causal-inner-product-has-d-degrees-freedom
IN premise — summaries/2026/08/24/park-2023-linear-representation-s4-experiments.md
Created 2026-08-24T17:11:03+00:00
The causal inner product is not unique: for d mutually causally separable concepts forming a basis, the metric has d degrees of freedom parameterized by a positive diagonal matrix D, with D=I_d being a canonical convention (yielding M=Cov(γ)⁻¹) rather than a theorem, and the Euclidean inner product is generally not a valid causal inner product unless M=I satisfies the constraint.
Summary
When concepts are causally independent, there is no single "correct" way to measure their geometric relationships; you get to choose among a family of valid metrics, and picking the standard identity scaling is a convenience convention rather than a forced result. This matters because the ordinary dot product most code defaults to is generally the wrong tool in this setting, so any system that assumes Euclidean geometry is silently making an unjustified choice that could distort causal inference.