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This paper investigates the geometric structures underlying arithmetic operations in Large Language Models (LLMs), revealing a critical disconnect between internal computations and their discrete outputs. By introducing the Iso-Raw-Sum Trajectory (IRST) and the Noisy Quantization Model, the authors explain how arithmetic errors arise from Geometric Slippages due to internal noise, which affects the carry potential during multi-operand addition. Their findings enhance the understanding of Probe Versatility, demonstrating how lightweight probes can effectively disentangle latent signals within a single activation vector, and they validate these insights through a novel geometric consistency check method.
Arithmetic errors in LLMs stem from geometric slippages in internal computations, revealing a surprising fragility in their handling of fundamental math.
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 geometric structure where representations are anchored by semantic digits and modulated by continuous carry fibers. We propose the Noisy Quantization Model to explain this geometry, framing arithmetic errors as Geometric 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 coexisting 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. Our code is available at https://github.com/RL-MIND/Shape-of-Addition.
