标题： 一种用于孪生素数问题的构造性启发式筛法 显示英文标题

标题： A Constructive Heuristic Sieve for the Twin Prime Problem

摘要： 孪生素数的定量分布仍然是数论中的一个核心开放问题。 本文提出了一种基于筛法理论原理的启发式模型，旨在从基本原理出发构建对孪生素数常数的解析近似。 该方法的核心，我们称之为“$f(t; z)$函数分析”，涉及将筛法的密度乘积表示为包含$f(t;z)$的无限级数之比，即素数倒数的基本对称多项式。 该框架为近似著名的哈代-利特尔伍德孪生素数常数提供了一条建设性的路径。 我们基于可验证的代码进行了详细且透明的数值分析，将截断级数近似与实证数据进行比较。 该模型的局限性，特别是系统性高估及其对级数截断的依赖性，被严格讨论。 这项工作的主要价值不在于提出一个更优的预测公式，而在于提供一种清晰、可分解且具有解析可处理性的启发式方法，以理解筛法常数的乘法结构。 显示英文摘要

摘要： The quantitative distribution of twin primes remains a central open problem in number theory. This paper develops a heuristic model grounded in the principles of sieve theory, with the goal of constructing an analytical approximation for the twin prime constant from first principles. The core of this method, which we term ``$f(t; z)$ function analysis,'' involves representing the sieve's density product as a ratio of infinite series involving $f(t;z)$, the elementary symmetric polynomials of prime reciprocals. This framework provides a constructive path to approximate the celebrated Hardy-Littlewood constant for twin primes. We present a detailed and transparent numerical analysis based on verifiable code, comparing the truncated series approximation to empirical data. The limitations of the model, particularly a systematic overestimation and its dependence on series truncation, are rigorously discussed. The primary value of this work lies not in proposing a superior predictive formula, but in offering a clear, decomposable, and analytically tractable heuristic for understanding the multiplicative structure of sieve constants.