Sheaf-Theoretic Architectures of Cellular Decision-Making: A Second Volume Synthesis of Topos-Theoretic Quantum Mechanics and Cellular Sheaf Laplacians
This paper synthesizes categorical logic, combinatorial topology, and cellular sheaf theory to formalize cellular decision-making, specifically synaptic transmission and mitochondrial threshold mechanisms. It moves beyond classical geometric and physical models—which struggle to explain phenomena like ion channel selectivity and probabilistic neurotransmitter release—by utilizing the Abramsky-Brandenburger framework for quantum contextuality and the Bohrification of algebraic quantum mechanics. By integrating the minimal extension sheaf of Barthel-Brasselet-Fieseler-Kaup (BBFK) and Jacob Wood's Combinatorial Hodge Index Theorem, the author demonstrates how biological thresholds remain topologically invariant despite fluid, molecular-level geometry. Finally, the deployment of Sheaf and Tarski Laplacians over cellular structures provides a computable model for how decentralized biological agents, such as SNARE complexes and MCU holocomplexes, achieve global consensus. KSCCN Mamun, SM (2026) ………………………………………….1 Sheaf-Theoretic Architectures of Cellular Decision-Making: A Second Volume Synthesis of Topos-Theoretic Quantum Mechanics and Cellular Sheaf Laplacians Also available on Academia.edu: https://www.academia.edu/attachments/133966008/download_file?s=portfolio
This paper synthesizes categorical logic, combinatorial topology, and cellular sheaf theory to formalize cellular decision-making, specifically synaptic transmission and mitochondrial threshold mechanisms. It moves beyond classical geometric and physical models—which struggle to explain phenomena like ion channel selectivity and probabilistic neurotransmitter release—by utilizing the Abramsky-Brandenburger framework for quantum contextuality and the Bohrification of algebraic quantum mechanics. By integrating the minimal extension sheaf of Barthel-Brasselet-Fieseler-Kaup (BBFK) and Jacob Wood's Combinatorial Hodge Index Theorem, the author demonstrates how biological thresholds remain topologically invariant despite fluid, molecular-level geometry. Finally, the deployment of Sheaf and Tarski Laplacians over cellular structures provides a computable model for how decentralized biological agents, such as SNARE complexes and MCU holocomplexes, achieve global consensus. KSCCN Mamun, SM (2026) ………………………………………….1 Sheaf-Theoretic Architectures of Cellular Decision-Making: A Second Volume Synthesis of Topos-Theoretic Quantum Mechanics and Cellular Sheaf Laplacians Also available on Academia.edu: https://www.academia.edu/attachments/133966008/download_file?s=portfolio
K SC This paper synthesises a multi-layered enquiry into quantum-enabled decision architectures within cellular systems. Beginning from the modular ultracold cryogenic and biomimetic principles that illuminate mitochondrial compartmentalisation, calcium microdomains and electron tunnelling, it proceeds to an examination of proposed quantum effects at neural synapses (SNARE-complex soliton tunnelling, ion-channel quantum transport, microtubule coherence) and culminates in the rigorous formalism of non-perturbative quantum transport models—transfer-matrix methods, nonequilibrium Green's functions and non-Markovian quantum Langevin equations. A unifying pattern emerges: hierarchical modular isolation of steep physical or electrochemical gradients coupled to controlled, distance-tuned exchange, operating at the quantum–classical boundary. This architecture is shown to support readable, threshold-governed decision processes (release or not; open or closed; flux or isolation). Ontological priority is accorded to the combinatorial and physical structures that exist independently of geometric realisation, while the epistemological trajectory traces progressive internalisation of geometric and quantum principles into cellular language. Limitations—decoherence, model reduction, absence of definitive in-vivo functional proof—are catalogued alongside concrete research vectors. Also available on Academia.edu: https://www.academia.edu/attachments/133931970/download_file?s=portfolio
This report examines the ontological foundations of quantum mechanics, challenging the notion that geometric space and classical subsystems are primordial axioms of nature. By integrating algebraic quantum mechanics, quantum mereology, discrete homology, and topos theory, the study demonstrates that these features emerge dynamically from purely quantum mechanical precursors. The synthesis utilizes the Gelfand-Naimark-Segal (GNS) construction to derive geometric Hilbert spaces from abstract algebraic relations, effectively relegating the Hilbert space to a derived and contingent structure rather than a fundamental one. The research further employs a "basic shapes" paradigm in discrete homology to efficiently compute the higher-order structures of complex networks, bypassing the exponential computational bottlenecks traditionally associated with high-dimensional topological analysis. By identifying a minimal universal set of atomic configurations, such as the triangular prism or the discrete hypercube, the dimensionality of the boundary matrix is collapsed, achieving median speedups of 47.5x over previous algorithms. KSCCN Mamun, SM (2026) ………………………………….1 Algebraic, Topological, and Mereological Foundations of Quantum Information: A Synthetic Framework for Emergent Geometry, Computational Physics, and Cosmology K S C Furthermore, the report establishes a link between subatomic gauge symmetries and cosmological dark energy through the Axion-Modulated Quintessence framework. Also available on Academia.edu: https://www.academia.edu/attachments/133038377/download_file?s=portfolio
This research report presents a comprehensive topos-theoretic architecture to achieve the rigorous unification of classical dissipative systems, exemplified by the Resistor-Inductor-Capacitor (RLC) circuit, and open quantum systems, represented by the qubit. Historically, these domains rely on disparate formalisms (ODEs and symplectic geometry versus complex Hilbert spaces and the Lindblad master equation). By deploying a foundational shift to Topos Theory, the work synthesizes Olivia Caramello's "Toposes as Bridges" methodology, the Time-Embedded Information Geometry (TEIG) framework, and the Isham-Döring topos foundation of quantum physics. The central finding is that classical resistance and quantum decoherence are revealed as strictly Morita-equivalent manifestations of a single geometric invariant residing within a unified Classifying Topos of Dissipative Dynamics. The architecture employs relative toposes to elegantly encapsulate infinite, chaotic environmental noise and utilizes geometric logic, supplemented by categorical Picard-Vessiot theory, to translate continuous differential equations into verifiable logical sequents. Also available on Academia.edu: https://www.academia.edu/attachments/132007728/download_file?s=portfolio
As artificial intelligence compute demands push silicon CMOS architectures toward insurmountable thermodynamic limits, the search for post-silicon computational paradigms has become an existential imperative. This paper presents an exhaustive analysis of emerging computational paradigms spanning room-temperature thermodynamic sampling units, superconducting quantum circuit heat engines, and living biological organoid intelligence. We examine how non-trivial quantum biological phenomena—including environment-assisted quantum transport (ENAQT), enzymatic quantum tunneling, and open-system DNA proton dynamics—enable biological neural networks to achieve radical energy efficiency and sample efficiency. Furthermore, we deploy a closed-system game-theoretic framework to model the strategic behavior of state actors and tech conglomerates competing in the post-silicon intelligence race, highlighting the bio-quantum breakaway strategy and the geopolitical, ethical, and dual-use implications of wetware computing. KSCCN Mamun, SM (2026) …………………….…1 KSCCN The Architecture of Post-Silicon Intelligence: Quantum Thermodynamics, Biological Computing, and the Game-Theoretic Race for the Next Computational Paradigm Also available on Academia.edu: https://www.academia.edu/attachments/133849341/download_file?s=portfolio
This paper synthesizes the "Anatomical Reviews" to redefine the quantum-classical transition through an intrinsic operator-algebraic framework. It moves away from standard environment-induced decoherence toward a model where classicality emerges from the sectorisation of infinite-dimensional Hilbert spaces within isolated systems. Utilizing the Avalanche Photodiode (APD) model, the study demonstrates that macroscopic definiteness arises from double-exponential divergence in state overlap, creating a rigorous structural transition. The work formalizes this transition as a geometric degeneracy in the quantum Fisher information metric, linking fractal avalanche geometry to Tomita-Takesaki modular theory and providing a falsifiable experimental foundation for the algebraic Heisenberg cut. KSCCN Mamun, SM (2026) ……………………………….1 Algebraic Sectorisation, Information Geometry, and the Quantum-Classical Transition A Synthesis of the Anatomical Reviews (Vol-VI) Also available on Academia.edu: https://www.academia.edu/attachments/133083183/download_file?s=portfolio
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