Represented Is Not Computed: A Causal Test of Candidate Algorithmic Intermediates in a Transformer

Structured prompts require integrating components according to task-relevant relations. How a network implements this integration is often hard to judge in language or vision, where those relations are rarely specified precisely enough to define a candidate internal algorithm. Arithmetic offers a cleaner setting. We study a Transformer trained on base-digit extraction: given N, B, and D, it must report the coefficient of BD in the base-B expansion of N. The closed-form solution, lfloor N/BD rfloor bmod B, provides explicit candidate algorithmic intermediates. Across three seeds, the model reaches 99.83% exact-answer accuracy on held-out number-base intersections, establishing reliable task competence. Linear probes decode the intermediates, making staged arithmetic computation plausible. Causal tests then separate representation from use: within the localized route from the stream with D as input to the output positions, behavior depends on early D-selective communication, independent of N and B. Relatedly, a sparse circuit search finds mostly separate N, B, and D routes that combine late rather than the staged route suggested by the probes. Thus, the model represents the intermediates that make the closed-form solution plausible, but the identified localized causal route does not transmit them to the output stream. This case shows that probe-based conclusions can diverge sharply from causal observations, even when explicit algorithmic hypotheses are available.