This document explores a fundamental paradigm shift in computational theory, moving beyond classical Boolean and quantum unitary architectures toward a generalized framework based on symplectic geometry. It establishes the "symplectic computational ontology," which utilizes the non-compact symplectic group to achieve enhanced expressivity and robustness compared to standard complex-Hilbert space models. The analysis systematically formalizes the quantum-symplectic duality, investigates non-linear dynamical regimes through canonoid transformations and bi-Hamiltonian structures, and extends these principles to infinite-dimensional field-theoretic and non-Archimedean
p-adic arithmetic spaces. Furthermore, it details how categorical tools, such as Fukaya categories and gentle algebras, can compress complex topological data into executable combinatorial algorithms. Finally, the volume addresses the profound geopolitical consequences of this technological transition, arguing that the emergence of advanced symplectic KSCCN Mamun, SM (2026) …………………………..1 The Symplectic Computational Ontology: From Infinite-Dimensional Geometries and Categorical Algebra to Geopolitical Sovereignty (Vol-I) K S C and categorical AI architectures necessitates new decolonial frameworks to protect epistemic and infrastructural sovereignty against systemic data extraction and digital colonialism. Also available on Academia.edu: https://www.academia.edu/attachments/133170160/download_file?s=portfolio
The problem of extending the insights and techniques of categorical quantum mechanics to infinite-dimensional systems was considered in (Coecke and Heunen, 2016). In that work the
CP∞-construction, which recovers the category of Hilbert spaces and quantum operations from the category of Hilbert spaces and bounded linear maps, was defined. Here we show that by a `horizontal categorification' of the
CP∞-construction, one can recover the category of all von Neumann algebras and channels (normal unital completely positive maps) from the 2-category
[W∗] of von Neumann algebras, bimodules and intertwiners. As an application, we extend Choi's characterisation of extremal channels between finite-dimensional matrix algebras to a characterisation of extremal channels between arbitrary von Neumann algebras.
The problem of extending the insights and techniques of categorical quantum mechanics to infinite-dimensional systems was considered in (Coecke and Heunen, 2016).In that work the CP ∞ -construction, which recovers the category of Hilbert spaces and quantum operations from the category of Hilbert spaces and bounded linear maps, was defined.Here we show that by a 'horizontal categorification' of the CP ∞construction, one can recover the category of all von Neumann algebras and channels (normal unital completely positive maps) from the 2-category of von Neumann algebras, bimodules and intertwiners.As an application, we extend Choi's characterisation of extremal channels between finite-dimensional matrix algebras to a characterisation of extremal channels between arbitrary von Neumann algebras.