The Shape of Addition: Geometric Structures of Arithmetic in Large Language Models

Abstract

Large Language Models exhibit paradoxical fragility in fundamental arithmetic, implying a disconnect between internal computation and discrete output. By analyzing the residual stream geometry during multi-operand addition, we identify the **Iso-Raw-Sum Trajectory (IRST)**, a topological manifold where representations are anchored by semantic digits and modulated by continuous carry fibers. We propose the **Noisy Quantization Model**, which frames arithmetic errors as *topological slippages* caused by internal neural noise pushing a continuous, latent *carry potential* across quantization thresholds. This geometric framework further elucidates *probe versatility*, explaining how lightweight probes can disentangle conflicting latent signals (such as ground truth versus hallucination) from a single activation vector. Finally, we validate these insights through a geometric consistency check method that effectively detects and corrects these quantization failures during inference.