Dynamic Rigid Fractal Spacetime Manifold Theory

This paper proposes an innovative framework, the Dynamic Rigid Fractal Spacetime Manifold Theory (DRFSMT), which integrates fractal and noncommutative algebra to provide a unified mathematical and physical foundation for quantum field theory, dark energy evolution, and fractal superconductivity. The core of the theory is the construction of dynamic fractal spacetime manifolds, whose mathematical foundation is based on local self-similarity and the scale invariance of fractal measures, and whose local anisotropic characteristics of spacetime are described by noncommutative derivative operators. The definition of fractal derivatives incorporates topological winding effects, reflecting the nontrivial geometric properties of fractal structures, and their dynamic behavior reduces to standard spacetime derivatives in the classical limit. In physical modeling, the evolution mechanism of dark energy is reinterpreted through dynamic modifications of fractal dimensions. The theory proposes that the equation of state for dark energy includes an evolutionary term for fractal dimensions as the universe expands, with key parameters derived from the variational principle of the fractal Einstein f ield equations, showing excellent agreement with high-redshift observational data. The evolution equation of fractal dimensions reveals the coupling between cosmic expansion and the multiscale characteristics of spacetime structures, providing a geometric origin for the phenomenon of dark energy. Meanwhile, in condensed matter physics, the microscopic mechanism of fractal superconductivity is elucidated through a curvature-coupled modified Ginzburg-Landau equation. The theory predicts a negative correlation between the superconducting critical temperature and fractal geometric curvature, which has been experimentally verified using Sierpinski-structured superconducting films, observing regular variations in critical current density with fractal dimensions. Regarding experimental validation, cosmological data analysis from DESI and JWST demonstrates that the dark energy model of this theory achieves a goodness-of-fit comparable to the standard ΛCDM model, while aligning better with observational details of redshift evolution. Fractal superconductivity experiments using scanning tunneling microscopy (STM) and superconducting quantum interference devices (SQUID) confirm the regulatory role of fractal curvature on superconducting phase transitions, with the observed trends in critical temperature consistent with theoretical predictions. These results highlight the universality of fractal spacetime theory in describing cross-scale physical phenomena. 1The breakthrough of DRFSMT lies in its mathematical consistency and physical unif ication: the recursive compactness theorem of rigid fractal manifolds ensures the rigor of its theoretical foundation, while incorporating dark energy, quantum field theory, and superconductivity into a unified geometric framework. Future research will focus on extending the fractal spacetime framework to high-energy quantum gravity and designing experiments to investigate multiscale coupling effects, further advancing the application of fractal geometry in fundamental physics.