Topological Emergence and Fundamental Proof of π——A Unified Arithmetic-Geometric Study Based on Calabi-Yau Threefold with $\mathbb{Z}_{8} \times \mathbb{Z}_{120}$ Symmetry

Abstract: To reveal the fundamental mathematical origin of the transcendental number π and break the disciplinary barrier between traditional number theory and differential geometry, this paper constructs a complete topological axiom system with zero manual tuning parameters, zero empirical fitting, and no preset external constants based on a Calabi-Yau threefold (CY₃) with $\mathbb{Z}_{8} \times \mathbb{Z}_{120}$ dual symmetry. By defining a uniquely constrained CY₃ manifold, constructing prime eight-ridge topological bands and homologous cycle pairing rules, adopting holomorphic period integral solution, Picard-Fuchs steady-state differential constraints, and high-dimensional holographic dimensionality reduction projection operators, this study realizes the fundamental unified proof of number theory and high-dimensional geometry. The core conclusion demonstrates that the CY₃ manifold satisfying the above topological symmetry constraints converges uniquely, compulsorily and stably to the standard value of π. Instead of being an axiomatic primitive transcendental constant, π is a steady-state topological fixed point generated by the low-dimensional holographic projection evolution of high-dimensional topological prime vortex interference fields. Complying with the four axiomatic criteria of self-consistency, uniqueness, falsifiability and endogeneity proposed by Inventiones Mathematicae and the Clay Mathematics Institute (CMI), this research provides a novel interdisciplinary paradigm for the topological origin research of fundamental mathematical constants and the cross unification of number theory and high-dimensional geometry.