Abstract Building upon the existing theoretical foundation of the Symmetric Primorial Recurrence (SPR) framework, this paper focuses on demonstrating specific applications of this framework in algorithm design, cryptography, physical simulation, and sustainable development technologies. Based on the exponentially decaying error, optimal candidate density, and fractal self-similarity of SPR recursion, we rigorously derive the time complexity lower bound for prime generation algorithms, the isogeny graph construction for post-quantum cryptography, the convergence of quantum chaotic spectral statistics, the discrete model for turbulent energy cascade, and the theoretical energy efficiency and retrieval complexity of sustainable development technologies including P-ALU, PAFS, Prime AI, and PRI. This paper extends the Zenodo preprint 1, providing detailed mathematical arguments for the applications of the SPR framework. Furthermore, this paper systematically elaborates on the conditional arguments of the SPR framework regarding core number-theoretic problems such as the BSD Conjecture, and P vs NP problem, and presents the complete proof of the Hardy-Littlewood Conjecture (Prime k-tuple Conjecture) by combining the Shao-Tao 2026 results with the SPR framework, thereby providing a unified theory of prime distribution and deducing the Twin Prime Conjecture and Goldbach’s Conjecture.
Mei Shenglin梅生林 (2026) studied this question.