The Mathematical Essence of the Universe (1) ——Number Selection Principle and Fundamental Theory of Qi

This paper, as the first in the "Mathematical Essence of the Universe" series, aims to establish a solid mathematical and conceptual foundation for the entire Qi theory system. We first systematically expound and rigorously prove the Principle of Optimal Filling (also termed "most uniform filling / maximum distancing principle") centered on the golden ratio Φ (approximately 1. 618). From four dimensions—planar geometric filling, three-dimensional sphere packing, phase distribution uniformity, recursive dynamical stability and structural stiffness—we demonstrate the universality of Φ as an extremal solution. On this basis, we further prove from the perspective of nonlinear recursive dynamics that the golden ratio conjugate φ = (√5−1) /2 ≈ 0. 618 is the optimal robustness parameter for systems under recursive coupling: its maximum Lyapunov exponent attains a global minimum when the coupling ratio equals φ. This property provides a profound mathematical basis for the phase transitions in cosmic evolution. Based on this, we propose the Principle of Mathematical Natural Selection (also referred to as Mathematical Selection Principle in some contexts). In the "chaotic sea" of mathematical structures at the very beginning of the universe, only configurations obeying the golden ratio Φ law, owing to their comprehensive optimality in both spatial filling uniformity and temporal dynamical robustness, can "survive" the competition and become, by a "survivorship bias", the underlying laws of the observable universe. This principle provides a deep "mathematical necessity" explanation for the numerous empirical relations involving Φ discovered in Appendix B. Furthermore, we argue that the cross-scale stability of the Φ law necessarily requires spacetime to be an independent, absolutely flat, and non-dynamical background stage, thereby establishing an absolute spacetime for the theory. Within this framework, we define the fundamental entity of the theory—the Qi—whose energy scale EQ = (Φ/2) EP is directly proved from the golden law. We detail the intrinsic Φ rhythm of the Qi and rigorously derive its golden spiral solution via a nonlinear oscillator equation. From this golden spiral solution we then construct the dynamical starting point of the photon cantilever–crystal cluster model. This theory has only one unknowable a priori assumption: the existence of an indivisible "minimal primordial fragment", whose essence lies beyond our cognitive boundary. Apart from this, all physical entities (all recursively composed matter originating from Qi), material structures, and their laws of motion are completely derived from the golden ratio number Φ—selected by the cosmic matter itself based on self-stabilization—and its mathematical optimality and robustness. This principle ensures the extreme parsimony of the theory, conforming to Occam's razor, while also providing a unique mathematical origin for all observable physical quantities. This paper completes the crucial transition from pure mathematical principles to the definition of fundamental physical entities, providing a self-consistent logical starting point and a mathematical toolbox for the subsequent construction of a complete material spectrum and interaction theory. Based on the above ideas, the entire theory will unfold in the following parts: Part 1: "The Mathematical Essence of the Universe 1 – Number Selection Principle and Fundamental Theory of Qi" – presents the mathematical foundation of the theory; Part 2: "The Mathematical Essence of the Universe 2 – The Five-Level Structural Hierarchy of the Material World" – constructs the five-level material spectrum of the universe; Part 3: "The Mathematical Essence of the Universe 3 – Gravity, Cosmic Evolution, and Future Perspectives" – presents the logic of cosmic evolution and proves the equivalence of this theory with relativity; Part 4: "The Mathematical Essence of the Universe 4 – Recursive Nested Dynamics" – provides the dynamical interpretation based on Parts 1–3, validated by the calculation of the hydrogen atom ground state energy; Part 5: "The Mathematical Essence of the Universe 5 – A Preliminary Physics of Everything" – integrates the entire theory into a unified framework. A Chinese version of this paper is also included in the uploaded files.