DOI: 10.5281/zenodo.18683263
title: Ω-SOVEREIGN™ — The Planetary Sovereign Continuum A Unified Mathematical, Constitutional, and Governance Operating Framework for Planetary-Scale Systems

description:  

Title:Ω-SOVEREIGN™: Mathematical Sovereign Continuum and Planetary Governance Architecture — Complete Canonical Framework



Description

This record presents the complete sovereign scientific and governance corpus integrating rigorous mathematical foundations, stochastic control theory, constitutional system architecture, and planetary-scale institutional design into a unified research framework. The collection establishes a formally structured continuum connecting advanced stochastic analysis, dynamical systems theory, risk-sensitive control, and global governance operating standards within a single coherent canonical model.

The work develops a mathematically closed framework built upon infinite-dimensional stochastic evolution equations, analytic semigroup theory, entropy-driven nonlinear functionals, Lyapunov stability structures, dynamic programming principles, and long-time ergodic behavior. These analytical components are systematically coupled with constitutional and operational governance architectures, producing a formally consistent model suitable for large-scale socio-technical and institutional systems analysis.

The mathematical volume (Parts I–VIII) provides full rigor at international research standards, including:



Functional analytic foundations in Hilbert and Sobolev spaces

Coupled SDE–PDE dynamical system formulation

Entropy functional admissibility and nonlinear regularity analysis

Mild solution theory and well-posedness in product Banach spaces

Generator characterization and Lyapunov stability with explicit decay rates

Dynamic Programming and HJB–Isaacs game-theoretic control framework

Jump-adapted numerical schemes with convergence guarantees

Long-time behavior, invariant measures, ergodicity, and exponential mixing


Together, these results establish existence, uniqueness, stability, and asymptotic structure of the governing stochastic dynamics, forming a mathematically closed and analytically verifiable system.

Complementary volumes extend the framework into constitutional and operational domains, defining interoperable governance standards, sovereign system architectures, and institutional operating kernels designed for planetary-scale coordination and resilience modeling.



Scientific Contributions



Unified stochastic–analytic foundation linking control theory, entropy dynamics, and governance systems

Fully rigorous infinite-dimensional stochastic framework with jumps

Explicit Lyapunov and spectral-gap based stability theory

Game-theoretic control rigor via viscosity solution methodology

Numerically stable approximation schemes with provable convergence

Existence and uniqueness of invariant measures with exponential mixing estimates

Risk-sensitive asymptotic stability formulation




Research Scope

This corpus operates at the intersection of:



Stochastic Analysis and SPDE Theory

Mathematical Control and Optimization

Dynamical Systems and Ergodic Theory

Numerical Analysis of Stochastic Systems

Systems Governance and Digital Statecraft Modeling




Structure of the Collection

The archive consists of six canonical volumes forming a unified publication set:



Planetary Sovereign Operating System (PSOS)

Planetary Constitutional Governance System (PCGS-Ω)

Global Governance Operating Standard (GGOS-∞)

Ω-SOVEREIGN™ Canon

UN Article-102 Ready Planetary Constitution Kernel

Ω-SOVEREIGN Mathematical Sovereign Continuum (Parts I–VIII Unified Monograph)




Methodological Standard

All mathematical results are presented under full analytical rigor consistent with international peer-review expectations in advanced applied probability, stochastic control, and numerical analysis literature. Proof structures rely on semigroup methods, variational formulations, generator analysis, and ergodic stability techniques.



Intended Use

The framework is designed as a foundational research reference for:



advanced mathematical modeling of complex adaptive systems,

stochastic governance and institutional dynamics research,

risk-sensitive control and long-horizon stability analysis,

interdisciplinary sovereign systems engineering.




Author

Dr. B. Mazumdar, D.Sc. (Hon.), D.Litt. (Hon.)Independent Researcher–ScholarAI Governance • Cybersecurity • Post-Quantum Cryptography • Digital StatecraftFounder, FAIR+D Canon — The De-Facto Global Standards Body

ORCID: https://orcid.org/0009-0007-5615-3558DOI: 10.5281/zenodo.18683263



Keywords

Stochastic evolution equations; SPDE; entropy functional; Lyapunov stability; invariant measure; ergodicity; dynamic programming; HJB–Isaacs equations; stochastic numerical analysis; governance systems modeling; sovereign systems architecture.