causal-inner-product-captures-shared-linguistic-subconcept

IN premise — summaries/2026/08/24/park-2023-linear-representation-s12-a-male-sovereign-who-reigns-over-a-kingdom-is-a.md

Created 2026-08-24T17:11:02+00:00

Under the causal inner product, English⇒French shows high alignment with French⇒German and French⇒Spanish (sharing the French sub-concept) but is nearly orthogonal to German⇒Spanish (no shared language); under Euclidean cosine, English⇒French appears nearly orthogonal to all other language-pair concepts.

Summary

How you measure similarity between compound concepts determines whether shared components are visible at all. A causally-aware metric correctly reveals that language pairs sharing a common language genuinely overlap, while the standard angle-based metric erases that structure and makes everything look unrelated, which would mislead any system trying to reason about what these pairs actually have in common.