sae-neighborhood-as-directsum-navigation
OUT derived (depth 6)
Created 2026-08-25T03:50:14+00:00 · Reviewed 2026-08-25T04:02:18+00:00
The SAE feature neighborhood (Golden Gate Bridge → Alcatraz → San Francisco → California) is not merely a navigation path within a single categorical polytope but the operational traversal algorithm for the full direct-sum decomposition of the semantic space: it simultaneously navigates the categorical subspace (discrete concepts) and the hierarchical orthogonal subspace (WordNet parent-child structure), validated across Gemma-2B and LLaMA-3-8B.
Justifications
SL — The SAE neighborhood was previously understood as navigating a single polytope; the direct-sum decomposition (Theorem 8 + Proposition 9) shows the full space is a product of categorical and hierarchical subspaces. The same SAE neighborhood traversal (e.g., Bridge→SF→California) is the concrete algorithm for navigating *both* subspaces simultaneously, making it the unified navigation primitive for the entire geometric structure.
Antecedents (all must be IN):
- OUT sae-neighborhood-as-polytope-navigation — SAE feature neighborhoods (e.g., Golden Gate Bridge → Alcatraz → San Francisco → California) are the operational navigation algorithm for the categorical polytope geometry: each SAE feature is a polytope vertex, the neighborhood structure is the polytope edge adjacency, and cross-model universality confirms this polytope is a shared semantic object rather than a model-specific artifact.
- IN park-2025-direct-sum-space-decomposition — The combination of polytope representations and hierarchical orthogonality (Theorem 8) implies the full representation space decomposes as a direct sum of orthogonal subspaces, one per level of the hierarchy.
Unless (any of these IN defeats this justification):
- IN sae-neighborhood-as-directsum-navigation-v2 — The SAE feature neighborhood (Golden Gate Bridge → Alcatraz → San Francisco → California) serves as an operational navigation algorithm for the categorical polytope geometry, with cross-model universality indicating a shared semantic object rather than a model-specific artifact. Combined with hierarchical orthogonality (Theorem 8), this suggests the full representation space may decompose as a direct sum of orthogonal subspaces, one per hierarchical level. However, the antecedents do not explicitly establish that the same neighborhood traversal simultaneously navigates both the categorical and hierarchical subspaces of such a decomposition; they confirm the categorical-polytope navigation and imply the structural decomposition separately.