Operads proposed as mathematical foundation for LLM question decomposition and consistency
Researchers propose operads — algebraic structures modeling many-in, one-out compositions — as a rigorous mathematical framework for question decomposition in LLMs. They define a 'questions operad' where QA models are interpreted as algebras, and introduce 'operadic consistency' as a measure of whether a model's answers agree across partial collapses of a decomposition tree. A companion empirical paper reports operadic consistency is strongly correlated with accuracy across twelve LLMs and four multi-hop QA datasets, outperforming temperature-based self-consistency baselines. The work attempts to give formal grounding to a widely-used but theoretically underspecified reasoning strategy.
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Operadic consistency: a label-free signal for detecting compositional reasoning failures in LLMs
Researchers introduce operadic consistency (OC), a label-free inference-time signal that checks whether an LLM's direct answer to a compositional query agrees with the answer produced by composing its own stated decomposition of that query. Evaluated across 12 instruction-tuned LLMs (4B–671B parameters) on four multi-hop QA datasets, OC achieves Pearson r ∈ [0.86, 0.94] with accuracy uniformly across all datasets, outperforming self-consistency, semantic entropy, and P(True) in cross-dataset robustness. At the per-question level, OC provides information beyond existing baselines and yields selective-prediction improvements (AUARC lifts +0.086–0.096, AUROC lifts +0.092–0.164) at equal sampling cost, with results extending to frontier thinking models using chain-of-thought decompositions.
Bounding Compositional Incoherence in Multi-Component LLM Agents
This paper formalizes a failure mode in multi-component LLM agent systems where individual components are locally probabilistically coherent but their composition violates basic probability axioms. The authors introduce the 'compositional residual' (eps*) as a runtime-computable measure of this incoherence, finding it positive in 33–94% of ensemble cliques across 1,876 tested configurations on a four-LLM panel. A hierarchical Boyle-Dykstra projection is proposed as a deterministic repair, and an anytime-valid e-process enables sequential monitoring. Notably, three intuitive LLM-side mitigations—retrieval, partition-aware prompting, and aggregator-LLM—each fail or regress.
Large-Scale Evaluation of LLM-Driven Formal Proof Search on Open Mathematical Problems
Researchers present the first large-scale evaluation of LLM-based formal proof search on genuinely open mathematical problems, using Lean as a verification backend. Their most capable agent autonomously resolved 9 of 353 open Erdős problems and proved 44 of 492 OEIS conjectures, at a cost of a few hundred dollars per problem. The system is already being deployed in active research across combinatorics, optimization, graph theory, algebraic geometry, and quantum optics. The study also compares agent architectures, finding that more sophisticated designs outperform simple generate-and-verify loops on the hardest problems.
Benchmarking study finds LLMs fail at counterintuitive probability problems despite strong standard performance
A new arXiv paper evaluates 8 state-of-the-art LLMs on discrete probability problems using two datasets: standard exercises (average accuracy 0.96) and counterintuitive exercises designed to trigger heuristic reasoning (average accuracy 0.59). The authors document token bias causing 20%+ performance drops when canonical problem formulations are disguised, and up to 34% degradation when misleading suggestions are embedded in prompts. The findings argue that current LLMs are not genuine probabilistic reasoners despite their success on advanced math benchmarks.
Canonical-Context On-Policy Distillation (CCOPD) for Multi-Turn LLM Consistency
This paper identifies 'self-anchored drift' as a key failure mode in multi-turn LLMs: when information is revealed incrementally across turns, models produce unsupported assumptions that distort final answers, even when the total evidence is identical to a single-prompt setting. The authors propose Canonical-Context On-Policy Distillation (CCOPD), which trains a student model on incremental multi-turn conversations to match the output distribution of a frozen teacher conditioned on the full clean prompt. Trained only on math conversations, CCOPD achieves a 32% average relative improvement on multi-turn (RAW-SHARDED) tasks and generalizes zero-shot to five out-of-domain task families while preserving single-prompt performance.
Tracing the Emergence of Human-Like Pragmatic Reasoning in LLMs Across Languages
Researchers conducted a population-matching experiment evaluating 25 LLMs on conditional inference tasks across four languages, comparing model behavior to matched human populations. The study finds that LLMs function as accurate semantic operators but systematically fail to capture pragmatic enrichments—context-sensitive inferences beyond literal logical meaning—that humans apply effortlessly. Model performance on pragmatic reasoning is not predicted by open vs. closed weights, training orientation, or architecture type, suggesting pragmatic reasoning remains an emergent and unreliable capability. The findings contribute to ongoing debates about whether LLMs reason like humans or merely approximate surface-level linguistic patterns.
Mechanistic analysis of how LLMs encode essay quality in internal representations
Researchers systematically probe the hidden representations of eight LLMs across three essay datasets (ASAP++, CSEE, ENEM) to understand how automated essay scoring (AES) works internally. Using linear probing, dimensionality reduction, and neuron-level analysis, they find essay quality is encoded in a linearly accessible form that emerges progressively across layers and partially transfers across prompts. Individual 'essay scoring neurons' are identified whose activations correlate with scores and respond to targeted interventions, with longer essays relying more on deeper layers. The work contributes to mechanistic interpretability of LLM-based scoring systems.
Post-hoc falsification operators for frozen small code models fail to beat Best-of-N in leakage-free evaluation
A measurement study evaluates 26 post-hoc operators (selection, verification, repair, elimination, portfolios) applied to frozen small code models (≤1.5B parameters) against a Best-of-N baseline under a strict leakage-free, matched-compute protocol. None of the semantic operators improves held-out accuracy over BoN, with the failure traced to three structural mechanisms: a coverage wall, a capability scissors, and a near-empty consensus trap. Two non-semantic operators do provide value: an expression-layer recovery method (M1) lifts DeepSeek-Coder-1.3B by +12 tasks on HumanEval+ (p=2.4e-4), and an adaptive consensus early-stop saves ~19% compute with no accuracy harm. The paper's core lesson is that harness quality and coverage measurement should precede investment in semantic post-hoc reasoning.