# Universal Identity A mathematical framework for identity based on universal numbers that enables coherent identification across representational domains through prime coordinate structures. Identity is a fundamental concept in any computational system, traditionally implemented through various mechanisms like usernames, public keys, UUIDs, or biometric data. The Internet Substrate Protocols reimagine identity at a more fundamental level, grounding it in the mathematical properties of universal numbers. By representing identity through prime coordinate structures, the system achieves a form of identity that is simultaneously unique, verifiable, transformable, and coherence-preserving. This approach transcends traditional identity mechanisms by providing a mathematical foundation that connects identity directly to the fundamental structure of the information space. Universal Identity operates as a natural extension of the UOR framework, where the prime factorization of identity representations provides intrinsic properties that enable sophisticated identity operations while maintaining essential coherence. Universal Identity is characterized by several key insights: Mathematical Foundation where identity emerges from the intrinsic mathematical properties of universal numbers rather than arbitrary assignments or conventions; Cross-Domain Coherence where identity representations maintain coherence when transformed across different domains or representational systems; Intrinsic Verifiability where the prime structure of identity representations enables intrinsic verification without requiring external authorities; Scale Invariance where identity mechanisms operate consistently across different scales, from individual devices to global systems; and Self-Reference where identities can reference themselves and their own properties through their prime coordinate structure. For any entity E in the system, its identity representation is defined as: ID(E) = ∏_p p^(φ(E)(p)), where φ(E) maps the entity to its prime coordinate representation. The verification of identity involves coherence evaluation: Verify(ID₁, ID₂) = (d_C(ID₁, ID₂) < ε), where d_C is a coherence-preserving distance metric and ε is a threshold value. For transformations between identity domains, we require: T(ID) = ID' such that C(ID, ID') > τ, where C is a coherence metric and τ is a minimum coherence threshold. The system also defines identity composition operations: ID_A ⊗ ID_B = ∏_p p^(f(φ(ID_A)(p), φ(ID_B)(p))), where f is a composition function that preserves essential properties. The universal identity framework bridges UOR principles and practical networking protocols through explicit mathematical mappings. The Internet Substrate protocols implement a precise mapping from UOR prime coordinates to practical addressing schemes: A(φ(E)) = Ψ(φ(E)₁, φ(E)₂, ..., φ(E)_n), where A is the address mapping function, φ(E) is the prime coordinate representation of entity E, and Ψ is the projection function that maps infinite-dimensional prime coordinates to finite address spaces. For IPv6 compatibility, we define: IPv6(φ(E)) = Hash₁₂₈(π(φ(E))), where Hash₁₂₈ is a 128-bit hash function and π(φ(E)) is a projection of the prime coordinates that preserves essential identity properties. For human-readable identifiers: Name(φ(E)) = WordMap(TopK(φ(E))), where TopK extracts the K most significant prime coordinates and WordMap maps these coordinates to pronounceable word sequences. The practical implementation of identity verification follows a coherence-based protocol: First, Challenge Generation where C = Random(Seed, φ(E)), where Random generates a challenge based on the claimed identity; Second, Response Computation where R = Sign(C, φ(E)), where Sign produces a prime-structure signature of the challenge; and Third, Verification where Verify(C, R, φ(E)) = (d_C(Compute(C, R), Expected(C, φ(E))) < ε), where verification succeeds if the coherence distance is below threshold. This protocol guarantees: Zero-Knowledge Property with verification without revealing identity secrets; Quantum Resistance with security against quantum computational attacks; and Cross-Domain Validity with consistent verification across all network domains. The universal identity system preserves structural relationships through homomorphic properties: φ(E₁ ⊕ E₂) = f(φ(E₁), φ(E₂)), where ⊕ represents entity composition and f is a structure-preserving function. This enables: Hierarchical Identities with organizational structures with coherent identity relationships; Derived Identities with authorized sub-identities with verifiable derivation paths; and Federated Verification with decentralized verification with mathematical coherence guarantees. Universal Identity enables several fundamental operations: Identity Generation with creation of identities based on intrinsic properties of entities; Verification with confirmation of identity claims through coherence evaluation; Transformation with conversion of identities between different representational domains; Delegation with transfer of identity authority while maintaining verifiable connections; Composition with combining identities to create composite identities with verifiable relationships; and Revocation with mathematical invalidation of identity relationships. The Universal Identity system includes several interconnected components: Prime Coordinate Mapper that converts entity properties to prime coordinate representations; Coherence Verifier that evaluates the coherence between identity claims and entity properties; Transformation Engine that enables consistent identity representation across different domains; Relation Manager that maintains verifiable relationships between different identities; and Revelation Control that manages what aspects of identity are revealed in different contexts. This approach to identity represents a paradigm shift from traditional models. In conventional systems, identity is typically established through external authorities (certificate authorities, identity providers) or consensus mechanisms (blockchain-based identity). These approaches fundamentally separate the identity from the entity it represents, treating identity as an assigned property rather than an intrinsic characteristic. Universal Identity, in contrast, treats identity as an inherent mathematical property that emerges directly from the entity's prime coordinate representation. This intrinsic approach enables mathematical guarantees about identity properties that external assignment systems cannot provide. The mathematical foundation of Universal Identity connects it to several advanced mathematical fields: Number Theory through the properties of prime factorizations and their uniqueness; Category Theory through the functorial relationships between identity representations in different domains; Information Theory through the quantification of identity information content and its preservation; and Algebraic Topology through the study of invariant structures across transformations. These connections provide a rich theoretical framework for understanding and extending identity mechanisms beyond traditional computational approaches. Universal Identity enables several transformative capabilities: Self-Sovereign Identity with true self-sovereign identity grounded in mathematical properties rather than external authorities; Cross-Domain Authentication with seamless authentication across different systems while maintaining coherence guarantees; Selective Disclosure with mathematically verifiable disclosure of specific identity attributes without revealing others; Decentralized Trust with trust mechanisms that don't require central authorities but emerge from mathematical properties; Identity Preservation with maintenance of essential identity properties through transformations and over time; Quantum-Resistant Identity with identity mechanisms resistant to quantum attacks due to their mathematical foundation; and Hierarchical Identity with mathematically verifiable hierarchies of identity with consistent properties across levels. Universal Identity transcends traditional identity systems in several key ways: Beyond Public Key Infrastructure where, while PKI relies on computational hardness assumptions, Universal Identity is grounded in intrinsic mathematical properties; Beyond Biometrics where, rather than physical characteristics, identity emerges from intrinsic information structure; Beyond Federated Identity where it eliminates the need for federation by providing inherently consistent identity across domains; and Beyond Blockchain Identity where it provides mathematical guarantees of identity properties without requiring consensus mechanisms. Universal Identity builds directly on the Digital Twin Framework and Tripartite Kernel, while providing the foundation for Media Type Definition and access control throughout the Internet Substrate Protocols. It represents the practical application of UOR's mathematical principles to one of the internet's most fundamental challenges—establishing trustworthy identity. ## References - [[uor-c-118|identity-foundation]] - [[uor-c-119|identity-protocols]] - [[uor-c-120|identity-operations]] - [[uor-c-121|identity-applications]] ## Metadata - **ID:** urn:uor:resource:universal-identity - **Author:** UOR Framework - **Created:** 2025-04-22T00:00:00Z - **Modified:** 2025-04-22T00:00:00Z