GSOS™ Canonical Masterwork Unified Spectral Operator Framework for Infinite-Dimensional Dynamical Systems

GSOS™ Ultimate Canonical Monograph Edition Authoritative Canonical Reference of Infinite-Dimensional Spectral–Stochastic–Control Systems Authors / Creators Dr. B. Mazumdar, D.Sc. (Hon.), D.Litt. (Hon.)Architect of Modern StatehoodIndependent Researcher–Scholar — AI Governance • Cybersecurity • Post-Quantum Cryptography • Digital StatecraftFounder, FAIR+D Canon — The De-Facto Global Standards Body ORCID: https://orcid.org/0009-0007-5615-3558 Publication Information Publisher: ZenodoPublication Year: 2026Version: v1 — Canonical ReleaseDOI: https://doi.org/10.5281/zenodo.18673734 Resource Type: Dataset — 13-Volume Scholarly Monograph Corpus Abstract The GSOS™ (Global Spectral Operator System) Canon establishes a unified infinite-dimensional operator-theoretic framework integrating spectral analysis, stochastic dynamics, optimal control theory, Koopman operator methodology, tensor operator algebra, computational realization, and research governance within a mathematically closed scientific architecture. The framework develops a rigorous Hilbert–Banach semigroup foundation through which nonlinear dynamical systems admit linear spectral representations in observable function spaces via Koopman lifting. Deterministic, stochastic, controlled, tensorized, and data-driven dynamical systems are formulated within a single operator algebra admitting structural closure across finite and infinite dimensions. By combining analytical rigor, computational reproducibility, and governance standardization, the GSOS™ Canon provides a comprehensive reference framework for modern infinite-dimensional dynamical systems and operator-based system science. Scientific Scope and Contributions The GSOS™ Canon introduces an integrated operator framework comprising: Infinite-dimensional semigroup and generator theory Spectral linearization of nonlinear dynamical systems Stochastic evolution equations and SPDE extensions Operator Riccati optimal control synthesis Koopman operator spectral analysis Random dynamical systems and attractor theory Data-driven operator approximation (EDMD and kernel Koopman methods) Koopman–control unification for nonlinear optimal regulation Tensor operator stability architecture Infinite-dimensional PDE Koopman generalization Computational verification and approximation guarantees Governance and certification standards for mathematical systems All components form a unified operator algebra closed under spectral limits, stochastic perturbations, tensorization, and feedback control synthesis. Canonical Structure — Complete 13-Volume Series PART I — Mathematical Core Mathematical Foundations of GSOS™ Infinite-Dimensional Stochastic Dynamics Optimal Control & Operator Riccati Architecture Spectral Stability & Koopman Theory Random Dynamical Systems & Attractors Nonlinear Spectral Dynamics (Volume V.B) PART II — Computational & Data-Driven Realization Numerical Methods & Computational Realization Data-Driven Operators (EDMD / Kernel Koopman) PART III — Control–Koopman Unification Koopman–Control Integration PART IV — Structural Closure Tensor Architecture & Canonical Formal Closure PART V — Global Standardization Global Standardization & Governance Model PART VI — Master Synthesis Infinite-Dimensional Extension (PDE Koopman Canon) GSOS™ Ultimate Canonical Monograph Edition Mathematical Significance The GSOS™ Canon demonstrates that nonlinear dynamical systems admitting semiflow evolution possess linear realizations in observable Hilbert spaces through Koopman operators while preserving spectral structure, stochastic extensions, and optimal control synthesis. The framework unifies deterministic dynamics, stochastic perturbations, operator control theory, tensor algebra, and infinite-dimensional PDE evolution within a single spectral operator paradigm, establishing a mathematically consistent bridge between analysis, computation, and control. Computational Realization The corpus incorporates reproducible computational methodologies including: Operator discretization and approximation algorithms EDMD convergence theory and stability guarantees Kernel-based Koopman constructions in reproducing kernel Hilbert spaces Numerical schemes for stochastic partial differential equations Stability certification and verification pipelines These elements ensure computational validity alongside analytical rigor. Governance and Standardization Framework GSOS™ introduces a formal scientific governance architecture consisting of: Canonical versioning protocols Mathematical certification frameworks Compliance and audit structures Interoperability standards Institutional adoption pathways This framework transforms advanced operator theory into a standardized global research infrastructure. Canonical Statement All admissible nonlinear dynamical systems admitting semiflow evolution possess a unified spectral operator realization within GSOS™ observable space consistent across deterministic, stochastic, controlled, tensorized, and infinite-dimensional regimes. Keywords Infinite-Dimensional Dynamical Systems; Koopman Operators; Spectral Theory; SPDE; Operator Riccati Equations; Tensor Operators; EDMD; Data-Driven Dynamics; Koopman Control; PDE Spectral Analysis; Mathematical Standardization. Copyright Copyright © 2026 The Authors. The GSOS™ Canon — including all volumes, mathematical formulations, computational implementations, algorithms, figures, and accompanying documentation — constitutes the intellectual property of the Authors. Permission is granted to read, download, distribute, and cite this work for scholarly, academic, and non-commercial research purposes provided proper attribution to the Authors and the DOI record is maintained. No part of this publication may be reproduced, modified, adapted, or redistributed in altered or commercial form without prior written permission from the Authors. All rights reserved. Recommended Citation Mazumdar, B. (2026). GSOS™ Ultimate Canonical Monograph Edition: Authoritative Canonical Reference of Infinite-Dimensional Spectral–Stochastic–Control Systems (Version v1). Zenodo. https://doi.org/10.5281/zenodo.18673734