# The Right Answer, the Wrong Direction: Why Transformers Fail at Counting and How to Fix It

arXiv: https://arxiv.org/abs/2605.03258 · HTML: https://mlsanity.com/paper/2605.03258/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 23 references that are in the map.
- First posted on arXiv 5 May 2026.

## Builds on (most cited first, 19 of 19)

- [Chain-of-Thought Prompting Elicits Reasoning in Large Language Models](https://mlsanity.com/paper/2201.11903/) (2022)
- [Mistral 7B](https://mlsanity.com/paper/2310.06825/) (2023)
- [Locating and Editing Factual Associations in GPT](https://mlsanity.com/paper/2202.05262/) (2022)
- [Pythia: A Suite for Analyzing Large Language Models Across Training and Scaling](https://mlsanity.com/paper/2304.01373/) (2023)
- [Understanding intermediate layers using linear classifier probes](https://mlsanity.com/paper/1610.01644/) (2016)
- [Transformer Feed-Forward Layers Are Key-Value Memories](https://mlsanity.com/paper/2012.14913/) (2020)
- [Toy Models of Superposition](https://mlsanity.com/paper/2209.10652/) (2022)
- [Progress measures for grokking via mechanistic interpretability](https://mlsanity.com/paper/2301.05217/) (2023)
- [Discovering Latent Knowledge in Language Models Without Supervision](https://mlsanity.com/paper/2212.03827/) (2022)
- [Towards Automated Circuit Discovery for Mechanistic Interpretability](https://mlsanity.com/paper/2304.14997/) (2023)
- [The Linear Representation Hypothesis and the Geometry of Large Language Models](https://mlsanity.com/paper/2311.03658/) (2023)
- [Designing and Interpreting Probes with Control Tasks](https://mlsanity.com/paper/1909.03368/) (2019)
- [Eliciting Latent Predictions from Transformers with the Tuned Lens](https://mlsanity.com/paper/2303.08112/) (2023)
- [Transformer Feed-Forward Layers Build Predictions by Promoting Concepts in the Vocabulary Space](https://mlsanity.com/paper/2203.14680/) (2022)
- [Using the Output Embedding to Improve Language Models](https://mlsanity.com/paper/1608.05859/) (2016)
- [Linearity of Relation Decoding in Transformer Language Models](https://mlsanity.com/paper/2308.09124/) (2023)
- [Do NLP Models Know Numbers? Probing Numeracy in Embeddings](https://mlsanity.com/paper/1909.07940/) (2019)
- [Jump to Conclusions: Short-Cutting Transformers With Linear Transformations](https://mlsanity.com/paper/2303.09435/) (2023)
- [A Mechanistic Interpretation of Arithmetic Reasoning in Language Models using Causal Mediation Analysis](https://mlsanity.com/paper/2305.15054/) (2023)

## Built on by (most cited first, 1 of 1)

- [Targeted Retrieval, Compact Representations: How CoT Reasoning Improves Long-Context Counting](https://mlsanity.com/paper/2609.38958/) (2026)

## Abstract (arXiv)

Large language models often fail at simple counting tasks, even when items to count are in the prompt. We investigate whether this failure occurs because transformers do not represent counts internally, or because they cannot convert representations to the correct output tokens. Across three model families: Pythia, Qwen3, and Mistral, ranging from 0.4B to 14B parameters, we find evidence for the second explanation. Linear probes recover the correct count from intermediate layers with R2>0.99, showing that the information is present. However, the internal directions that encode counts are nearly orthogonal to digit-token output-head rows (|cos| ≤ 0.032). In other words, the model stores the count in a form that the digit logits do not naturally read out. We localize this failure with two interventions. Updating only the digit rows of the output head (36,864 parameters) substantially improves constrained digit prediction (60.7--100.0% on four tasks), but it does not fix unconstrained generation (0%); we do not claim that digit-row repair fixes open-ended text. By contrast, small LoRA on attention Q/V (7.67M parameters) improves upstream routing and achieves 83.1%±7.2% in true greedy autoregressive generation (deployable fix). Logit-lens at layer 35 (entity counting; correct-digit rank): (i) median over 3 seeds drops from order-104 to 1; (ii) seed 42 shows 54,332 → 838 (median top-1 while one seed stays far below). Norm, logit-lens, and cross-task analyses generalize the bottleneck to counting, addition, and list length; nulls on MMLU and GSM8K and limited DROP transfer. These results identify counting failure as a geometric readout bottleneck, not an internal-representation failure: the model knows the count but the output pathway is misaligned with tokens needed to express it.