# Prime Coordinates A powerful mapping system that transforms objects into vectors in an exponent space, revealing their intrinsic structure and converting multiplicative relationships to additive ones. Prime coordinates provide a powerful mapping system that transforms objects into vectors in an exponent space, revealing their intrinsic structure. For any object x in a domain with unique factorization, the prime-coordinate map φ assigns to x its vector of prime exponents: φ(x) maps each prime p to its exponent in the factorization of x; the result is a vector in the exponent space (typically Z^P for finite products); and this mapping is a homomorphism where φ(x·y) = φ(x) + φ(y). The prime-coordinate representation has several remarkable properties: Injectivity, where distinct objects map to distinct coordinate vectors; Linearization, where multiplication becomes vector addition in coordinate space; Dimension reduction, where infinite objects are represented by finite vectors; and Base independence, where representation is intrinsic, not tied to numeral systems. Viewing prime coordinates geometrically reveals deep structure: Each prime corresponds to an orthogonal dimension; Objects become points in a discrete lattice; Divisibility relations become geometric containment; and Prime powers trace rays along coordinate axes. This coordinate system transforms the multiplicative structure into an additive vector space, making complex patterns of factorization visually and algebraically accessible. ## References - [[uor-c-142|prime-coordinate-map]] - [[uor-c-143|coordinate-properties]] - [[uor-c-144|coordinate-geometry]] - [[uor-c-145|coordinate-significance]] ## Metadata - **ID:** urn:uor:resource:prime-coordinates - **Author:** UOR Framework - **Created:** 2025-04-22T00:00:00Z - **Modified:** 2025-04-22T00:00:00Z