explanations remain unfaithful: fluent prose with correct structured claims, but describing unrelated algorithms; a controlled pretrained-vs-from-scratch comparison shows the gap is not capacity-driven. Synthetic activation patching confirms causal influence (73-89% vs. 31% baseline), FEVER reveals that evidence-only pooling isolates genuine evidence sensitivity at the cost of raw accuracy, and per-claim analysis shows that consistency loss disproportionately benefits fine-grained claims over binary ones. These results establish that consistency losses are effective diagnostics and representation-shaping tools, but not sufficient conditions for faithful reasoning.

Know Your Limits : On the Faithfulness of LLMs as Solvers and Autoformalizers in Legal Reasoning

Large Language Models (LLMs) achieve strong performance on reasoning tasks, but whether this reflects faithful logical inference or heuristic approximation remains unclear. We study this question in legal entailment by comparing three paradigms, including pure LLM classification, LLM-based Formal Reasoning, and solver-based Formal Reasoning using the Z3 SMT solver, on a re-annotated subset of ContractNLI across five LLMs. Our re-annotation reveals a systematic and measurable gap between pragmatic legal interpretation and strict formal entailment, where a substantial proportion of legally sound inferences are not formally grounded without additional unstated assumptions. While introducing formal structure improves accuracy, with LLM-based Formal Reasoning achieving the highest benchmark performance, we show that this gain does not imply faithful reasoning. We identify three recurring failure modes: scope laundering, where LLMs report solver-inconsistent classifications without executing the underlying formal reasoning, producing conclusions that appear logically grounded but are not; implicit constraint blindness, where LLMs overlook logical constraints present in formal representations; and program synthesis failures, where LLMs generate incorrect Z3 code despite structured prompting. Critically, scope laundering persists across all models, raising serious concerns about the faithfulness of LLM-based formal reasoning as a proxy for symbolic execution. These results reveal a fundamental gap between benchmark accuracy and logical faithfulness.