# Objects Without Morphisms: What LLMs for Mathematics Do Not Represent

arXiv: https://arxiv.org/abs/2610.03551 · HTML: https://mlsanity.com/paper/2610.03551/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 18 of its 25 references that are in the map.
- First posted on arXiv 2 Oct 2026.

## Builds on (most cited first, 18 of 18)

- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [Self-Consistency Improves Chain of Thought Reasoning in Language Models](https://mlsanity.com/paper/2203.11171/) (2022)
- [Measuring Mathematical Problem Solving With the MATH Dataset](https://mlsanity.com/paper/2103.03874/) (2021)
- [Scaling LLM Test-Time Compute Optimally can be More Effective than Scaling Model Parameters](https://mlsanity.com/paper/2408.03314/) (2024)
- [Solving Quantitative Reasoning Problems with Language Models](https://mlsanity.com/paper/2206.14858/) (2022)
- [Large Language Monkeys: Scaling Inference Compute with Repeated Sampling](https://mlsanity.com/paper/2407.21787/) (2024)
- [LLM Evaluators Recognize and Favor Their Own Generations](https://mlsanity.com/paper/2404.13076/) (2024)
- [GSM-Symbolic: Understanding the Limitations of Mathematical Reasoning in Large Language Models](https://mlsanity.com/paper/2410.05229/) (2024)
- [Experts, Errors, and Context: A Large-Scale Study of Human Evaluation for Machine Translation](https://mlsanity.com/paper/2104.14478/) (2021)
- [Draft, Sketch, and Prove: Guiding Formal Theorem Provers with Informal Proofs](https://mlsanity.com/paper/2210.12283/) (2022)
- [CheckEval: A reliable LLM-as-a-Judge framework for evaluating text generation using checklists](https://mlsanity.com/paper/2403.18771/) (2024)
- [NaturalProofs: Mathematical Theorem Proving in Natural Language](https://mlsanity.com/paper/2104.01112/) (2021)
- [Reliable Evaluation and Benchmarks for Statement Autoformalization](https://mlsanity.com/paper/2406.07222/) (2024)
- [Putnam-AXIOM: A Functional and Static Benchmark for Measuring Higher Level Mathematical Reasoning in LLMs](https://mlsanity.com/paper/2508.08292/) (2025)
- [The Faithfulness Gap: Certifying Semantic Equivalence Between Natural-Language and Formal Mathematical Statements](https://mlsanity.com/paper/2606.16541/) (2026)
- [Operads for complex system design specification, analysis and synthesis](https://mlsanity.com/paper/2101.11115/) (2021)
- [Does My Embedding Reflect That A = B? Evaluating Mathematical Equivalence in Embedding Models](https://mlsanity.com/paper/2606.23959/) (2026)

## Abstract (arXiv)

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.