riesz-map-as-unified-operational-framework-v2
IN premise
Created 2026-08-25T04:32:40+00:00
The Riesz map under the causal (whitened) inner product serves as a unifying canonical isomorphism that both (a) renders embedding and unembedding semantically equivalent as a well-defined algebraic object in a model-independent vector space, and (b) provides a coherent geometric translation between ROME's key (MLP input activation) and value (MLP output projection) representations, reframing what appear as two independent weight matrices as a single dual-geometric object. In this sense, the Riesz map offers a shared algebraic framework under which semantic measurement and parametric editing can be understood as related views of one underlying structure, rather than as fully independent mechanisms.
Summary
This claim says that a single mathematical tool (the Riesz map) can make "reading what a model knows" and "writing new knowledge into a model" into two faces of the same operation rather than two unrelated tricks. Practically, that means the system no longer has to treat measurement and editing as independent ad-hoc mechanisms; they share one underlying structure, which gives the framework a consistent geometric logic and reduces the risk of the two sides drifting out of alignment.
Dependents
These beliefs depend on this one:
- OUT riesz-map-as-unified-operational-framework — The Riesz map under the causal inner product is the unique canonical object that simultaneously defines the evaluation coordinate system (cosine/Spearman as inner-product measurement), the editing mechanism (ROME's key-value bridge via pre-/post-activation isomorphism), and the semantic equivalence of embedding and unembedding—making all three operations different projections of a single canonical structure.