加法的形态：大型语言模型中算术的几何结构

摘要

大型语言模型在基础算术运算中表现出矛盾的脆弱性，暗示内部计算与离散输出之间存在脱节。通过分析多操作数加法过程中的残差流几何结构，我们识别出等原始和轨迹（IRST）——一种表示由语义数字锚定并由连续进位纤维调制的几何结构。我们提出噪声量化模型来解释这一几何形态，将算术错误归因于几何滑移，即内部神经噪声推动连续的潜在进位势跨越量化阈值。该几何框架进一步阐明了探针多功能性，解释了轻量探针如何从单一激活向量中解开共存的潜在信号（如真实值与幻觉）。最后，我们通过一种几何一致性检查方法验证这些洞见，该方法能在推理过程中有效检测并纠正这些量化失败。我们的代码可在 https://github.com/RL-MIND/Shape-of-Addition 获取。

English

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.