Source
Modell, Rubin-Delanchy and Whiteley (2025), LIT-935, read in NOTE-802: Definitions 1–2, Hypotheses 1–2, Proposition 1, Theorem 1 with its proof and Lemma 1 (App. B), and the tests of §§2.3 and 3.1.
What was actually shown
The setting. A feature is a compact metric space (Z_f, d_f). Its unit representation direction is v_f(x) = ϕ_f(z_f(x)) on S
Proposition 1. If ϕ_f is continuous and invertible (Hypothesis 1), it is a homeomorphism onto its image M_f. So the representation has the feature’s topology: an interval gives a curve, a circle a loop, m points m points.
Theorem 1. Suppose also that CosSim(ϕ_f(z), ϕ_f(z′)) = g_f(d_f(z, z′)²) whenever d_f(z, z′) ≤ ε, with g_f ∈ C² and g_f′(0) < 0 (Hypothesis 2). Then for every path η of finite length, L(ϕ_f ∘ η) = √(−2g_f′(0)) L(η). The proof partitions the path finely. On unit vectors, ½‖γ_{t_i} − γ_{t_{i−1}}‖² = g(0) − g(d²) = −g′(0)d² − ½g″(c)d⁴, and |α − β| ≤ |α² − β²|^{1/2} bounds each step’s error by |g″(c)|^{1/2}d². The summed error is at most C·L(η)·max d, which vanishes with the mesh.
What could have failed: locally reliable cosines might have recovered geometry only up to a distortion that accumulates along a path. They do not, because the error per step is of second order in the step.
The tests. These are the hypotheses checked on models, not the theorem.
- GPT-2-small’s years, rank along a neighbour graph: Kendall 0.97, Spearman above 0.99. That supports homeomorphism to an interval.
- Geodesic against |x − y| fails. Against log(2019 − year): ρ = 0.99, ξ = 0.84.
- Dates in a text-embedding model against the circular metric: ρ = 0.97.
- Colours: “weakly plausible”.
What this does not say
- Not that models satisfy the hypotheses. Both are posited. The paper’s account of why manifolds curve (§2.4) is informal. Its tests are correlations on three features, after SAE extraction or projection onto a few principal components.
- Not that the metric can be read off without a candidate. Theorem 1 makes the intrinsic geometry available in the representation. The paper tests it only against metrics it supplies by hand, and says this does not scale.
- Not d_f itself. What path length recovers is the length metric d_f induces. Where Z_f is not a length space, the chord metric of a circle for example, the representation recovers arc length instead. This is the record’s precision; the paper states the theorem for path lengths and draws the shortest-path conclusion informally.
- Not a group. A loop comes back as a circle with arc length. Nothing says whether it carries ℤ/12, SO(2) or no action at all.
- Not that topology fixes geometry. The years example is the paper’s own: linear and logarithmic scales are homeomorphic, and only the metric test separates them.
Connections
- QUESTION-058. Answers its “read off without supplying the group” in part, for topology and metric only, and only under the two hypotheses.
- THEORY-022. The months and weekdays it reads as 2-D irreducibles weave through a third principal component in Mistral 7B (LIT-935, Fig. 2).
- THEORY-018. Hypothesis 2 presupposes an inner product. Where training fixes none, the theorem holds for the one the analyst picks, if its hypothesis holds for it.
- THEORY-019. Both make the same point from two sides. Degeneracy does not identify a symmetry, and geometry recovered here does not identify one either.
- CLAIM-135 and CLAIM-193. Identification up to a homeomorphism, then a scale, with the candidate space supplied by the analyst.
Note of 2026-10-20: Theorem 1 and the shortest-path step, re-derived
Re-derived for the promotion audit (batch 051, ADR-047 v2), under Hypotheses 1 and 2. Write c = √(−2g′(0)) > 0.
- One step. For d = d_f(z, z′) ≤ ε, put u = ϕ(z) and v = ϕ(z′) on the unit sphere. A vector’s cosine with itself is 1, so g(0) = 1, and ‖u − v‖² = 2(1 − g(d²)) = c²d² − g″(ξ)d⁴ for some ξ in (0, d²). With α = ‖u − v‖, β = cd and K = max |g″| on [0, ε²]: |α² − β²| ≤ Kd⁴, and since |α − β|² ≤ |α − β|(α + β), |α − β| ≤ √K·d².
- Paths. For a path η and γ = ϕ ∘ η, refinement raises both sums of step lengths. Uniform continuity of η makes every step at most δ ≤ ε on fine partitions, and then |Σα_i − cΣd_i| ≤ √K·δ·Σd_i. Along common refinements of near-optimal partitions, with δ → 0, L(γ) = c·L(η) when L(η) is finite. When it is infinite, α_i ≥ cd_i − √K·d_i² ≥ (c/2)d_i for small steps, so L(γ) is infinite too.
- Shortest paths. A continuous injection from a compact space into a Hausdorff one is a homeomorphism onto its image (Proposition 1). So every path in M_f is ϕ ∘ η for a path η in Z_f, and its length is c·L(η). The on-manifold geodesic distance is c times the length metric that d_f induces. That metric is d_f itself only on a length space, as the body says.