# Universal Valuations The coherent system of valuations on universal numbers derived from their prime-coordinate structure, including prime-indexed valuations and the Archimedean meta-valuation. ## Definition Universal numbers support a coherent system of valuations derived from their prime-coordinate structure: Prime-Indexed Valuations For each prime `p`, there exists a valuation `v_p` on `𝕌` defined by: `v_p(α) = Re(e_p)` where `e_p` is the p-th coordinate in the prime-coordinate representation of α. Archimedean Meta-Valuation The complex embedding induces an Archimedean valuation: `v_∞(α) = log|z|` where `z` is the complex value of α. Product Formula These valuations satisfy a generalized product formula that extends the classical formula of algebraic number theory: Theorem 4 (Universal Product Formula): For any non-zero universal number `α ∈ 𝕌*`: `v_∞(α) + ∑_{p prime} v_p(α)·log(p) = 0` This formula establishes a fundamental conservation principle across all valuations, unifying the perspectives of complex and p-adic analysis. ## Mathematical Formulation $ \text{For each prime } p, \text{ there exists a valuation } v_p \text{ on } \mathbb{U} \text{ defined by:} $ $ v_p(\alpha) = \text{Re}(e_p) $ $ \text{where } e_p \text{ is the p-th coordinate in the prime-coordinate representation of } \alpha. $ $ \text{The complex embedding induces an Archimedean valuation:} $ $ v_\infty(\alpha) = \log|z| $ $ \text{where } z \text{ is the complex value of } \alpha. $ $ \text{Theorem 4 (Universal Product Formula): For any non-zero universal number } \alpha \in \mathbb{U}^*: $ $ v_\infty(\alpha) + \sum_{p \text{ prime}} v_p(\alpha) \cdot \log(p) = 0 $ ## Related Concepts - [[uor-c-034|universal-number]] - [[uor-c-035|universal-prime-representation]] ## Metadata - **ID:** urn:uor:concept:universal-valuations - **Code:** UOR-C-045