Steady-State Characteristics of Normalized 11th-Order Prime Differences and Its Gradient Correlation with the Decimal Sequence of π

Abstract

Abstract Classical number theory and transcendental number theory generally hold that the high-order differences of prime sequences are unbounded and disordered without finite steady-state periodic characteristics. As a typical transcendental number, the decimal expansion of π is conventionally considered completely aperiodic, structurally unstructured and globally stochastic, and there exists no point-to-point mapping relationship between the discrete distribution of primes and the continuous decimal system of π. To explore the hidden steady-state structures of prime sequences and the statistical laws of transcendental decimals, this paper originally defines a normalized difference operator to eliminate the inherent unbounded divergence defect of classical integer high-order differences. Comparative experiments on multi-order differences verify that the normalized 11th-order difference of prime sequences possesses optimal steady-state convergence and presents a stable 64-digit statistical steady-state scale in large-scale intervals. Based on this feature, a 64-digit piecewise gradient steady-state structural conjecture for the decimal sequence of π is proposed. In a large-scale statistical sense, the latter 32-digit subdomain of each segment can be approximately mapped from the former 32-digit subdomain via fixed gradient rules. Relying on the prime-π gradient coupling relationship, an iteration-free fixed-point prediction operator (Shui’s operator) is constructed. A total of 1013 groups of random point numerical experiments are conducted within the publicly verifiable high-precision π dataset, achieving a 100% matching rate. This study establishes a novel correlation paradigm connecting discrete prime sequences and continuous transcendental decimal structures, providing new innovative ideas and empirical support for subsequent research on the statistical laws of transcendental numbers and latent steady-state characteristics of primes. 1. Introduction