# When Tokenization is Secretly Output Supervision

arXiv: https://arxiv.org/abs/2609.01386 · HTML: https://mlsanity.com/paper/2609.01386/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 104 of its 151 references that are in the map.
- First posted on arXiv 1 Sep 2026.

## Builds on (most cited first, 30 of 104)

- [The Llama 3 Herd of Models](https://mlsanity.com/paper/2407.21783/) (2024)
- [Neural Machine Translation of Rare Words with Subword Units](https://mlsanity.com/paper/1508.07909/) (2015)
- [Gemma 2: Improving Open Language Models at a Practical Size](https://mlsanity.com/paper/2408.00118/) (2024)
- [SentencePiece: A simple and language independent subword tokenizer and detokenizer for Neural Text Processing](https://mlsanity.com/paper/1808.06226/) (2018)
- [Pythia: A Suite for Analyzing Large Language Models Across Training and Scaling](https://mlsanity.com/paper/2304.01373/) (2023)
- [Math-Shepherd: Verify and Reinforce LLMs Step-by-step without Human Annotations](https://mlsanity.com/paper/2312.08935/) (2023)
- [Progress measures for grokking via mechanistic interpretability](https://mlsanity.com/paper/2301.05217/) (2023)
- [ProcessBench: Identifying Process Errors in Mathematical Reasoning](https://mlsanity.com/paper/2412.06559/) (2024)
- [GSM-Plus: A Comprehensive Benchmark for Evaluating the Robustness of LLMs as Mathematical Problem Solvers](https://mlsanity.com/paper/2402.19255/) (2024)
- [Future Lens: Anticipating Subsequent Tokens from a Single Hidden State](https://mlsanity.com/paper/2311.04897/) (2023)
- [Tokenization counts: the impact of tokenization on arithmetic in frontier LLMs](https://mlsanity.com/paper/2402.14903/) (2024)
- [MathBench: Evaluating the Theory and Application Proficiency of LLMs with a Hierarchical Mathematics Benchmark](https://mlsanity.com/paper/2405.12209/) (2024)
- [Tokenization Is More Than Compression](https://mlsanity.com/paper/2402.18376/) (2024)
- [MARIO: MAth Reasoning with code Interpreter Output -- A Reproducible Pipeline](https://mlsanity.com/paper/2401.08190/) (2024)
- [Transformers represent belief state geometry in their residual stream](https://mlsanity.com/paper/2405.15943/) (2024)
- [Towards Robust Mathematical Reasoning](https://mlsanity.com/paper/2511.01846/) (2025)
- [MathGenie: Generating Synthetic Data with Question Back-translation for Enhancing Mathematical Reasoning of LLMs](https://mlsanity.com/paper/2402.16352/) (2024)
- [Step-level Value Preference Optimization for Mathematical Reasoning](https://mlsanity.com/paper/2406.10858/) (2024)
- [Why do language models perform worse for morphologically complex languages?](https://mlsanity.com/paper/2411.14198/) (2024)
- [Unpacking Tokenization: Evaluating Text Compression and its Correlation with Model Performance](https://mlsanity.com/paper/2403.06265/) (2024)
- [A Survey of Mathematical Reasoning in the Era of Multimodal Large Language Model: Benchmark, Method & Challenges](https://mlsanity.com/paper/2412.11936/) (2024)
- [Are All Languages Equally Hard to Language-Model?](https://mlsanity.com/paper/1806.03743/) (2018)
- [Bridging the Novice-Expert Gap via Models of Decision-Making: A Case Study on Remediating Math Mistakes](https://mlsanity.com/paper/2310.10648/) (2023)
- [Do language models plan ahead for future tokens?](https://mlsanity.com/paper/2404.00859/) (2024)
- [ChatGLM-Math: Improving Math Problem-Solving in Large Language Models with a Self-Critique Pipeline](https://mlsanity.com/paper/2404.02893/) (2024)
- [MuggleMath: Assessing the Impact of Query and Response Augmentation on Math Reasoning](https://mlsanity.com/paper/2310.05506/) (2023)
- [DocMath-Eval: Evaluating Math Reasoning Capabilities of LLMs in Understanding Long and Specialized Documents](https://mlsanity.com/paper/2311.09805/) (2023)
- [Evaluating Mathematical Reasoning of Large Language Models: A Focus on Error Identification and Correction](https://mlsanity.com/paper/2406.00755/) (2024)
- [CHAMP: A Competition-level Dataset for Fine-Grained Analyses of LLMs' Mathematical Reasoning Capabilities](https://mlsanity.com/paper/2401.06961/) (2024)
- [LLM Critics Help Catch Bugs in Mathematics: Towards a Better Mathematical Verifier with Natural Language Feedback](https://mlsanity.com/paper/2406.14024/) (2024)

## Abstract (arXiv)

Tokenization in language models is treated by default as an input preprocessing decision. We argue that this framing is incomplete: in autoregressive models, tokenizer granularity determines what the model must resolve in a single forward pass, and therefore the supervision signal it receives. This affects both the difficulty of the learning problem and the representations that emerge inside the model. We test this in a controlled experiment on numeric reasoning with a novel decoupling of input and output tokenization. As the output supervision view predicts, differences in task performance, training dynamics, and model internals are induced by output tokenization and largely invariant to input tokenization. This may matter in practice, because models with different tokenization strategies differ not only in input representation but in the task they were trained on. Comparisons between models may thus partly reflect task definition rather than ability. A survey of 120 recent *CL papers on numeric reasoning confirms that this is rarely acknowledged: only about 10% report the numeric tokenization of the models they evaluate, while 69% compare across tokenization, and thus supervision, regimes without reporting it. While prior work documents that tokenization consistently affects model performance, there is no principled account of why. We argue that framing tokenization as output supervision provides that account.