Padding to k: A Minimum Count in the Schema Makes Open-Weight Models Pad Their Extraction Lists

Counting the items in a bracketed list looks trivial, yet open-weight chat models often get it wrong. Prior work usually blames input bottlenecks such as subword fragmentation or attention dilution, which predict that different models should fail in roughly the same way. Across seven instruct models on identical prompts, however, wrong answers form distinct modes: Qwen and Gemma 27B often flip odd lengths to a nearby even integer, OLMo concentrates errors on a few mid-sized integers, and Llama tends to under-count. These modes are useful labels rather than a stable family law (Gemma 9B does not reproduce Gemma 27B's odd-to-even drop), and heavier subword fragmentation does not make counting harder on our benchmark. When the model answers incorrectly, a linear probe can usually still recover the true count from the residual stream. Matching the same odd-to-even error also does not imply the same late-MLP magnitude fix: scaling a late MLP output helps Qwen modestly but is near null on Gemma 27B under the same protocol, while residual steering can move both only by trading odd gains for even losses. These results caution against transferring that magnitude fix across models without a transfer check.

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.

Pipelines that require structured outputs from hosted language models often specify a Pydantic schema in the prompt, validate the response, and retry when validation fails. Although native structured-output modes provide an alternative, application-side validation remains important for enforcing custom constraints. Yet pipeline reliability is often summarized by a single validity rate, which obscures why outputs fail and which interventions may help. We introduce a fine-grained taxonomy of schema-conformance failures and a deterministic classifier that assigns failure categories from model responses and Pydantic validation errors. We apply this framework to 3,997 generation attempts across nine OpenAI and Anthropic model/API configurations, using questions from HotpotQA and FinanceBench. Our results show that similar validity rates can mask different failure patterns, and that dominant failure categories vary with schema constraints and task. Markdown-fenced JSON is the most common failure: among 681 fenced failures, only 48.0% pass validation after fence removal, while the remainder conceal additional violations, predominantly field-length violations. In an exploratory prompt ablation, variants that suppress fences appear to shift failures toward length violations, but the shift is consistent with exposing violations that were already present, illustrating how aggregate validity rates can obscure changes in failure composition. We derive four engineering practices for validator, schema, retry and evaluation design, three supported by our data and one, on retry feedback, proposed but not tested, and release the classifier, completions, and annotations.