A Sub Quantum Mathematical Framework : Category Theoretic and Algebraic Geometric Foundations (IEEE)

Yoon A Limsuwan

Yoon A Limsuwan

changzheng zhou, ZIQING zhou

This paper establishes the theoretical framework of Observer Algebraic Geometry, modeling the observer as a coherent sheaf structure on a noncommutative algebraic variety. Based on the area spectrum of loop quantum gravity and the spectraltriple of noncommutative geometry, we prove that the algebraic dimension of theminimal observer unit is approximately 62.5, a value determined by the BarberoImmirzi parameter. Within this framework, the observation act is reinterpreted asa mirror equivalence between the derived category and the Fukaya category, andthe observation backreaction is described by a nonlocal operator whose algebraicstructure leads to a fundamental bound on the cognitive-physical uncertainty. Thetheory predicts an observer horizon at the Planck scale, a sheaf cohomology criterion for the emergence of consciousness in AI systems, and characteristic oscillationmodes in the cosmic microwave background radiation. This framework incorporatesthe observer into the geometrization program, providing a rigorous mathematicalformulation for the quantum measurement problem.

Yoon A Limsuwan

Kai Xu

The Global Hilbert Space Boundary Screening Theory (GHS-BST) defines physical reality as the projection of informational structures through boundary screening. This paper provides the rigorous mathematical foundations for the theory. We employ category theory to reformulate the global Hilbert space \mathcal{H}{\text{global}} and the boundary screening mechanism, and topos theory as the mathematical framework for information ontology. We construct the information category \mathcal{I}, whose objects are all possible informational structures and whose morphisms are transformations between informational structures. We prove that the five axioms of GHS-BST are consistent within this mathematical framework, and demonstrate that Hilbert space is a realization of the information category under specific representational conditions. Furthermore, we introduce the von Neumann algebraic criterion for indecomposability: \mathcal{H}{\text{global}} is indecomposable if and only if its associated operator algebra \mathcal{A}_{\text{global}} has trivial center (i.e., is a Type III factor). This provides a rigorous mathematical definition of "informational primacy." Keywords: Category Theory; Topos Theory; Information Ontology; von Neumann Algebra; Type III Factor; GHS-BST; Mathematical Foundations

Ritabrata Biswas, Zhumayev Asset ZHOLDYBAYULY, Satyajit Pal +4 more