nucleation

CLAIM-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

4 min read

Properties10
StatusProposed
Version4
Rolethesis
Warrantinterpretive
Tagsrepresentation-learning, mathematics
Linepragmatic-transport
Bears onQUESTION-031 — When does a chosen family of observational probes separate the communicative realizations a task needs told apart, and when does it collapse distinct ones, beyond the cases in which the task's target and the probes' outcome laws are linear in one state?
Rests onCLAIM-264 — Two objects can agree on every probe in a restricted family without being isomorphic, so equivalence under a probe family is coarser than identity unless the family is rich enough, 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
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
ComplementsCLAIM-209 — Probes and statistical models recover only the distinctions identifiable from the relational evidence they are given, and a signature sufficient for a stated family of communicative decisions outside the observed probes can be identified when the full latent factorization is not, CLAIM-142 — Two communicative realizations can agree on every probe in a restricted family without being structurally identical, so observational equivalence is relative to the probes, and only independently justified probes evaluated on held-out cases are evidence that structure was preserved

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The claim

QUESTION-031 asks when a chosen family of probes separates the realizations a task needs told apart, and when it collapses distinct ones. CLAIM-142 says observational equivalence is relative to the probes. For one important family, linear readouts, the record already holds an exact answer:

  • The kernel decides. “What a regularized linear readout can decode from a representation is a function of its normalized kernel” (THEORY-008, proved identities). Two representations with the same normalized kernel are indistinguishable to every such probe, 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”, and a chosen one is fixed by stipulation (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 realizations that differ only by a symmetry of the observed data, and a distinction drawn beyond that comes from the analyst, not from the probes. That gives CLAIM-142’s relativity a precise form in this case.

What it does not say

  • It does not cover nonlinear probes, behavioural probes or human judges, which is most of what the pragmatic-transport line uses. It answers the linear half of QUESTION-031.
  • The four results are about learned representations. That communicative realizations, read through a probe family, have a kernel playing the same role is the filer’s step.
  • THEORY-019 also bears on QUESTION-036, toward a negative answer, as that question’s note records.

“What it does not say” counts human judges among the probes this result does not cover. PR #121’s round of 2026-10-10 contests that placing. CLAIM-142 version 2, on which this claim rests, says “The judges are the criterion (CLAIM-217), not a probe”, while CLAIM-206 treats human same-kind judgements as “themselves one probe family”; that dispute is open. On the first reading this result bounds what an analyst’s linear family can separate when tested against the judges’ standard; on the second it leaves out one probe family among others. CLAIM-217 also addresses QUESTION-031: “QUESTION-031 asks when a family is rich enough; this says rich enough relative to what.” This claim’s “comes from the analyst, not from the probes” is the linear-case form of CLAIM-209’s first clause, now related by complements, and of the part CLAIM-204 says the analyst can certify.

Note of 2026-10-13: the general content is filed separately

Under ADR-040, the linear-readout result is filed on its own as CLAIM-269, in the formal-apparatus line, resting on no claim. This claim now rests on it for the general content, and keeps CLAIM-142, for carrying the result to communicative realizations. Claims in other lines that need only the general content cite CLAIM-269, so setting pragmatic-transport aside does not take it from them.

Note of 2026-10-19: the warrant

The linear-readout result is CLAIM-269’s, in the formal-apparatus line, and its defeat condition is that claim’s. This claim keeps the step that communicative realizations read through a probe family have a kernel that plays the same role. That step is argued, not proved, and whether human judges are a probe family at all is the open dispute between CLAIM-142 and CLAIM-206. So the warrant is interpretive.

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

Its premises are now Active (CLAIM-264, CLAIM-269), and its defeat condition was restated today to its own step: that a linear family used on communicative realizations reads them through one representation with a kernel. The restated test has not yet been weighed against the record’s probe families on communicative realizations, so nothing yet says whether any of them meets it.

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