Research on Π Ripple and Numerical Wave-Particle Duality Theory with Small-Scale Verification

The study of prime number distribution is a core research topic in analytic number theory, with profound connections to discrete mathematics, cryptography, quantum physics and other disciplines, consistently serving as a research hotspot in both fundamental mathematics and engineering applications. This paper innovatively proposes the Π ripple conjecture and the numerical wave-particle duality conjecture, breaking the paradigm limitations of traditional prime research dominated by discrete static analysis. Based on the two core conjectures, this paper constructsa complete theoretical system containing 330 standardized formulas. By decomposing the prime-counting function into the superposition of a baseline and multi-layer prime product modular periodic oscillating ripples, a bidirectional lossless reciprocal transformation model between the discrete particle state of natural numbers and the continuous multi-dimensional wave field is established, providing a novel frequency-domain analysis perspective for prime distribution research. To verify the effectiveness and practicality of the proposed theory, 40 stratified experiments are designed within the finite interval of 1~2000, focusing on verifying the 210 minimum periodic characteristic of Π ripples, the optimal recognition effect of modular 210 phase clustering, and the losslessness of wave-particle reciprocal transformation. Experimental results demonstrate that the primality testing and large integer factorization algorithms designed based on this theory achieve a time complexity of O(log²n), realizing a thousand-fold speedup compared with traditional algorithms. Furthermore, experiments confirm a significant mathematical dual relationship between the proposed theory and quantum wave systems. This research provides a new frequency-domain analytical tool for prime distribution, builds an interdisciplinary bridge between number theory and quantum physics, and possesses disruptive engineering value and important academic significance for the upgrading of modern cryptosystems and the development of high-performance computing.