A Structural Proof of the Goldbach Conjecture via Center Value Framework, Factorial Combinatorial Density, and Dynamic Completion Model

Abstract This paper develops a unified structural framework to establish the inevitability of the Goldbach conjecture. Traditional analytic methods (Hardy–Littlewood circle method, Vinogradov＇s weak conjecture, Chen Jingrun＇s theorem) and modern computational veri fications provide partial evidence but fail to demonstrate structural necessity. We introduce the center value framework, prime gap generation, and a dynamic completion model to show right-biased coverage of even numbers. Furthermore, by defining 𝑛 as the number of available primes and 𝑛! as the factorial-logarithmic scale of additive combinations, we prove that the density of prime-pair sums grows at a rate significantly exceeding prime thinning. This “density advantage” ensures that every even integer 2𝑛 is structurally de composable into a prime pair, combining historical primes with newly generated primes. A number-line visualization and modular arithmetic constraints (mod 12, mod 60) reveal predictable prime distribution patterns, integrating empirical evidence into a coherent struc tural proof. The framework extends naturally to twin primes, bounded gaps, and multi prime additive conjectures, offering a unified language for additive number theory and new directions for future research. 摘要 本文建立一個統一的結構性框架，以證明哥德巴赫猜想的必然性。傳統解析方 法（如Hardy–Littlewood 圓法、維諾格拉多夫弱猜想、陳景潤定理）以及現代計算 驗證雖提供部分支持，但未能揭示結構上的必然性。本研究提出中心值框架、質 數間隙生成與動態完備模型，展示偶數的右偏覆蓋。此外，透過將𝑛定義為可用 質數的數量，並將𝑛!視為加法組合的階乘對數尺度，我們證明質數對之和的密度 增長顯著超過質數稀疏化的速度。這種「密度優勢」確保所有偶數2𝑛都能結構性 地分解為質數對，結合歷史質數與新生成質數。數線視覺化與模12、模60的算術 約束揭示質數分布的可預測性，將經驗性證據轉化為一致的結構性證明。此框架自 然擴展至雙質數、有界缺口與多質數加法猜想，為加法數論提供統一語言與未來研 究的新方向。