The Zhu-Liang Theorem of Recursive Nested Functions: Reconstructing Dark Matter

Dark matter is one of the central puzzles in contemporary cosmology: observational evidence such as galactic rotation curves, gravitational lensing, and large-scale structure formation indicates the existence of invisible matter, yet its particle nature has not been detected to date. Based on the Zhu-Liang Theorem of Recursive Nested Functions, this paper incorporates cosmological systems into the cognitive category $\Cog$, demonstrating that dark matter effects correspond to condensation structures of deep recursive elements that cannot be directly projected, and their gravitational influence arises from corrections of deep recursive elements to shallow-layer spacetime geometry. Dark matter particle candidates (such as WIMPs and axions) are interpreted as ``projection phantoms'' of deep recursive elements under specific conditions, while modified gravity theories (e.g., MOND) correspond to dynamical approximations of recursive elements at different layers. This reconstruction not only provides a recursive ontological foundation for dark matter but also reveals its intrinsic unity with dark energy and quantum gravity phenomena.