riesz-map-as-canonical-semantic-isomorphism

OUT derived (depth 4)

Created 2026-08-25T03:13:05+00:00 · Reviewed 2026-08-25T04:28:09+00:00

The Riesz map under the causal (whitened) inner product is the unique canonical isomorphism that makes embedding and unembedding semantically identical, thereby elevating "meaning" from a model-specific activation pattern to a well-defined algebraic object in a model-independent vector space

Justifications

SL — The Riesz map (base) provides the mathematical unification of embedding and unembedding; covariance-geometry-as-operational-semantic-space (depth-3) establishes that this geometry IS the semantic coordinate system. Together: the Riesz map is not merely a convenience but THE canonical identification, giving meaning a precise algebraic definition independent of the specific model.

Antecedents (all must be IN):

IN park-2023-riesz-map-unifies-embedding-unembedding — Under the causal inner product, the Riesz map γ̄ ↦ ⟨γ̄, ·⟩_C sends each unembedding representation γ̄_W exactly to its embedding representation λ̄_W; in the transformed space with A = M^{1/2}, embeddings and unembeddings are literally equal (g̃_W = l̃_W) and the Euclidean inner product becomes the causal one.
OUT covariance-geometry-as-operational-semantic-space — The covariance/whitening geometry (second-moment matrices) is the operational definition of semantic coordinate space in LLMs: it simultaneously parameterises feature interpretation (SAE decoder space, Park polytopes), similarity evaluation (cosine→Spearman pipeline), and knowledge modification (ROME rank-one updates), and this structure converges across model families.

Dependents

These beliefs depend on this one:

OUT riesz-map-as-key-value-semantic-bridge — The Riesz isomorphism under the causal inner product is the mathematical "compiler" that translates between ROME's key representation (MLP input activation) and value representation (MLP output projection), unifying the parametric knowledge storage mechanism as a single dual-geometric object rather than two independent weight matrices.
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.
OUT riesz-map-unifies-full-logical-structure — The Riesz map under the causal inner product is the unique mathematical object that simultaneously proves cross-model convergence is necessary (not contingent), enables the polytope/orthogonal space decomposition, and provides the canonical embedding↔unembedding isomorphism—collapsing three independent observations into theorems of a single structural premise (over-complete superposition in a shared substrate).