Why it is a question
THEORY-243 now states an existence result. In the identity network x ↦ x(1, 0, 1) ↦ (0, 2, 1)·h, aligning the input with the third unit and aligning it with (h₁ + h₂)/√2 pass every interchange intervention on the input equally (Makelov et al., LIT-968; Wu et al., LIT-961). So interventions on that variable alone do not fix which direction mediates it. The general reading, that the mediating direction is fixed only relative to a chosen decomposition into units, needs more than one constructed case.
An answer is one of the following:
- A trained-network tie. In a trained network, two alignments of the same high-level variable that differ by a change of basis crossing component boundaries and agree on every interchange intervention drawn from the data distribution. Then a showing of what, if anything, separates them.
- A uniqueness bound. A proof that, for a stated class of networks and high-level models, alignments that agree on all such interventions for every aligned variable are unique up to a stated symmetry.
Further cases where one alignment scores higher than another cannot settle it, since the question is about ties under a fixed set of interventions.
Note of 2026-10-26: units fixed by symmetry, and a benchmark that cannot see ties
Two works read on 2026-10-26 for T24 of the exchange “Experimental Infrastructure Review” (QUESTION-275) bear on which decomposition into units fixes a mediating direction.
- Units fixed by the data’s symmetries (THEORY-981). Musat, Pimentel, Zucchet and Hofmann compute, for attention-only transformers on block-list inductive tasks with fixed orthogonal write maps, the subspace of weights the data’s symmetry group fixes, and gradient descent never leaves it. Its basis of elementary token and position actions is a decomposition chosen neither by convention (Wu et al.’s reading, THEORY-802) nor by privileging the network’s components (Makelov et al.’s), but by the task’s symmetry group. It is not an answer of either kind this question names: it says nothing about interchange interventions, and it holds only inside the idealization. It suggests where a uniqueness bound might be sought, for networks whose training the data’s symmetries confine.
- A benchmark that scores one alignment at a time (THEORY-968). MIB’s causal-variable track ranks localization methods by mean interchange-intervention accuracy, alignment by alignment. A tie between two basis-related alignments would show as two equal scores and is never looked for. Its winning method, distributed alignment search, is trained against the causal model it is scored on.
The question is unchanged.
Note of 2026-10-27: ties by construction, not ties found
The core circuits works, read on 2026-10-26 at the owner’s request, add four facts about the instruments. None is a trained-network tie of the kind the question asks for.
- DAS fixes subspaces, not directions. In Geiger et al.’s Definition 3 (LIT-1551) the rotation enters an intervention only through the projector onto the learned subspace, so rotations that select the same subspaces give identical interventions. DAS alignments are therefore unique at most up to rotations within the learned subspaces: a stated symmetry, but of the instrument, not of networks. The observation is the record’s (NOTE-1192), not the paper’s.
- Causal scrubbing ties by construction basis-related hypotheses that induce the same partitions of the data (THEORY-978). So does any method that scores nodes by their effect on one metric under one family of contrasts, sparse feature circuits among them (NOTE-1187).
- The basis changes the verdict. DAS’s toy and its trained-network results (THEORY-983) are THEORY-243’s converse: a learned rotation passes where the neuron basis fails.
- A wide search can manufacture alignment. On frozen random networks DAS’s IIA rises above chance once the layer is wide (THEORY-1002), so a tie found by a supervised search in a wide layer would need that control.
The source of the basis freedom in transformers, the residual stream’s lack of a privileged basis, is stated in Elhage et al.’s framework (LIT-1534; THEORY-802’s note of 2026-10-27). The neighbouring question, whether two dictionaries of equal fidelity give the same validated circuit, is filed as QUESTION-274. The question is unchanged.