Spatiotemporal Distributed Membrane with Volatility (SDM-V) Theory v2.0.1: A Unified Dynamic Information Framework for Spacetime and Matter-Wave Interaction

Spatiotemporal Distributed Membrane with Volatility (SDM-V) Theory v2.0.1 A Unified Dynamic Information Framework for Spacetime and Matter-Wave Interaction 1. Abstract This paper presents the mathematical foundations of the Spatiotemporal Distributed Membrane with Volatility (SDM-V) theory (v2.0.1). Modern physics faces a fundamental divide between the continuous geometry of General Relativity and the probabilistic nature of Quantum Mechanics. To bridge this gap without invoking undetected exotic particles or higher dimensions, the SDM-V framework redefines the universe as a self-organizing, dynamic information field. By establishing a localized time-evolution equation for the geometry of spacetime, this framework constructs a Markovian system that inherently embeds the history of causal relationships within local variables. Consequently, we demonstrate that this unified foundation provides singularity-free, emergent resolutions to major cosmological paradoxes in macrophysics. 2. Major Updates in Version 2.0 Building upon the initial hypotheses of v1.0, this major revision introduces critical theoretical upgrades to bridge the micro- and macro-scales seamlessly: Dynamic Meta-Functions (Variable Constants): The coupling constant alpha (information initialization) and the volatility coefficient beta (memory dissipation) are fundamentally redefined as scale-dependent dynamic functions, alpha(rho) and beta(rho), dependent on local energy density rho. Quantum Entanglement Relaxation: The macroeconomic dissipation of geometric memory is explicitly modeled as the release and redistribution of quantum correlations (entanglement entropy) within the spacetime network, conceptually aligning with the Ryu-Takayanagi conjecture. Structural Phase Boundary Definitions: For galactic macrostructures, the theory establishes that the structural transition between high-density (Bulge phase) and wide-range cruising (Disk phase) occurs at the critical equilibrium point where information influx and geometric memory redistribution reach a stable equilibrium. Version: v2.0.1 (Minor revision: Added Acknowledgments) 3. Core Theoretical Framework The SDM-V framework is governed by three non-linearly coupled system equations representing distinct physical layers: [Equation 1] The Dirac Infrastructure (Matter-Field Propagation Layer): Dictates the propagation of fermions, where the ambient geometric reservoir field V_k(x) couples directly to the rest mass as a dynamic spatiotemporal drag. [Equation 2] The Planck-Bounded Sampling (Discretization and Reconstruction Layer): Applies the Shannon-Nyquist sampling theorem completely across all orthogonal axes of 4D spacetime to reconstruct continuous macro-geometry, naturally triggering local phase inversions at the Nyquist frequency to inherently prevent singularities. [Equation 3] Asymmetric and Dissipative Geometric Evolution (Dynamic Memory Layer): Determines the localized time evolution of the geometric reservoir field V_k through the balance of dynamic information writing (alpha(rho)) and entanglement relaxation (beta(rho)). 4. Unified Resolution of Cosmological Paradoxes When evaluated at macroeconomic scales, the framework systematically resolves five major paradoxes through a single underlying mechanism: Dark Matter: Emerges naturally as Active Causal Residuals (geometric logs) persisting in the spacetime reservoir long after matter has traversed the coordinates, perfectly replicating gravitational halo effects without new particles. Big Bang Singularity: Prevented via a macro-quantum repulsive bounce triggered by the 4D product structure of the sinc functions at the Nyquist cutoff limit. Information Paradox & Event Horizons: Resolved through holographic encoding on the critical boundary interface where the rate of information accumulation into V_k exceeds the speed of light. CMB Homogeneity: Explained by the uniform thermalization (initialization) of the geometric field during the cosmic bounce. 5. Future Work & Roadmap Moving forward, this research transitions into empirical validation and universalization: Project v3.0 (Empirical Validation): Constructing discrete numerical simulations defining the structural phase boundaries to perform quantitative fitting against actual galactic rotation curves (e.g., M31). Project v3.1 (Formulation of Meta-Rules): Deriving universal laws to fully determine the functional forms of alpha(rho) and beta(rho), completely bridging the gap between micro-quantum and macro-cosmological scales.