This paper, as the fourth in the "Mathematical Essence of the Universe" series, aims to establish Recursive Nesting Dynamics—a unified dynamical framework connecting the intrinsic motion of Qi to macroscopic physical phenomena. In the first three parts, we derived the necessity of absolute spacetime from the Φ law, and established the mathematical foundations, material spectrum, and gravitational/cosmological framework of Qi theory. In Part 2, we constructed a seven-level material structure spectrum from Qi to black holes, clarifying that photons are core planar crystal clusters + crystal cluster clouds (two-dimensional recursive nested structures) and massive particles are three-dimensional recursive nested structures. However, a unified dynamical description from the two-dimensional golden spiral motion of Qi to observable recursive phenomena has been missing. This paper systematically proposes the Principle of Recursive Nesting Dynamics: any stable material structure is formed by hierarchical recursive nesting obeying the golden ratio Φ; each level maintains its own golden ratio rhythmic motion while being nonlinearly coupled with adjacent levels. This principle contains two branches: two-dimensional recursive systems describe massless particles (e. g. , photons), giving rise to wave equations in the continuum limit; three-dimensional recursive systems describe massive particles (e. g. , electrons, protons), naturally yielding a recursive amplitude field equation (identical in form to the Schrödinger equation) in the macroscopic limit. Notably, the core planar crystal cluster of a two-dimensional recursive system is decorated with golden spiral arms (the ground state has two arms, and the number can increase in excited states), while the surrounding crystal cluster cloud provides the quasi-continuity of the energy spectrum. As the first quantitative verification of Recursive Nesting Dynamics, we compute the ground state energy of the hydrogen atom. Because the precise information of the Qi recursive hierarchical states is currently not fully known, we adopt an "anchoring" method: we anchor the basic physical state using experimental values such as the electron mass, and then compute upward to higher material levels using recursion theory. All parameters are uniquely determined by the Φ constraints of the recursive nested structure. No experimental data from hydrogen were used in fitting. Under the continuum limit approximation (replacing discrete recursive sums by continuous integrals) and a rigorous calculation including tidal potentials to infinite order, the theoretical ground state energy of hydrogen is -13. 59842 eV. Compared with the CODATA 2022 recommended value -13. 59844 eV, the deviation is only +0. 02 meV, corresponding to a relative precision of 1. 5×10^-6 (0. 00015%). For massless particles, we establish equations for a two-dimensional multi-level coupled oscillator system, and prove that in the continuum limit the wave equation and dispersion relation ω = ck of photons naturally emerge. From the symmetry of the recursive structure, we derive the photon spin 1 and an additional orbital angular momentum contribution. The fine-structure peak ratios λ₍+₁/λₙ = Φ (error <0. 3%) observed in photovoltaic EQE experiments directly verify the photon frequency recursion ω₍+₁/ωₙ = Φ. The current precision of 0. 00015% is sufficient to validate the Recursive Nesting Dynamics framework at atomic scales. The remaining tiny deviation of 0. 02 meV can be attributed to several higher-order physical effects not yet included (nonlinearity of recursive kinetic energy, intrinsic oscillation contact potential, Möbius topological boundary corrections, non-sphericity of the internal quark structure of the proton, etc. ) as well as conceptual contamination of the anchor parameters by relativistic effects. This paper provides a qualitative analysis of these effects within the absolute spacetime framework and points out three directions for future precision improvement. Recursive Nesting Dynamics offers a unified perspective for understanding the nature of massive and massless particles, and lays a solid foundation for the eventual unification of recursive field theory and crystal cluster modulated gravity. A Chinese version of this paper is also included in the uploaded files.
Lin Hao (Sat,) studied this question.