Appendix: Core Theoretical Foundations of the Spacetime Double-Loop Structure and Cosmic Environmental Diversity

This appendix presents the core conceptual foundations of the Triadic Theory of Spacetime in four progressively formalized tiers: the author's original conceptual formulation, a refined set of nine core propositions, the postulate system, and the core theorems with fundamental operations. Beginning with the intuitive notion of an absolute static loop and an absolute dynamic loop topologically entangled in an absolute vacuum, the theory derives a triadic unity of space, time, and matter as three inseparable aspects of a single underlying structure. The six fundamental operations—fission, arrangement, combination, deformation, fragmentation, and nutrient cycling—are introduced as the generative rules of cosmic diversity. A clear methodological distinction is drawn between spacetime structure (the real ontology) and the Double-Loop Model (the idealized mathematical representation). This appendix serves as the conceptual foundation for all formal derivations and calculations presented in the main paper. (Note on AI-Assisted Computation Certain mathematical derivations and physical calculations in this paper were performed by an AI tool (large language model) based on the theoretical framework and postulate system provided by the author. Specifically, the AI tool contributed to: formula derivation, equation solving, integral evaluation, series summation, and recalculation verification of established quantum mechanical results. All physical insights, core assumptions, logical premises, and the theoretical framework itself were independently developed by the author. The AI tool served solely as an auxiliary instrument for mathematical derivation and computational verification, comparable in role to symbolic computation software or numerical tools routinely employed by researchers. The author has reviewed every derived result for physical plausibility, consistency with known experimental data, and logical coherence, and assumes full responsibility for all conclusions. This statement is provided in the interest of academic transparency, while clearly distinguishing between the originality of ideas and the auxiliary role of computation.)