Quantitative Extremes, Factor Structure,and Arithmetic ProgressionGeneralization of Load Uniformity inDiscrete Periodic Folding

Abstract (English): This paper investigates the load distribution rules of a one-dimensional finite discrete set under a periodic folding mapping. First, for continuous integer sets, we provide an exact explicit formula for the load variance V(L) and prove that the global upper bound is 1/4; this upper bound can be exactly attained when the cardinality is an odd number (including odd primes). Second, by introducing the concept of a "factor set", we establish a connection between the uniform state and the divisor function τ(N+1), proving that the uniform state proportion D(X) ≥ 2/X, with equality if and only if X is prime. Finally, we generalize the model to a general arithmetic progression point set and derive the algebraic relationship between its full-slot uniform state conditions and the greatest common divisor gcd(d,L). The conclusions of this paper provide an elementary quantitative framework for understanding the algebraic asymmetry in discrete folding. 中文摘要: 本文研究了一维有限离散集在周期折叠映射下的负载分布规律。首先,针对连续整数集,本文给出了负载方差 V(L) 的精确显式公式,并证明了全局上界为 1/4;当基数为奇数(含奇素数)时,该上界可被精确达到。其次,本文通过引入“因子集”概念,将均匀态与除数函数 τ(N+1) 建立联系,证明了均匀态比例 D(X) ≥ 2/X,且等号当且仅当 X 为素数。最后,本文将模型推广至一般的等差数列点集,推导了其全槽均匀态条件与最大公约数 gcd(d,L) 的代数关联。本文的结论为理解离散折叠中的代数不对称性提供了一个初等的定量框架。 Note: The full text is provided in both English (Folding_Variance_Extremes_EN.pdf) and Chinese (Folding_Variance_Extremes_CN.pdf). The complete LaTeX source codes are also included as attachments.