Paper Detail
Liuyuan Wen, Xun Zhu, Lihao Huang, Wenbin Li, Yang Gao
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.
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@misc{wen2026shape,
title = {The Shape of Addition: Geometric Structures of Arithmetic in Large Language Models},
author = {Liuyuan Wen and Xun Zhu and Lihao Huang and Wenbin Li and Yang Gao},
year = {2026},
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 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 Slippa},
url = {https://huggingface.co/papers/2606.03645},
keywords = {residual stream, Iso-Raw-Sum Trajectory, Noisy Quantization Model, Geometric Slippages, Carry Potential, probe versatility, geometric consistency check, quantization failures, code available, huggingface daily},
eprint = {2606.03645},
archiveprefix = {arXiv},
}
{}