# UOR Generalization The generalization of the Universal Object Reference framework to various mathematical domains beyond integer factorization. The UOR framework can be generalized to various mathematical domains beyond integer factorization. Theorem (General UOR Existence): Let D be any UFD (or GCD domain satisfying factorization axioms). Then the prime-coordinate functor φ: D^× → ℤ^P is well-defined, exact, and admits an extension to polynomial rings D[x], localizations D_S, and formal power series D[[x]], preserving unique factorization in each case. Concretely, UOR applies whenever a domain admits a notion of irreducible generators P and a UFD-like factorization property. Key examples include: - Polynomial Rings D[x]: Irreducible monic polynomials in D[x] serve as primes; φ encodes factor degrees and coefficients via multi-index valuations. - Combinatorial Monoids: In incidence algebras, connected components decompose uniquely, with φ capturing component counts. - Localization & Sheaf Domains: Local UFDs obtained via inverting primes (e.g. local rings at maximal ideals) carry local φ maps, gluing into sheaves over Spec(D). In each context, φ remains injective, multiplicative-additive, and coherence‑minimizing, yielding a universal coordinate atlas. This universality hints at a grander vision: UOR as a spectrum-level invariant extending beyond first-order domains into derived categories, spectral stacks, and even the realm of homotopy types, where "primes" become indecomposable objects and φ a homotopical coordinate functor. ## References - [[uor-c-031|general-uor-existence]] - [[uor-c-032|polynomial-ring-extension]] - [[uor-c-033|combinatorial-monoid-extension]] ## Metadata - **ID:** urn:uor:resource:generalization - **Author:** UOR Framework - **Created:** 2025-04-22T00:00:00Z - **Modified:** 2025-04-22T00:00:00Z