AAIS Phase III: Relational Pre-Metric Geometry and Nonlinear Evolution Operators

AAIS Phase III Mathematics — Relational Spacetime Formalism This record presents the formal structure of Phase III Mathematics within the AAIS framework, focusing on the emergence of spacetime as a relational and constraint-based construct. Phase III replaces background-dependent spacetime assumptions with a relational model built on: • Accessibility relation 𝒜 defining state connectivity • Effective dimension D_eff emerging from constraint density • Relational time parameter τ derived from ordering consistency • Interaction dynamics reformulated as constraint alignment processes Spacetime is treated as an emergent structure arising from informational and topological relations rather than a fundamental geometric background. This phase forms the critical bridge between the scalar-geometric foundations established in Phase I, the perturbative regimes of Phase II, and the operator-algebraic and spectral formulations developed in later stages of the AAIS hierarchy. It provides the relational abstract overlay upon which co-equal physical projections and the full nonlinear evolution framework are constructed.