# Universal Transform Definition The formal mathematical definition of transformations between different dimensional spaces that preserve information coherence through operations on prime coordinates. ## Definition For a signal `S` with universal number representation `φ(S)`, the universal transform `T` between dimensional spaces `D₁` and `D₂` preserves the prime coordinate structure: `T: φₐ(S) → φᵦ(S)` Where `φₐ` represents the coordinate mapping in dimension `D₁` and `φᵦ` represents the mapping in dimension `D₂`. The transform satisfies the coherence preservation property: `‖φₐ(S)‖ₑ = ‖φᵦ(S)‖ₑ` Where `‖·‖ₑ` is the essential norm measuring the information content. For continuous transformations between parameter spaces, we can define: `T_θ: φ(S) → φ_θ(S)` Where `θ` represents a continuous parameter controlling the transformation. This continuous transformation satisfies: `∂/∂θ ‖φ_θ(S)‖ₑ = 0` Ensuring that the essential information content remains invariant along the transformation path. ## Mathematical Formulation $ \text{For a signal } S \text{ with universal number representation } \phi(S)\text{, the universal} $ $ \text{transform } T \text{ between dimensional spaces } D_1 \text{ and } D_2 \text{ preserves} $ $ \text{the prime coordinate structure:} $ $ T: \phi_a(S) \to \phi_b(S) $ $ \text{Where } \phi_a \text{ represents the coordinate mapping in dimension } D_1 \text{ and} $ $ \phi_b \text{ represents the mapping in dimension } D_2\text{.} $ $ \text{The transform satisfies the coherence preservation property:} $ $ \|\phi_a(S)\|_e = \|\phi_b(S)\|_e $ $ \text{Where } \|\cdot\|_e \text{ is the essential norm measuring the information content.} $ $ \text{For continuous transformations between parameter spaces, we can define:} $ $ T_\theta: \phi(S) \to \phi_\theta(S) $ $ \text{Where } \theta \text{ represents a continuous parameter controlling the transformation.} $ $ \text{This continuous transformation satisfies:} $ $ \frac{\partial}{\partial\theta} \|\phi_\theta(S)\|_e = 0 $ $ \text{Ensuring that the essential information content remains invariant along} $ $ \text{the transformation path.} $ ## Related Concepts - [[uor-c-091|transform-properties]] - [[uor-c-092|transform-mechanics]] - [[uor-c-005|coherence-norm]] ## Metadata - **ID:** urn:uor:concept:universal-transform-definition - **Code:** UOR-C-090