Abstract The fundamental dichotomy between continuous geometry and discrete arithmetic constitutes a core bottleneck restricting the development of modern fundamental mathematics. As a global core invariant connecting topological geometry and discrete number theory, Pi (Π) has long been defined in academia as a single transcendental constant. This definition completely ignores its inherent virtual-real dual topological structure and hierarchical nested arithmetic properties, resulting in a millennium theoretical fault: geometric systems lack discrete anchor points, while arithmetic systems lack continuous substrates. Based on the rigorous frameworks of standard topology, real and complex analysis, elementary number theory, and computational mathematics, this paper constructs a trinity novel axiom system integrating the virtual-real dichotomous structure of Pi, 16-decimal-digit rigid real truncation, and arithmetic hierarchical topological nesting. This study strictly distinguishes the real-domain discrete metric component and the imaginary-domain continuous transcendental component of Pi, adopts the 16 significant digits of double-precision floating-point as the rigid boundary surface between the infinite continuous topological field and the finite discrete arithmetic system, and establishes three core axioms: topological shell nesting homeomorphism, virtual-real coupling conservation, and modular hierarchical topological phase locking. Supported by 86 sets of basic core formulas and 100 sets of advanced extended formulas, this paper systematically characterizes the steady-state projection, hierarchical nesting, error compensation, and phase evolution mechanism from continuous topology to discrete arithmetic, and reveals the underlying mathematical laws of prime hierarchical distribution, modular periodic steady state, and closed-loop topological invariance from the topological essence. The research indicates that the finite discrete arithmetic system is not an artificial approximate fitting, but a steady-state projection structure formed by the 16-bit rigid truncation of the infinite continuous topological field of Pi. The virtual-real dual ontology of Pi serves as the system core, and the third-order arithmetic nested topological shells act as the implementation carrier. The coupling of the two realizes the self-consistent unification of continuous geometry and discrete arithmetic, fills the underlying connection gap between the two core branches of fundamental mathematics, and provides a novel axiomatic paradigm for the great unification of fundamental mathematics and the breakthrough of core problems in analytic number theory.
xiaogang shui (Sat,) studied this question.
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