riesz-isomorphism-unifies-embedding-and-unembedding-spaces
IN premise — summaries/2026/08/24/park-2023-linear-representation-s2-the-linear-representation-hypothesis.md
Created 2026-08-24T17:11:02+00:00
Under the causal inner product, the Riesz isomorphism (Theorem 3.2) maps each unembedding representation γ̄_W to its corresponding embedding representation λ̄_W via ⟨γ̄_W, ·⟩_C = λ̄_Wᵀ, collapsing the two separate representation spaces into a single unified geometric framework.
Summary
The system tracks two kinds of representations of each variable — an "embedding" form and an "unembedding" form — but this result shows they are not actually separate spaces. Under the system's causal inner product, one is just the geometric mirror of the other, so they can be treated as a single unified space, which means any reasoning done in one form automatically carries over to the other without needing to track two parallel bookkeeping systems.