[2602.22600] Transformers converge to invariant algorithmic cores

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Abstract:Training selects for behavior, not circuitry: many weight configurations can implement the same function. Studying any single trained neural network thus risks describing accidents of one training run rather than the computation itself. This work shifts focus from what transformers happen to do to what they must do by extracting algorithmic cores, compact subspaces that are necessary and sufficient for a task and that recur across independently trained models. Here, Algorithmic Core Extraction (ACE) is introduced to isolate these subspaces, causally validate them, and recover the algorithms they implement, from synthetic tasks to large pretrained models. Markov-chain transformers embed 3D cores in nearly orthogonal subspaces yet recover identical transition spectra. Modular-addition transformers form compact cyclic cores at grokking that later inflate under continued regularization, redundantly distributing the same computation across many functionally equivalent modes. This functional redundancy accelerates the transition from memorization to generalization, yielding an inverse scaling law for grokking time. In six language models spanning two orders of magnitude in scale (GPT-2 Small/Medium/Large, LLaMA-3.1, Gemma-2, and Qwen2.5), subject-verb agreement is governed by a single, steerable axis that aligns across architectures. Flipping this axis inverts grammatical number throughout open-ended generation. For indirect object identification, a multi-step task, the same six models route computation through compact multi-dimensional cores at two sites, linked by a linear operator that transfers across model families. Together these results suggest that beneath the apparent complexity of trained transformers lies a simpler, shared computational structure, and that targeting invariants rather than parameterizations may offer a more tractable path to mechanistic understanding and control.