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Algebraic Phase Theory VI: Duality, Reconstruction, and Structural Toolkit Theorems

Joe Gildea

Abstract

We prove that any functorial finite-depth reconstruction framework based on representation theory satisfies a strict dichotomy: it either collapses into a rigid regime or necessarily admits intrinsic structural boundaries. In the non-rigid case, reconstruction is uniquely determined up to canonical boundary collapse, and no boundary-free reconstruction theory with the same formal properties can exist. We establish that algebraic phases are \emph{information-complete}. Any phase satisfying the axioms of Algebraic Phase Theory (APT) is uniquely determined, up to intrinsic phase equivalence, by its filtered representation category together with its boundary stratification. Reconstruction is exact on rigidity islands and fails globally only through canonical and unavoidable boundary phenomena. We further show that the axioms of APT force a minimal structural toolkit, including canonical finite generation, rigidity--obstruction equivalence, finite-depth boundary detectability, and the existence of universal obstruction objects. These results apply uniformly across all phase models developed in the APT series. Taken together, the results of this paper complete the foundational development of Algebraic Phase Theory and position it as a canonical and inevitable reconstruction framework extending classical duality theories beyond rigid and semisimple regimes.

Algebraic Phase Theory VI: Duality, Reconstruction, and Structural Toolkit Theorems

Abstract

We prove that any functorial finite-depth reconstruction framework based on representation theory satisfies a strict dichotomy: it either collapses into a rigid regime or necessarily admits intrinsic structural boundaries. In the non-rigid case, reconstruction is uniquely determined up to canonical boundary collapse, and no boundary-free reconstruction theory with the same formal properties can exist. We establish that algebraic phases are \emph{information-complete}. Any phase satisfying the axioms of Algebraic Phase Theory (APT) is uniquely determined, up to intrinsic phase equivalence, by its filtered representation category together with its boundary stratification. Reconstruction is exact on rigidity islands and fails globally only through canonical and unavoidable boundary phenomena. We further show that the axioms of APT force a minimal structural toolkit, including canonical finite generation, rigidity--obstruction equivalence, finite-depth boundary detectability, and the existence of universal obstruction objects. These results apply uniformly across all phase models developed in the APT series. Taken together, the results of this paper complete the foundational development of Algebraic Phase Theory and position it as a canonical and inevitable reconstruction framework extending classical duality theories beyond rigid and semisimple regimes.
Paper Structure (7 sections, 21 theorems, 67 equations, 1 figure)

This paper contains 7 sections, 21 theorems, 67 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Structural Framework and the Limits of Classical Reconstruction
  3. Duality Principles and Testing Objects
  4. Intrinsic Reconstruction and Inevitability
  5. Toolkit Theorems Forced by the Axioms
  6. A Flagship Reconstruction Example over $\mathbb{F}_2[u]/(u^2)$
  7. Conclusion

Key Result

Theorem 2.1

There exist functorial filtered representation categories, arising from finite-depth phase interaction and equipped with intrinsic defect stratification and finite termination, for which no reconstruction procedure based on Tannakian duality, Lie-theoretic methods, or operator-algebraic duality reco

Figures (1)

  • Figure 1: Structural organisation of the six papers. Papers I--III develop the analytic-to-algebraic extraction, establish canonical defect and filtration, and identify Frobenius and quantum phenomena. Papers IV--V introduce categorical structure, boundary behaviour, rigidity islands and deformation theory. Paper VI completes the programme with duality and reconstruction.

Theorems & Definitions (53)

  • Theorem 2.1
  • proof
  • Theorem 3.1
  • proof
  • Remark 3.2
  • Definition 3.3
  • Theorem 3.4
  • proof
  • Corollary 3.5
  • proof
  • ...and 43 more