Related papers: A Structural Criterion for the Applicability of Al…
We develop Algebraic Phase Theory (APT), an axiomatic framework for extracting intrinsic algebraic structure from phase based analytic data. From minimal admissible phase input we prove a general phase extraction theorem that yields…
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…
We develop the representation theory intrinsic to Algebraic Phase Theory (APT) in regimes where defect and canonical filtration admit faithful algebraic realisation. This extends the framework introduced in earlier work by incorporating a…
We complete the foundational architecture of Algebraic Phase Theory by developing a categorical and $2$-categorical framework for algebraic phases. Building on the structural notions introduced in Papers~I-III, we define phase morphisms,…
We develop a general boundary calculus for algebraic phases and use it to formulate an intrinsic structural framework for deformation and obstruction phenomena. Structural boundaries are shown to be finitely detectable and canonically…
Since Ref. [1] shows the emergence of non-Abelian fusion rules in some examples of a class of Abelian models, but does not prove whether these rules also exist in other cases, the purpose of this paper is to present such proof emphasizing…
We study the structure of local algebras in relativistic conformal quantum field theory with phase boundaries. Phase boundaries are instances of a more general notion of boundaries that give rise to a variety of algebraic structures. These…
We introduce an adiabatic perturbation theory for quantum systems with degenerate energy spectra. This perturbative series enables one to rigorously establish conditions for the validity of the adiabatic theorem of quantum mechanics for…
We develop the quantum component of Algebraic Phase Theory by showing that quantum phase, Weyl noncommutativity, and stabiliser codes arise as unavoidable algebraic consequences of Frobenius duality. Working over finite commutative…
A deductive system is structurally complete if its admissible inference rules are derivable. For several important systems, like modal logic S5, failure of structural completeness is caused only by the underivability of passive rules, i.e.…
We consider an example of a system with two degrees of freedom admitting separation of variables but having a subset of codimension 1 on which the 2-form defining the symplectic structure degenerates. We show how to use separation of…
The Bethe Ansatz provides exact solutions for certain interacting quantum many-body systems, yet its success is confined to narrow regimes and breaks down abruptly outside them. Despite extensive developments in integrable systems, a…
Different types of reasoning impose different structural demands on representational systems, yet no systematic account of these demands exists across psychology, AI, and philosophy of mind. I propose a framework identifying four structural…
In this paper, the algebraic Bethe ansatz with periodic boundary conditions is used to investigate trigonometric vertex models associated with the fundamental representations of the non-exceptional Lie algebras. This formulation allow us to…
We study structural limitations of purely algebraic reasoning in the analysis of arithmetic dynamical systems. Rather than addressing the truth of specific conjectures, we introduce a fragment - relative notion of algebraic refutability for…
Recent studies have unveiled new possibilities for discovering intrinsic quantum phases that are unique to open systems, including phases with average symmetry-protected topological (ASPT) order and strong-to-weak spontaneous symmetry…
Materials science has adopted the term of auxetic behavior for structural deformations where stretching in some direction entails lateral widening, rather than lateral shrinking. Most studies, in the last three decades, have explored…
We show that the operational quantum formalism -- the Born rule, L\"uders state updating, quantum interference, non-Markovian effective dynamics, and Bell inequality violation at the Tsirelson bound $2\sqrt{2}$ -- arises within…
We introduce a perturbative approach to solving the time dependent Schroedinger equation, named adiabatic perturbation theory (APT), whose zeroth order term is the quantum adiabatic approximation. The small parameter in the power series…
We propose a novel constructive framework for approaching the Hodge Conjecture via explicit degenerations. Building on limiting mixed Hodge structures (LMHS), we formulate a criterion under which a rational class of type (p, p) on a smooth…