Elevation of Prime Gaps to Vector Topological Distances on Calabi-Yau Three-Folds: Discrete-Continuous Geometric Embedding within the PDSM-NT Framework

Abstract As a core research object in analytic number theory, the prime gap $$g_n=p_{n+1}-p_n$$ has long been regarded as a one-dimensional non-negative integer scalar. Within the axiomatic framework of the PDSM-NT primitive vortex number theory and wormhole space theory, this paper proposes the paradigm-shifting elevation of scalar prime gaps to vector topological distances $$\boldsymbol{\Delta}_{n,\mathrm{shui}}^{\mathcal{Q}}$$ on the Calabi-Yau three-fold $$CY_3$$. By constructing the Shui-type cohomological lattice embedding operator$$\boldsymbol{\Phi}_{\mathrm{shui}}$$, prime sequences are mapped to the lattice points of $$H^2(CY_3,\mathbb{Z})$$, and the difference vectors between adjacent lattice points are defined as topological distances. This approach thoroughly resolves the core contradiction between the global boundedness of geodesic distances on compact manifolds and the unbounded growth of prime gaps. On this basis, an innovative π-phase normalization mechanism is introduced for second-order system upgrading, which performs complex geometric phase smoothing on discrete topological vectors, eliminates discrete lattice noise, and achieves strict alignment between discrete number-theoretic structures and $$CY_3$$ holomorphic fields and period integrals. This paper verifies the well-posedness, order-preserving property, and asymptotic statistical equivalence of the embedding, and establishes an exact correspondence between the distribution of phase-normalized topological distance magnitudes and the Gallagher Poisson distribution conjecture. This work opens up anew interdisciplinary research path integrating complex geometry, string theory and modular forms for prime distribution studies, realizing a leap-forward iteration from discrete number theory to continuous high-dimensional complex geometric field theory. Keywords: prime gap; Calabi-Yau manifold; vector topological distance; PDSM-NT; cohomological lattice embedding; Shui-type operator