space-decomposition-under-superposition
OUT derived (depth 5)
Created 2026-08-25T03:13:06+00:00 · Reviewed 2026-08-25T03:24:16+00:00
The full LLM semantic space admits a clean algebraic decomposition into categorical polytope subspaces (discrete concepts) and hierarchical orthogonality subspaces (graded taxonomic structure) within the covariance-geometric framework, providing a complete account of how discrete and graded meaning coexist in a single over-complete vector space
Justifications
SL — park-2025-direct-sum-space-decomposition provides the decomposition theorem (categorical ⊕ hierarchical); superposition-covariance-editability-triangle (depth-4) provides the geometric framework that makes the decomposition well-defined. Together: the superposition structure is not a source of ambiguity but of clean algebraic decomposition, unifying discrete and graded semantics in one space.
Antecedents (all must be IN):
- 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.
- OUT superposition-covariance-editability-triangle — Superposition, covariance whitening, and rank-one editability form a closed logical triangle in which each property necessitates the others: over-complete superposition requires covariance separation for feature addressability, covariance separation defines the geometric space in which rank-one updates are well-defined, and the boundedness of rank-one editing confirms the addressable space is finite.
Dependents
These beliefs depend on this one:
- OUT riesz-map-unifies-full-logical-structure — The Riesz map under the causal inner product is the unique mathematical object that simultaneously proves cross-model convergence is necessary (not contingent), enables the polytope/orthogonal space decomposition, and provides the canonical embedding↔unembedding isomorphism—collapsing three independent observations into theorems of a single structural premise (over-complete superposition in a shared substrate).
- OUT sae-resolution-of-space-decomposition — SAE's expansion-ratio scaling (broad→specific features) is the operational resolution mechanism for the algebraic space decomposition: coarse SAEs resolve the polytope hulls (categorical structure), fine SAEs resolve individual vertices (entity-level features), and the neighborhood adjacency graph is the polytope edge structure visible at whichever resolution is chosen.