Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.03551

Objects Without Morphisms: What LLMs for Mathematics Do Not Represent

Yanli Wang·Suijin Wang·Xiaopeng Yuan·Haohan Wang
View original source ↗Read source document ↗Export metadata ↓

Abstract

Large language models (LLMs) have reached expert-level performance on competition mathematics largely through the volume of search placed around them: candidate solutions are sampled in quantity and retained only when an external criterion accepts them. Such a procedure improves the outcome that survives it while leaving untouched what the model represents. We examine that question where no external criterion exists: translating statements between the dialects of neighbouring subfields, where fidelity turns on the level of generality at which content is asserted. The source leaves that level implicit in its vocabulary, so a faithful translation must recover it from the relation between the theories. We introduce an instrument that codes truth, content and scope in separate blind queues, with a judge-free measure of whether a rewrite states the hypothesis implicit in its source, and establish its sensitivity with a planted-positive control. Across seven models from four families, translating towards the general framing widens the domain of quantification in 60.6% of rewrites and narrows it in none; translating towards the concrete framing narrows it in 28.3% and widens it in 0.3%. The hypothesis that would prevent it is stated in 21.6% of model rewrites and 4.2% of human statements. Capability does not govern the asymmetry: it appears in every model tested, and the most capable widens least. It replicates on the half of the benchmark held out by a pre-registered rule, and on statements written by mathematicians. Instructing a model to state every hypothesis it requires raises that rate but not its sensitivity to direction. We argue that these systems have acquired an object-level correspondence between subfield vocabularies without the constraint under which a translation between theories carries hypotheses to hypotheses.

Explore related subjects

cs.LG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Yanli Wang, Suijin Wang, Xiaopeng Yuan, Haohan Wang. 2026-10-02. Objects Without Morphisms: What LLMs for Mathematics Do Not Represent. https://arxiv.org/abs/2610.03551

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Robust Adversarial Quantification via Conflict-Aware Evidential Deep Learning

Reliability of deep learning models is critical for deployment in high-stakes applications, where out-of-distribution or adversarial inputs may lead to detrimental outcomes. Evidential Deep Learning, an efficient paradigm for uncertainty quantification, models predictions as Dirichlet distributions of a single forward pass. However, EDL is particularly vulnerable to adversarially perturbed inputs, making overconfident errors. Conflict-aware Evidential Deep Learning~\mbox{(C-EDL)} is a lightweight post-hoc uncertainty quantification approach that mitigates these issues, enhancing adversarial and OOD robustness without retraining. C-EDL generates diverse, task-preserving transformations per input and quantifies representational disagreement to calibrate uncertainty estimates when needed. C-EDL's conflict-aware prediction adjustment improves detection of OOD and adversarial inputs, maintaining high in-distribution accuracy and low computational overhead. Our experimental evaluation shows that C-EDL significantly outperforms state-of-the-art EDL variants and competitive baselines, achieving substantial reductions in coverage for OOD data (up to $\approx55\%$) and adversarial data (up to $\approx90\%$), across a range of datasets, attack types, and uncertainty metrics.

cs.LG↗
arXiv2026

Differential Privacy as a Perk: Federated Learning over Multiple-Access Fading Channels with a Multi-Antenna Base Station

Federated Learning (FL) is a distributed learning paradigm that preserves privacy by eliminating the need to exchange raw data during training. In its prototypical edge instantiation with underlying wireless transmissions enabled by analog over-the-air computing (AirComp), referred to as \emph{over-the-air FL (AirFL)}, the inherent channel noise plays a unique role of \emph{frenemy} in the sense that it degrades training due to noisy global aggregation while providing a natural source of randomness for privacy-preserving mechanisms, formally quantified by \emph{differential privacy (DP)}. It remains, nevertheless, challenging to effectively harness such channel impairments, as prior arts, under assumptions of either simple channel models or restricted types of loss functions, mostly considering (local) DP enhancement with a single-round or non-convergent bound on privacy loss. In this paper, we study AirFL over multiple-access fading channels with a multi-antenna base station (BS) subject to user-level DP requirements. Despite a recent study, which claimed in similar settings that artificial noise (AN) must be injected to ensure DP in general, we demonstrate, on the contrary, that DP can be gained as a \emph{perk} even \emph{without} employing any AN. Specifically, we derive a novel bound on DP that converges under general bounded-domain assumptions on model parameters, along with a convergence bound with general smooth and non-convex loss functions. Next, we optimize over receive beamforming and power allocations to characterize the optimal convergence-privacy trade-offs, which also reveal explicit conditions in which DP is achievable without compromising training. Finally, our theoretical findings are validated by extensive numerical results.

cs.LG↗
arXiv2026

Demystifying LLM-as-a-Judge: Analytically Tractable Model for Inference-Time Scaling

Recent developments in large language models have shown advantages in reallocating a notable share of computational resource from training time to inference time. However, the principles behind inference time scaling are not well understood. In this paper, we introduce an analytically tractable model of inference-time scaling: Bayesian linear regression with a reward-weighted sampler, where the reward is determined from a linear model, modeling LLM-as-a-judge scenario. We study this problem in the high-dimensional regime, where the deterministic equivalents dictate a closed-form expression for the posterior predictive mean and variance. We analyze the generalization error when training data are sampled from a teacher model. We draw $k$ inference-time samples and select via softmax at a temperature applied to a quadratic reward. When the reward is not too different from the teacher, the generalization error decreases monotonically with increasing inference time samples $k$. However, the specific reward that optimizes inference-time selection generally differs from the teacher. In contrast, substantial reward misspecification induces a finite optimal $k$ beyond which more sampling can increase the generalization error. For fixed $k$, there exists an optimal sampling temperature. We experimentally verify these facts in large language model inference with an additional large language model as a judge. In the "best-of-$k$" limit with the teacher as reward, we theoretically show that the generalization error decays as $Θ(1/k^2)$ and determine the leading coefficient via extreme value theory. These formulas delineate domains where scaling inference-time computation is provably preferable to collecting more data. Finally, we demonstrate that when task difficulty increases, the previously mentioned advantage of inference-time compute degrades.

cs.LG↗