# Unique Factorization The essential foundation of the UOR framework that guarantees every non-zero non-unit element can be expressed as a product of primes in an essentially unique way, ensuring unambiguous object representation. The principle of Unique Factorization forms the essential foundation of the UOR framework, guaranteeing that objects have unambiguous prime representations. In a Unique Factorization Domain, every non-zero non-unit element x can be expressed as a product of irreducible elements (primes) in an essentially unique way: x = u · p₁ · p₂ · ... · pₙ, where u is a unit, each pᵢ is prime, and the representation is unique up to reordering of factors and multiplication by units. The power of this theorem lies in its uniqueness clause: if x = u · p₁ · p₂ · ... · pₙ = v · q₁ · q₂ · ... · qₘ, where u and v are units and the pᵢ and qⱼ are primes, then: n = m (same number of prime factors), and after reordering, pᵢ and qᵢ are associates (differ by a unit) for each i. This principle applies across various domains, as seen in examples such as: Integers, where every integer > 1 factors uniquely into primes (e.g., 60 = 2² · 3 · 5); Polynomials, where p(x) = (x-1)²(x+2) has unique factorization into irreducibles; and Gaussian Integers, where a+bi factors uniquely using Gaussian primes. Unique factorization ensures that: Every object has exactly one prime-coordinate representation; The mapping to prime-coordinate space is well-defined; and Objects can be unambiguously compared and analyzed. This principle establishes the foundation for the entire coherence framework by guaranteeing that each object's intrinsic structure can be captured uniquely in its prime spectrum. ## References - [[uor-c-138|factorization-theorem]] - [[uor-c-139|uniqueness-guarantee]] - [[uor-c-140|factorization-examples]] - [[uor-c-141|factorization-significance]] ## Metadata - **ID:** urn:uor:resource:unique-factorization - **Author:** UOR Framework - **Created:** 2025-04-22T00:00:00Z - **Modified:** 2025-04-22T00:00:00Z