AAIS Mathematical Framework (Phase I): Geometric and Functional-Analytic Foundations of Nonlinear Dissipative Evolution Equations on Compact Riemannian Manifolds

This document constitutes the technical documentation for Phase I of the AAIS Mathematical Framework. It presents a systematic development of nonlinear dissipative evolution equations on compact Riemannian manifolds without boundary, using geometric functional analysis and the theory of infinite-dimensional dynamical systems. The work covers variational reconstruction of generators, semigroup well-posedness and dissipativity, spectral properties of linearized operators, construction of global attractors, and quantitative estimates on their fractal dimension. Standard techniques from Sobolev theory on manifolds, semigroup theory, and dynamical systems are employed throughout. The manuscript also outlines the logical organization and dependency structure of the Phase I components. It is intended as an archival and preprint record for reference and reproducibility purposes.