{"id":"arxiv:2510.00184","title":"Why Can't Transformers Learn Multiplication? Reverse-Engineering Reveals Long-Range Dependency Pitfalls","date":"2025-09-30","section":"Reasoning and the \"physics\" of language models","quality":0.062547,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/7e395b2ae5bfc2a4d8bfc1e8669b4f324739b6a9b11161cba150bfdffc31746c/","source_url":"https://arxiv.org/abs/2510.00184","text_available":true,"authors":["Xiaoyan Bai","Itamar Pres","Yuntian Deng","Chenhao Tan"],"outgoing":[{"source":"arxiv:2510.00184","target":"arxiv:2301.05217","evidence":"id","id":"arxiv:2301.05217","title":"Progress measures for grokking via mechanistic interpretability","url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/f71e3b65a8b3f35beb171d17f6c65878e0914227a0042ccba59b88952554baef/"},{"source":"arxiv:2510.00184","target":"arxiv:2309.14316","evidence":"id","id":"arxiv:2309.14316","title":"Physics of Language Models: Part 3.1, Knowledge Storage and Extraction","url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/b87376d90e99d2bb590e1a02ed50a55c90881591ed7d3e00c3d5be2aaacbb2af/"},{"source":"arxiv:2510.00184","target":"arxiv:2405.14838","evidence":"id","id":"arxiv:2405.14838","title":"From Explicit CoT to Implicit CoT: Learning to Internalize CoT Step by Step","url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/a242b6148f6f84466ce4315ac4937278586a242dd3f847146e142f3c7e27c39c/"},{"source":"arxiv:2510.00184","target":"arxiv:2407.20311","evidence":"id","id":"arxiv:2407.20311","title":"Physics of Language Models: Part 2.1, Grade-School Math and the Hidden Reasoning Process","url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/de5db7bc6b929af401b5b83c845823505a7823c25987b0e455231d2e4fcec3e4/"},{"source":"arxiv:2510.00184","target":"arxiv:2410.21272","evidence":"id","id":"arxiv:2410.21272","title":"Arithmetic Without Algorithms: Language Models Solve Math With a Bag of Heuristics","url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/397b826ba289bd4f14f457562942cb092c38550a4c49451e95ffcc7b376c8cb0/"},{"source":"arxiv:2510.00184","target":"arxiv:2502.00873","evidence":"id","id":"arxiv:2502.00873","title":"Language Models Use Trigonometry to Do Addition","url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/7766b2640c6b5ab53a295e8d60920cb2419bfbbd812810d9b35d4b7c323922ed/"}],"incoming":[{"source":"arxiv:2512.16902","target":"arxiv:2510.00184","evidence":"id","id":"arxiv:2512.16902","title":"In-Context Algebra","url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/5aa7087a88619eafd156aad9217e16f6d1ce284c65534322239be0e02f04e553/"}],"text_parts":[{"start":0,"end":15939,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/7e395b2ae5bfc2a4d8bfc1e8669b4f324739b6a9b11161cba150bfdffc31746c/part-1.html"},{"start":15939,"end":31923,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/7e395b2ae5bfc2a4d8bfc1e8669b4f324739b6a9b11161cba150bfdffc31746c/part-2.html"},{"start":31923,"end":41094,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/7e395b2ae5bfc2a4d8bfc1e8669b4f324739b6a9b11161cba150bfdffc31746c/part-3.html"}],"contents":[{"title":"Why Can’t Transformers Learn Multiplication? Reverse-Engineering Reveals Long-Range Dependency Pitfalls","offset":0,"level":1,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/7e395b2ae5bfc2a4d8bfc1e8669b4f324739b6a9b11161cba150bfdffc31746c/part-1.html#text"},{"title":"Abstract","offset":107,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/7e395b2ae5bfc2a4d8bfc1e8669b4f324739b6a9b11161cba150bfdffc31746c/part-1.html#text"},{"title":"1 Introduction","offset":1809,"level":2,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/7e395b2ae5bfc2a4d8bfc1e8669b4f324739b6a9b11161cba150bfdffc31746c/part-1.html#text"},{"title":"2 Experiment Setup, Training ICoT, Notations","offset":4837,"level":2,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/7e395b2ae5bfc2a4d8bfc1e8669b4f324739b6a9b11161cba150bfdffc31746c/part-1.html#text"},{"title":"Notations.","offset":8755,"level":5,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/7e395b2ae5bfc2a4d8bfc1e8669b4f324739b6a9b11161cba150bfdffc31746c/part-1.html#text"},{"title":"3 Comparing the Mechanisms of ICoT versus SFT","offset":9223,"level":2,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/7e395b2ae5bfc2a4d8bfc1e8669b4f324739b6a9b11161cba150bfdffc31746c/part-1.html#text"},{"title":"3.1 Long-range dependencies in multi-digit multiplication","offset":9273,"level":3,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/7e395b2ae5bfc2a4d8bfc1e8669b4f324739b6a9b11161cba150bfdffc31746c/part-1.html#text"},{"title":"3.2 Evidence of Long-Range Dependencies in ICoT","offset":11156,"level":3,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/7e395b2ae5bfc2a4d8bfc1e8669b4f324739b6a9b11161cba150bfdffc31746c/part-1.html#text"},{"title":"3.3 Encoding Long-Range Dependencies via Attention Trees","offset":14072,"level":3,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/7e395b2ae5bfc2a4d8bfc1e8669b4f324739b6a9b11161cba150bfdffc31746c/part-1.html#text"},{"title":"4 Feature Geometry of ICoT","offset":17791,"level":2,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/7e395b2ae5bfc2a4d8bfc1e8669b4f324739b6a9b11161cba150bfdffc31746c/part-2.html#text"},{"title":"4.1 Digit-wise Multiplications as Minkowski Sums","offset":17921,"level":3,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/7e395b2ae5bfc2a4d8bfc1e8669b4f324739b6a9b11161cba150bfdffc31746c/part-2.html#text"},{"title":"4.2 Embedding Digits on a Pentagonal Prism via Fourier Bases","offset":20672,"level":3,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/7e395b2ae5bfc2a4d8bfc1e8669b4f324739b6a9b11161cba150bfdffc31746c/part-2.html#text"},{"title":"5 Pitfalls of Learning: Lack of Long-Range Dependency","offset":23751,"level":2,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/7e395b2ae5bfc2a4d8bfc1e8669b4f324739b6a9b11161cba150bfdffc31746c/part-2.html#text"},{"title":"6 Learning Multiplication Without ICoT","offset":25131,"level":2,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/7e395b2ae5bfc2a4d8bfc1e8669b4f324739b6a9b11161cba150bfdffc31746c/part-2.html#text"},{"title":"6.1 Does the model with auxiliary loss learn the same mechanism as ICoT?","offset":27685,"level":3,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/7e395b2ae5bfc2a4d8bfc1e8669b4f324739b6a9b11161cba150bfdffc31746c/part-2.html#text"},{"title":"7 Related Work","offset":28477,"level":2,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/7e395b2ae5bfc2a4d8bfc1e8669b4f324739b6a9b11161cba150bfdffc31746c/part-2.html#text"},{"title":"Studying Transformers with Arithmetic Tasks.","offset":28496,"level":5,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/7e395b2ae5bfc2a4d8bfc1e8669b4f324739b6a9b11161cba150bfdffc31746c/part-2.html#text"},{"title":"8 Conclusion","offset":30308,"level":2,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/7e395b2ae5bfc2a4d8bfc1e8669b4f324739b6a9b11161cba150bfdffc31746c/part-2.html#text"},{"title":"Acknowledgments","offset":31071,"level":4,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/7e395b2ae5bfc2a4d8bfc1e8669b4f324739b6a9b11161cba150bfdffc31746c/part-2.html#text"},{"title":"Reproducibility Statement","offset":31565,"level":3,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/7e395b2ae5bfc2a4d8bfc1e8669b4f324739b6a9b11161cba150bfdffc31746c/part-2.html#text"},{"title":"Appendix A Training Details","offset":31798,"level":2,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/7e395b2ae5bfc2a4d8bfc1e8669b4f324739b6a9b11161cba150bfdffc31746c/part-2.html#text"},{"title":"A.1 ICoT Training","offset":31923,"level":3,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/7e395b2ae5bfc2a4d8bfc1e8669b4f324739b6a9b11161cba150bfdffc31746c/part-3.html#text"},{"title":"A.2 Standard Fine-Tuning","offset":34810,"level":3,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/7e395b2ae5bfc2a4d8bfc1e8669b4f324739b6a9b11161cba150bfdffc31746c/part-3.html#text"},{"title":"Appendix B Fourier structure in model’s weights, activations","offset":35595,"level":2,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/7e395b2ae5bfc2a4d8bfc1e8669b4f324739b6a9b11161cba150bfdffc31746c/part-3.html#text"},{"title":"Appendix C Per Token Gradients and Losses: 12-Layer Model","offset":39356,"level":2,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/7e395b2ae5bfc2a4d8bfc1e8669b4f324739b6a9b11161cba150bfdffc31746c/part-3.html#text"},{"title":"Appendix D Attention Patterns of All Models","offset":39677,"level":2,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/7e395b2ae5bfc2a4d8bfc1e8669b4f324739b6a9b11161cba150bfdffc31746c/part-3.html#text"},{"title":"Appendix E LLM Usage","offset":40041,"level":2,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/7e395b2ae5bfc2a4d8bfc1e8669b4f324739b6a9b11161cba150bfdffc31746c/part-3.html#text"},{"title":"Instructions for reporting 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