Euclidean Framework for Recursive Feedback Dynamics and Multifractal Analysis

In contemporary theoretical physics, the treatment of spacetime as a Lorentzian manifold has been pivotal in relativistic frameworks. However, for systems governed by recursive feedback dynamics and multifractal geometries, a Euclidean spacetime framework R d offers distinct advantages. By adopting a d-dimensional Euclidean space, we decouple temporal and spatial coordinates, enabling a mathematically tractable analysis of non-local interactions and memory effects. This approach simplifies the study of recursive feedback dynamics, where past states recursively influence present configurations—a feature critical to systems ranging from social networks to quantum gravity. The Euclidean framework inherently supports fractional calculus, allowing us to model memory-dependent processes (via fractional time derivatives) and long-range spatial interactions (via fractional Laplacians). This abstraction is particularly suited to multifractal systems, where scale invariance and hierarchical structures emerge naturally from recursive feedback loops. By treating time as an additional spatial dimension, we can understand the treatment of temporal and spatial non-locality, a necessity for systems exhibiting both memory effects and spatial heterogeneity.