Repeated-Token Counting Reveals a Dissociation Between Representations and Outputs

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 $R^2>0.99$, showing that the information is present. However, the internal directions that encode counts are nearly orthogonal to digit-token output-head rows ($|\cos| \leq 0.032$). 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%$\pm$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-$10^4$ to 1; (ii) seed 42 shows $54{,}332 \to 838$ (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.

Large language models are highly capable of answering difficult questions by retrieving, recombining, and attending to information in long contexts. For agentic tasks, an additional capability is required: the preservation of an exact state while repeatedly applying rules. We find that this reliability is absent across language models. To demonstrate, we query 126 leading model variants with the task of counting a long string of repeated characters, and we find they all cannot accurately count above a model-dependent, syntax-sensitive counting capacity threshold. Failures are abrupt and persist even with increasing model size, inference time computation, and external tool. Mechanistic probing indicates that models use a finite number of internal states to mimic counting as a rule and fail once these states are exhausted. Furthermore, such states are the basis for performing complex tasks beyond counting. These results indicate that fundamentally new model architectures are required for autonomous agents to achieve truly reliable rule following capabilities.

Large language models (LLMs) exhibit failure modes on seemingly trivial tasks. We propose a formalisation of LLM interaction using a deterministic multi-tape Turing machine, where each tape represents a distinct component: input characters, tokens, vocabulary, model parameters, activations, probability distributions, and output text. The model enables precise localisation of failure modes to specific pipeline stages, revealing, e.g., how tokenisation obscures character-level structure needed for counting tasks. The model clarifies why techniques like chain-of-thought prompting help, by externalising computation on the output tape, while also revealing their fundamental limitations. This approach provides a rigorous, falsifiable alternative to geometric metaphors and complements empirical scaling laws with principled error analysis.