Concept Subspaces Compute Beyond the Logit Lens: A Weights-Only Test for Locating Representations Upstream of Readout

Aojie Yuan, Zhiyuan Julian Su, Haiyue Zhang, Zijian Su

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Abstract

A concept subspace's effect on model behavior does not establish how it relates to the output readout. We introduce a two-sided geometric diagnostic that measures an extracted subspace's overlap with the dominant right-singular directions of the unembedding matrix, evaluated against output-oriented positive controls. Given an extracted basis, the raw diagnostic requires only model weights. Our testbed is the Format-Agnostic Reasoning Subspace (FARS), a ten-dimensional basis extracted from eighteen reasoning concepts expressed in six surface forms. Across nine rank-matched estimators and twenty-six models, four activation-derived concept estimators carry only 0.38--0.80% mean energy in the top-ten readout span. Final-layer PCA carries 3.56%, exceeding FARS in 25 of 26 models. A same-layer next-token control, evaluated using a fitted linear translator for depth matching, carries approximately thirteen times more energy than FARS, with separation in all 25 tested models. Re-extracting FARS on ten disjoint concepts yields 62--100% cross-format retrieval across twenty-four generative models, demonstrating transfer of the extraction procedure rather than a fixed basis. A complementary four-model, three-seed intervention study finds model-dependent source-directed effects that remain well below full-vector replacement. Together, the geometry and intervention controls distinguish concept structure from dominant readout directions while limiting claims of causal sufficiency.

Citation

@article{yuan2026concept,
  title     = {Concept Subspaces Compute Beyond the Logit Lens: A Weights-Only Test for Locating Representations Upstream of Readout},
  author    = {Yuan, Aojie and Su, Zhiyuan Julian and Zhang, Haiyue and Su, Zijian},
  journal   = {arXiv preprint arXiv:2609.39263},
  year      = {2026},
  eprint    = {2609.39263},
  archivePrefix = {arXiv},
  doi       = {10.48550/arXiv.2609.39263},
  url       = {https://arxiv.org/abs/2609.39263}
}