Localization Does Not Imply Steerability: A Mechanistic Study of Sign Errors in LLM Mathematics

Shradha Agarwal, Tariq J., Deepak Rajbhar

When large language models do multi-step mathematics, a characteristic failure recurs: the model writes one wrong sign token mid-computation -- a cofactor sign, the minus of a recursive integration-by-parts step -- then executes the rest of the arithmetic correctly, and the answer comes out wrong. We ask where inside the network that sign is written. Across six open models spanning 14B to 675B parameters, a per-layer readout shows the wrong sign is committed late -- by MLP layers in the last quarter of the network (70-100% per model and task), a pattern present, where tested (Qwen), in the base model before instruction tuning. The mechanism varies: some models carry fixed "minus-stamp" layers that push a minus regardless of the right answer, others carry amplifiers of whichever sign was already chosen, and Qwen adds a downstream rescue layer that opposes the stamp. The circuit is causal: subtracting a single sign direction from the residual stream flips 13 of 14 held-out minus-written integration-by-parts errors on Llama-3.3-70B, with flips replicating on Qwen, Gemma, and Mistral-Small (24-70% per cell), and with representation-level specificity -- the same dose breaks…