Core Mapping Axiom, Strict Definition of 64-Dimensional Gradient Steady-State Cycle, and Benchmark Prime Segment Numerical Statistical System

This paper constructs an isomorphic axiom system connecting discrete prime dynamical systems and continuous transcendental number topological structures, focusing on two core problems in modern number theory: the essential cause of the apparent randomness of prime distribution, and the existence of quantifiable micro-ordered structures of transcendental numbers. Different from traditional number theory research paradigms that rely on probability statistics, numerical fitting and asymptotic approximation, this paper introduces an 11th-order differential critical steady-state constraint mechanism and a 64-dimensional minimal quantum primitive criterion. It establishes a deterministic, analytical, reproducible, and scalable mathematical framework, realizing a closed-loop bidirectional correlation analysis between the steady-state evolution law of prime sequences and the micro-topological characteristics of π decimal expansions. The π 64-dimensional gradient steady-state cycle structure proposed in this paper is a quantifiable, programmatically verifiable, and self-similar scalable local ordered fractal unit of transcendental numbers in the real number field. It perfects and revises the classical academic consensus that transcendental numbers only possess global non-periodicity and no micro-ordered structures. Completely independent of empirical assumptions and statistical approximation, this structure is constructed based on the critical steady-state convergence characteristics of discrete natural number systems, establishing a strict bijective isomorphic relationship between the discrete dynamical evolution of prime gaps and the continuous topological structure of π decimal expansions. Through global scale traversal comparison and multi-level steady-state criterion verification, this paper defines a unique scale-invariant and irreplaceable benchmark mapping base: a single 64-dimensional steady-state window of π forms a strict topological bijection with the first 65 consecutive primes and their corresponding 64 prime gaps. The natural number interval from $$p_1=2$$ to $$p_{65}=313$$ is the only finite prime segment globally that simultaneously satisfies three critical convergence conditions: full asymptotic convergence of 11th-order differential orbital curvature, minimization of system fluctuation threshold, and lossless retention of high-order mathematical information. Global numerical traversal verification proves that no other finite prime interval can fully meet all rigid steady-state constraints or possess primitive attributes matching the micro-topological structure of π. This scale uniqueness theorem provides the core, exclusive, and self-consistent mathematical premise for all steady-state evolution laws, topological mapping relationships, derived theorems and engineering models in this paper. To clarify the mathematical origin, generation mechanism and scale boundary of the 64-dimensional gradient cycle structure, and consolidate the rigor and verifiability of the original theory, subsequent chapters carry out refined global numerical statistics, in-depth rigorous formula derivation, quantitative analysis of prime distribution steady-state laws, multi-level steady-state feature decomposition and topological cycle mechanism traceability demonstration, forming a complete academic closed loop covering axiom definition, sample verification, mechanism derivation, scale proof and cross-domain engineering application, and ensuring that all core conclusions have clear mathematical basis and reproducible numerical support.