Mirror-Shell Symmetry and Structural Projection Theory: A Mathematical Framework for Asymmetry Measurement, Geometric Invariance, Structural Dynamics, and Topological Analysis

This research presents a unified mathematical framework for Mirror-Shell Symmetry and Structural Projection Theory (SPT). The study introduces the A_{BY} asymmetry index as a quantitative measure of structural deviation from symmetry and develops a geometric formulation based on invariant relationships, projection operators, phase-space representations, and conservation principles.

The framework is progressively extended from geometric symmetry and apparent solar motion to structural preservation, prediction, and comparative analysis. An applied case study in Andalusian architecture is presented as an independent test of the stability and applicability of the proposed structural indicator, while maintaining a distinction between mathematical regularity and claims of historical intentionality or physical causation.

The final part develops Structural Field Dynamics and Topological Analysis, including periodic structural fields, continuity relations, compatible fluxes, path-ordered matrix evolution, Jacobi determinant identities, global closure conditions, and holonomy. These developments provide mathematical criteria for distinguishing local structural invariance from global structural closure.

The work is organized as a five-part unified research framework combining mathematical formulation, geometric analysis, structural projection, empirical application, and topological field analysis.