Beyond the Conductor Blind Spot: Eigenfunction Identifiability and Planning in Non-Gaussian Worlds

The companion certificate The Conductor Blind Spot proved a sharp, finite fact: on a categorical head whose outcomes carry a cyclic ring structure, the second-order curvature class — the Fisher form shared by natural gradient, K-FAC, Adam's empirical Fisher, and gradient-based attribution — is constitutionally unable to represent an order-three coupling (the Amari–Chentsov cubic) that the head's own information geometry carries. That statement is finite, exact, and lives on a discrete index. This paper asks, and answers, the continuous counterpart: when a representation is learned by the generic self-supervised recipe — pull positive pairs together, keep the embedding whitened — what does it recover, and when is the second-order (linear) picture of that representation complete? We show that the population optimum of this objective recovers, in each latent coordinate, the slowest eigenfunction of the pair-generating transition operator, up to a rotation: a clean, model-free consequence of the Ky Fan maximum principle (a statement about the optimum, not about what a finite encoder trained by gradient descent reaches). That eigenfunction — the slow-feature chart, the coordinate-free object the recovery is "up to" — is a straight line exactly when the latent law is Gaussian in the observed coordinate, and a known, monotone, curved coordinate otherwise. Three theorems follow. (i) Exact recovery: the optimum equals the slow-eigenfunction chart up to a block rotation, so a linear probe recovers the latent factors if and only if the world is Gaussian in that coordinate, and is provably lossy otherwise (other downstream targets may still be linearly readable). (ii) Approximate recovery: under inexact whitening (\ (\) ) and inexact alignment (\ (\) ), the chart is recovered within \ ( (+2/) ²\), where \ (\) is the slow-feature spectral gap; the guarantee degrades gracefully and is governed by a checkable condition, not a tunable. (iii) Planning: because the recovered chart is a diffeomorphism, every optimal-control problem on the true latent transports exactly into the learned coordinates — planning in the representation equals planning in the world, with the warp applied when it is not the identity. The unifying statement is that the conductor blind spot and the failure of linear probing share a trigger of the same kind — departure from second-order sufficiency (\ (₃ 0\) ) — and are quantitatively linked near the Gaussian; they are two registers of one boundary, offset by one in their order index and coinciding at the Gaussian corner. This is a correspondence of model classes that we measure — the finite order-three object regenerated exactly, the lock measured as a square law — not a single diagnostic run end-to-end across both. We give deterministic, exact-arithmetic and numerical certificates for every claim; the finite-categorical certificate independently regenerates the conductor blind spot's published counts, and a cross-register certificate measures the two registers locking together — the recovery-map curvature scales as the square of the leading distributional cumulant, vanishing with it at the Gaussian. A related signature is visible, unforced, in a pretrained language model, where it appears as a middle-layer phenomenon: the heavy non-Gaussian directions of the residual stream tend to be those whose recovered slow feature is curved (a positive but imperfect rank correlation, not an identity — and an observational check well outside the theorem's hypotheses). The consequences for practice — interpretability (linear probes are provably partial off-Gaussian), optimization (the same order-two ceiling as the blind spot), and the auditing of learned world models (identifiability holds, but only up to a recoverable nonlinear chart) — are drawn out explicitly.