# Easy to anticipate, hard to compute: boundary dependence finds the computed outputs that entropy patching misses

arXiv: https://arxiv.org/abs/2610.11790 · HTML: https://mlsanity.com/paper/2610.11790/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 17 of its 17 references that are in the map.
- First posted on arXiv 8 Oct 2026.

## Builds on (most cited first, 17 of 17)

- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [Measuring Mathematical Problem Solving With the MATH Dataset](https://mlsanity.com/paper/2103.03874/) (2021)
- [Mixture-of-Depths: Dynamically allocating compute in transformer-based language models](https://mlsanity.com/paper/2404.02258/) (2024)
- [OpenWebMath: An Open Dataset of High-Quality Mathematical Web Text](https://mlsanity.com/paper/2310.06786/) (2023)
- [Byte Latent Transformer: Patches Scale Better Than Tokens](https://mlsanity.com/paper/2412.09871/) (2024)
- [Rho-1: Not All Tokens Are What You Need](https://mlsanity.com/paper/2404.07965/) (2024)
- [Tokenization counts: the impact of tokenization on arithmetic in frontier LLMs](https://mlsanity.com/paper/2402.14903/) (2024)
- [Efficient Transformers with Dynamic Token Pooling](https://mlsanity.com/paper/2211.09761/) (2022)
- [Dynamic Chunking for End-to-End Hierarchical Sequence Modeling](https://mlsanity.com/paper/2507.07955/) (2025)
- [SpaceByte: Towards Deleting Tokenization from Large Language Modeling](https://mlsanity.com/paper/2404.14408/) (2024)
- [MrT5: Dynamic Token Merging for Efficient Byte-level Language Models](https://mlsanity.com/paper/2410.20771/) (2024)
- [From Bytes to Ideas: Language Modeling with Autoregressive U-Nets](https://mlsanity.com/paper/2506.14761/) (2025)
- [Fast Byte Latent Transformer](https://mlsanity.com/paper/2605.08044/) (2026)
- [TokEval: A Tokenizer Evaluation Suite](https://mlsanity.com/paper/2608.18062/) (2026)
- [ByteSpan: Information-Driven Subword Tokenisation](https://mlsanity.com/paper/2506.18639/) (2025)
- [Scratchpad Patching: Decoupling Compute from Patch Size in Byte-Level Language Models](https://mlsanity.com/paper/2605.09630/) (2026)
- [EntropyMoE: Entropy-Aware Sparse Expert Routing for Tokenizer-Free LLMs](https://mlsanity.com/paper/2608.06398/) (2026)

## Abstract (arXiv)

Byte-level language models such as the Byte Latent Transformer (BLT) group bytes into patches and run their large global model once per patch. BLT starts a patch where a small model's next-byte entropy is high, so global compute goes where the next byte is hard to predict. We show that this rule has a systematic blind spot: positions whose type is predictable but whose value must be computed, such as the number after "=" in a worked math solution. Under tight patch budgets, entropy-triggered layouts skip these positions, and accuracy on them collapses. In Meta's BLT-1B with patch starts on 10% of bytes, the entropy rule puts a patch start at 16% of the computed results in GSM8K solutions and gets 19.0% of them exactly right; a boundary after each "=" at the same patch count gets 51.8%, and entropy combined with a label-free boundary-dependence signal gets 67.1% (default layout at 26% of bytes: 76.8%). The gap survives adapting BLT-1B to the budget with low-rank fine-tuning (32.9% vs 72.7%, three runs per rule, paired p < 1e-200) and grows with model size in byte models trained from scratch at a 10% budget: at 1M, 12M and 50M parameters, boundary dependence beats entropy on final answers by -1.6, +10.1 and +19.8 points, and at 50M it gets 35.9% of computed results against 13.9% (3 seeds each). BLT's entropy-jump rule helps neither target at 50M. The entropy trigger of Scratchpad Patching is likewise indistinguishable from random scratchpads on final answers (5.6% vs 5.9%, 5 seeds), while answer-start scratchpads give 38.1%. The effect is specific to computed values: copies and lookups gain little, and values the model cannot compute gain nothing. Boundary dependence, the rise in the model's own loss when a patch start is removed, measured per two-byte context, finds these positions without labels: combined with entropy it beats the hand-written rule on computed results.