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LIT-935: The Origins of Representation Manifolds in Large Language Models

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Properties5
StatusActive — read 2026-10-14 (NOTE-802) in full; worth reading for Theorem 1: if a feature is a compact metric space mapped continuously and injectively onto the unit sphere, and cosine similarity is locally a C² decreasing function of squared feature distance, path lengths on the representation equal the feature's up to the scale √(−2g′(0)), so geodesics recover its intrinsic geometry. Read it knowing that both hypotheses are assumed rather than derived, that the title's "origins" are an informal argument, and that the tests supply a candidate metric by hand: GPT-2's years are homeomorphic to a linear and a logarithmic scale, and only the metric test separates them. Held as THEORY-284.
Version2
Tagsrepresentation-learning, anthology-candidate
Cited as “paper” byNOTE-802 — Modell, Rubin-Delanchy and Whiteley — The Origins of Representation Manifolds in Large Language Models
Cited as “source” byTHEORY-284 — If a feature's values form a compact metric space carried continuously and injectively onto the unit sphere, and cosine similarity is locally a C² decreasing function of squared feature distance, then every path on the representation manifold is √(−2g′(0)) times as long as the feature path it images, so on-manifold shortest paths recover the feature's intrinsic length metric up to one scale; a homeomorphism alone does not fix it

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Alexander Modell, Patrick Rubin-Delanchy and Nick Whiteley (2025), arXiv:2505.18235.

Key takeaways

From the reading, NOTE-802.

  • Two hypotheses, two levels. Hypothesis 1: a continuous invertible map ϕ_f from the feature’s compact metric space onto the sphere, which makes the representation homeomorphic to the feature (Proposition 1). Hypothesis 2: locally, CosSim(ϕ_f(z), ϕ_f(z′)) = g_f(d_f(z, z′)²) with g_f ∈ C² and g_f′(0) < 0.
  • Theorem 1. Under both, L(ϕ_f ∘ η) = √(−2g_f′(0)) L(η) for every rectifiable path η. Shortest paths on the manifold measure the feature up to a unit, without knowing g. The proof (App. B) is a partition argument with a second-order error per step, and checks out. It recovers the length metric d_f induces, which is d_f on a length space.
  • The evidence.
    • GPT-2-small layer-7 years: in order along a neighbour-graph curve (Kendall 0.97, Spearman above 0.99).
    • Not isometric to |x − y|. Isometric to log(2019 − year) (ρ = 0.99, ξ = 0.84).
    • Dates in a text-embedding model: ρ = 0.97 against the circular metric. Colours: weakly plausible.
    • Mistral 7B’s month and weekday loops weave through a third principal component that the published 2-D view hides.
  • What it does not do. It gives no mechanism for either hypothesis. §2.4’s argument from linear readability is informal. It supplies each candidate metric by hand, which the authors call “clearly not scalable”. It recovers a metric, not a group.

Standing in the record

Filed on 2026-10-12 from the owner’s papers-feed for QUESTION-058, and read on 2026-10-14 in NOTE-802. Theorem 1 is held as THEORY-284, which answers QUESTION-058 in part: the intrinsic metric of a feature can be read off a representation without the group, given two hypotheses the paper assumes. Its Fig. 2 qualifies THEORY-022’s reading of the month and weekday circles as single two-dimensional irreducibles.

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