# Euclidean Structure The natural geometric structure inherited by prime-coordinate space from the coherence norm and inner product, providing measures of complexity and similarity. ## Definition The prime-coordinate space inherits a natural Euclidean structure from the coherence norm and inner product. The coherence norm ||φ(x)|| measures the intrinsic complexity of an object. The inner product ⟨φ(x),φ(y)⟩ quantifies the degree of prime-factor similarity. Angles between coordinate vectors represent compositional relationships. Orthogonal objects have no common prime factors. ## Mathematical Formulation $ ||φ(x)|| $ $ ⟨φ(x),φ(y)⟩ $ ## Related Concepts - [[uor-c-020|prime-space-geometry]] - [[uor-c-005|coherence-norm]] - [[uor-c-308|coherence-inner-product]] ## Metadata - **ID:** urn:uor:concept:euclidean-structure - **Code:** UOR-C-021