Duality, Reconstruction, and Structural Toolkit Theorems in Algebraic Phase Theory

We study finite-depth reconstruction frameworks based on representation theory and show that non-rigid reconstruction behaviour is naturally accompanied by intrinsic structural boundaries. Within the finite-depth setting considered in this paper, reconstruction is controlled up to boundary equivalence by the associated filtered representation data together with boundary stratification. We show that algebraic phases satisfying the axioms of Algebraic Phase Theory (APT) are reconstructible up to intrinsic phase equivalence from their filtered representation categories together with their boundary structure. Reconstruction proceeds without boundary collapse on rigidity islands, while globally the remaining ambiguity is governed by intrinsic boundary phenomena. We further study a collection of structural consequences associated with the axioms of APT, including finite generation phenomena, rigidity and obstruction behaviour, finite-depth boundary detectability, and obstruction structures arising from boundary layers. These results apply across the phase models developed in the APT series. Taken together, the results of this paper further develop Algebraic Phase Theory as a structural framework for studying reconstruction, duality, rigidity, and boundary behaviour beyond rigid or semisimple settings.