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 p₁=2 to p₆₅=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.
xiaogang shui (Tue,) studied this question.