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CLAIM-269: A regularized linear readout can tell apart only representations that differ in normalized kernel, and kernels fix a representation only up to the symmetry of what is observed, so no such readout supplies a distinction beyond that symmetry

3 min read

Properties9
StatusActive
Version5
Rolethesis
Warrantdemonstrated
Tagsrepresentation-learning, mathematics
Lineformal-apparatus
Rests onCLAIM-379 — On a finite set of stimuli, regularized linear readouts with ridge b > 0 cannot tell apart two centred representations of the same width exactly when one is an orthogonal transformation of the other, and the unregularized CCA member cannot tell apart representations with the same column space, which differ by an invertible linear map, CLAIM-385 — If a symmetric operator on ℝⁿ commutes with an orthogonal representation of a compact group, each eigenspace is a sum of real irreducibles and each multiplicity a sum of multiples of their dimensions, so SO(2) and ℤ/n for n ≥ 3 force paired eigenvalues; conversely the spectrum fixes neither the group nor a basis inside an eigenspace
GroundsTHEORY-008 — On a finite set of stimuli, what a regularized linear readout can decode from a representation is a function of its normalized kernel, and CKA, CCA and GULP are averages of readout agreement, THEORY-004 — A representation is determined by its centred kernel up to a translation and an orthogonal transformation, and the rotation-aligned distance between two representations equals the Bures distance between their kernels, THEORY-723 — For the final embedding and unembedding of a softmax model, the output distribution fixes the representations only up to an invertible linear map, and whether closeness in distribution bounds representational dissimilarity depends on the distance: not for KL in the saturation regime, but yes for the log-likelihood variance distance, THEORY-018 — Softmax training identifies a language model's final-layer representation only up to an invertible linear map, so no inner product on it is intrinsic, and Park, Choe and Veitch's causal inner product is fixed by a stipulation, THEORY-019 — Symmetry forces spectral degeneracy but degeneracy does not identify a symmetry: an operator commuting with a group has eigenspaces built from its real irreducibles, so abelian rotation groups force pairs over ℝ, and the spectrum fixes neither the group nor a basis inside a multiplet
SupportsCLAIM-193 — A linear probe family can tell apart only realizations whose representations differ in normalized kernel, and kernels fix a representation only up to the symmetry of what is observed: any distinction beyond that is supplied by the analyst

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The claim

For one important probe family, linear readouts, the separating power is known exactly:

  • The kernel decides. What a regularized linear readout can decode from a representation is a function of its normalized kernel (THEORY-008). Two representations with the same normalized kernel are indistinguishable to every such readout, whatever basis they are written in. CKA, CCA and GULP are averages of readout agreement.
  • Kernels fix representations only up to symmetry. A centred kernel fixes a representation up to a translation and an orthogonal map (THEORY-004), and a softmax model’s output distribution fixes its final layer only up to an invertible linear map (THEORY-723).
  • Anything further is stipulated. Softmax training identifies a final layer only up to an invertible linear map, so no inner product on it is intrinsic (THEORY-018).
  • The spectrum does not recover the symmetry. Symmetry forces degeneracy, but degeneracy does not identify a symmetry (THEORY-019).

So a linear probe family separates exactly what the kernel separates. It collapses representations that differ only by a symmetry of the observed data. A distinction drawn beyond that comes from something other than the readouts, such as a basis-dependent probe or the analyst’s choice of inner product (THEORY-018).

Where the record applies it

CLAIM-193 carries the result to communicative realizations read through a probe family, resting on CLAIM-142 for the relativity of observational equivalence there. It now also rests on this claim for the linear result.

What it does not say

  • It does not cover nonlinear probes, behavioural probes or human judges.
  • It does not say any domain other than learned representations has a kernel playing the same role. That is each application’s step.

Note of 2026-10-19: the readout half is held Active

The first bullet above, with what follows from it through THEORY-004, is now CLAIM-379, filed Active with its derivation. For ridge b > 0 the kernel K_X and the Gram matrix determine each other, so readouts collapse exactly the orthogonal orbit of a centred representation; the CCA member collapses the invertible-linear orbit. What this claim adds, and why it stays Proposed, is the fourth bullet: that spectral degeneracy does not identify the symmetry of the observed data (THEORY-019).

Note of 2026-10-29: what holds it back

No premise is now Proposed: THEORY-019 is Active since the twelfth status review, and the fourth bullet also rests on CLAIM-385. What is left is a check of the claim’s own last step, that a distinction beyond the kernel’s symmetry “is supplied by the analyst”, which may be interpretive rather than demonstrated.

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