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LIT-322: Not All Language Model Features Are One-Dimensionally Linear

2 min read

Properties5
StatusActive — read in full 2026-09-29 (NOTE-273); worth reading as the first causal demonstration that a pretrained language model computes with an irreducibly two-dimensional (circular) feature, and for a clean statistical definition of feature reducibility. Read the discovery as largely manual: circles for days, months and years, found by eye among about 2,500 inspected clusters, with the automatic score ranking them 9th, 28th and 15th of 1,000. Read "irreducible" as statistical, not representation-theoretic.
Version2
Tagsrepresentation-learning, anthology-candidate
Cited as “paper” byNOTE-273 — Not All Language Model Features Are One-Dimensionally Linear
Cited as “source” byTHEORY-022 — The reverse-engineered grokking network computes modular addition by multiplying characters of ℤ/p at a few frequencies, the same real two-dimensional irreducibles later found as circles in language models; in both the group comes from the task, not the network, THEORY-183 — When the co-occurrence of words on a latent continuum depends only on their separation, a spectral word embedding places them on the Fourier modes of that continuum, so the circles and rippled curves of months, years and places come from a symmetry of the corpus statistics

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Joshua Engels, Eric J. Michaud, Isaac Liao, Wes Gurnee, Max Tegmark (2024), arXiv preprint (a later conference venue is unverified here) — ARXIV-2405.14860

Standing in the record

Registered on 2026-09-29 at the owner’s request, because the owner’s prior-art novelty map (a private working document on concepts as projectors, the linear representation hypothesis and their neighbours) cites it: §5 “scooped again?” item 3: feature-geometry/non-linear-features groups who could reach the sectors/irrep-multiplet predictions first. It is seeded from metadata and published descriptions and has not been read here.

Identification notes: arXiv v1 title was “Not All Language Model Features Are Linear”; current title adds “One-Dimensionally”. Not held in the anthology (anthology hit on “Engels” is co-authorship of ANTH-LIT-568). ICLR 2025 venue from recollection — verify before filing if venue is recorded.

It was filed Deferred, unread. NOTE-273 is the close reading of 2026-09-29, and it placed the work: Active — worth reading as the first causal demonstration that a pretrained language model computes with an irreducibly two-dimensional (circular) feature, and for a clean statistical definition of feature reducibility. Read the discovery as largely manual: circles for days, months and years, found by eye among about 2,500 inspected clusters, with the automatic score ranking them 9th, 28th and 15th of 1,000. Read “irreducible” as statistical, not representation-theoretic.

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