park-2024-inner-product-zero-high-probability
IN premise — summaries/2026/08/24/park-2024-categorical-hierarchical-concepts-sR-references-chunk-2.md
Created 2026-08-25T02:58:26+00:00
The algebraic derivation in Eqs. G.3–G.5 shows the inner product between the child–parent difference vector and the parent vector equals 0 with high probability as sequence length grows.
Summary
As sequences grow longer, each new step in the process points in a direction that is essentially perpendicular to the state it is building from, and that near-perpendicularity becomes more reliable. In practice, this means the system accumulates genuinely new, non-redundant information at each step rather than just re-amplifying what already exists, giving a structural guarantee about how representations evolve over length.