nucleation

CLAIM-187: Whether two systems compute the same way can be tested by quotienting the full admissible symmetry group and measuring what remains: a residual barrier marks a different mechanism, and a feature with no counterpart blocks the comparison

4 min read

Properties8
StatusProposed
Version2
Rolethesis
Warrantinterpretive
Tagsrepresentation-learning, loss-landscapes
Linepragmatic-transport
Rests onCLAIM-263 — Whether two trained networks compute the same way can be tested by quotienting the architecture's full symmetry group and measuring what remains: a barrier that survives alignment marks a different mechanism, connection after alignment means shared feature structure, and the test cannot run where features have no counterpart, CLAIM-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
GroundsTHEORY-112 — Aligning independently trained solutions over the architecture's symmetries removed the linear barrier between them for MNIST MLPs and 32×-wide ResNet-20s on CIFAR-10, two-thirds of it for an ImageNet ResNet-50, and not all of it for VGG-16 or ResNet-20 at 1× width, the barrier after weight matching falling with width; for 6- to 8-layer ViTs and GPT-2s, permutations alone removed 63–90% of it, and adding an orthogonal map on the residual stream, learned against the loss at the path's midpoint, took it to zero or near zero except on BookCorpus (0.42), THEORY-116 — A linear barrier between two minima that survives the removal of symmetries marks a difference in the input attributes the two models rely on: they compute by different mechanisms, THEORY-111 — In image classifiers, between two networks linearly connected by spawning from a shared checkpoint or by permutation alignment (MLPs, and VGG-16, ResNet-20 and ResNet-50 on CIFAR-10 and Tiny-ImageNet), nearly every layer's features at three points on the line were proportional to the interpolation of the endpoints' features, and pairs that were not linearly connected were far from it, THEORY-113 — Merging ResNets trained from different initialisations on disjoint halves of a label set by permuting one onto the other and averaging fell far short of the ensemble; their cross-model feature correlation fell with depth, merging only the early layers recovered most of the ensemble's accuracy, and merging features within each model as well as across both helped at 4× width and above; Stoica et al. attribute the failure to features with no counterpart in the other model, THEORY-115 — Linear connectivity from a shared, already-stable start holds only when the objectives the runs optimise share structure, and breaks as that structure is removed

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The claim

The pragmatic-transport line asks, in several forms, what is invariant under admissible transformations and whether a kind can be identified by it (QUESTION-027, QUESTION-029, QUESTION-036). The mode-connectivity results the record holds are a worked protocol for exactly that, measured:

  1. Quotient the full group. Once the architecture’s symmetries are factored out, most of the linear barrier between independently trained solutions disappears. The group must be the full one: for transformers it includes an orthogonal map on the residual stream, since permutations alone leave a barrier (THEORY-112).
  2. Read the residue. A barrier that survives the removal of symmetries “marks a difference in the input attributes the two models rely on: they compute by different mechanisms” (THEORY-116).
  3. Inside a basin, structure interpolates. In the image classifiers measured, along a low-loss linear path nearly every layer’s features were proportional to the interpolation of the endpoints’ (THEORY-111), so connectedness after alignment is sameness of feature structure.
  4. Know when the test cannot run. Merging fails, Stoica et al. argue, where features have no counterpart in the other model (THEORY-113), and connectivity holds only when the objectives share structure (THEORY-115).

Read across, two communicative realizations would count as the same kind when, after quotienting the admissible transformations, they are connected with no residual barrier on the probes that matter. A residual barrier would be the invariant that separates kinds. A realization with no counterpart, CLAIM-129’s topic kept while the act is lost, is the case where the comparison cannot be made, not one where it fails.

What it does not say

  • It does not say communicative kinds have a loss landscape. The protocol needs an admissible group, a path between realizations and a measure along it. Whether these can be defined for communicative realizations is open, and it is the first defeater.
  • It does not answer QUESTION-027 or QUESTION-036. It gives them a template with measured examples.
  • It inherits CLAIM-142’s caution. A barrier is measured on probes, and what the probes cannot see they cannot separate (CLAIM-193).

Note of 2026-10-12: a case for the protocol

THEORY-108’s phase III is a ready case for this protocol. Independently trained solutions there are poorly connected on a fitted curve, and THEORY-108 itself asks whether the barrier survives permutation alignment. If it does, the protocol reads phase III as a different mechanism from the connected phases, which THEORY-108’s accuracy and output-similarity differences would corroborate. If it does not, the barrier was a symmetry the curve family failed to quotient. Notes on THEORY-108 and THEORY-109 record the tension this would settle.

Note of 2026-10-12: an assessed result on the first defeater

“What it does not say” leaves open whether an admissible group can be defined for communicative realizations. PR #121’s CLAIM-249, granted and Active, assesses that point: “The transformations the line studies are not group elements”, and among stochastic maps on finite observation spaces the invertible core is the relabellings. So the group “has to be a symmetry group of the situation that the analyst chooses”, with its action on each medium specified independently, and TERM-008’s transport is directed and need not be invertible. That does not show no group can be posed; it says the group is chosen, not read off the realizations. CASE-057 is a ready case: a swap of the two persons of a dyad, a group of order 2, which CLAIM-249 notes has no automorphism to move its characters.

Note of 2026-10-13: the protocol is filed separately, and the premise re-pointed

Under ADR-040 the protocol for trained networks is filed on its own as CLAIM-263, the first claim in the learned-structure line, and this claim now rests on it. What stays here is the carry-over to communicative realizations, which the summary already calls the claim’s weak point. The claim also needed only the relativity of observational equivalence to probes, so its rests_on names CLAIM-264, the general form, in place of CLAIM-142. The body’s citation of CLAIM-142 stands.

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