Mathematical Foundations of the Higher-Dimensional Information Field (Part 5): Rigorous Formulation of Projection Kernels and Nonlinear Dynamics

This paper presents the mathematical and theoretical foundations of a higher-dimensional information field as Part 5 of the Entropy-Induced Collapse (EIC) Model Series (2025). It rigorously formulates how an information field, Φ(x,t), emerges as a non-equilibrium fifth force mediated by entropy gradients and projected onto 3+1 spacetime. Extending previous works, this paper introduces a consistent framework for projection kernels, activation functions, and fractal stochastic dynamics underlying the information field. Key contributions of this work include: 1. Formal derivation of projection kernels and activation functions that selectively project the higher-dimensional information field Φ_H(y) based on local entropy gradients. 2. Mathematical formulation of the Lagrangian and field equations for Φ(x,t), where entropy gradients serve as a dynamic source of “information energy.” 3. Integration of fractal stochastic differential equations to model non-Markovian and long-memory effects in entropy dynamics. 4. Comparative analysis with existing fifth-force theories, highlighting how this model uniquely avoids Equivalence Principle violations by coupling to entropy gradients rather than mass. 5. Experimental prospects for verifying the model through short-, mid-, and long-term tests, including MEMS sensors, Casimir effect, and cosmological observations. 6. Ethical statement emphasizing that all aspects of this research are solely intended for peaceful scientific development and must never be used for military or harmful purposes. This work establishes a novel theoretical and mathematical foundation for a fundamentally new type of interaction, potentially bridging quantum measurement, gravity, and non-equilibrium physics through the lens of information dynamics. Keywords: Fifth Force, Entropy-Induced Collapse, Information Field, Projection Kernel, Fractal Stochastic Dynamics, Non-Markovian Process, Atomki Anomaly, Muon g-2, Non-Equilibrium Physics, Quantum Measurement, Modified Gravity, Information Energy, Peaceful Use