superposition-covariance-editability-triangle-v2
IN premise
Created 2026-08-25T03:28:28+00:00
Superposition, covariance geometry, and rank-one editability form an interdependent framework in which each concept provides structural context for the others: over-complete superposition in the residual stream is the primary structural condition that makes covariance/whitening the canonical tool for feature isolation and targeted rank-one edits, covariance geometry serves as the operational coordinate space in which those rank-one updates are formulated, and the structural bounds on such edits—limited to single-fact local corrections rather than novel multi-hop associations—reflect the distributed, locally non-consolidated nature of the underlying superposed representation.
Summary
In a transformer, knowledge is stored by densely packing many features into shared dimensions, and that packing is exactly what lets you use the statistical structure of those dimensions to locate and correct a single stored fact. The catch is that the same structure that makes those local fixes possible also caps them: you can nudge one fact already encoded in the superposition, but you cannot manufacture a brand-new multi-step association through the same mechanism, because the information simply isn't consolidated in a way that supports that.
Dependents
These beliefs depend on this one:
- OUT superposition-covariance-editability-triangle — Superposition, covariance whitening, and rank-one editability form a closed logical triangle in which each property necessitates the others: over-complete superposition requires covariance separation for feature addressability, covariance separation defines the geometric space in which rank-one updates are well-defined, and the boundedness of rank-one editing confirms the addressable space is finite.