Elevation of Prime Gaps to Vector Topological Distances on Calabi-Yau Three-folds: A Discrete-Continuous Geometric Embedding Study Based on the PDSM-NT Framework

Abstract The prime gap $$g_n=p_{n+1}-p_n$$ is a fundamental physical and mathematical quantity that characterizes the asymptotic distribution behavior of prime numbers in analytic number theory. In traditional research systems, prime gaps are simply defined as one-dimensional discrete scalars that only reflect the numerical difference between adjacent primes, without carrying inherent topological morphology, geometric structural attributes or high-dimensional evolutionary degrees of freedom. This inherent limitation confines existing prime distribution research to simple analytic estimation and empirical statistical fitting, failing to explore the essential geometric origins and field-theoretic evolutionary mechanisms of prime number distribution. Based on the original PDSM-NT primitive vortex number theory framework and wormhole space axiomatic system, this study proposes a fundamental paradigm innovation for prime number research: the traditional scalar prime gap is strictly elevated to an intrinsic vector topological distance defined on the ontology of pure compact Calabi-Yau three-folds $$CY_3$$, thoroughly breaking away from the limitations of extrinsic cohomology lattice approximation and finite field truncation models adopted in previous arithmetic-geometric studies. Supported by the original $$\pi$$-eigen conformal phase scaling mechanism and strictly abiding by the complete axiomatic system of standard $$CY_3$$ manifolds (including compact Kähler property, Ricci flatness and vanishing first Chern class), this study fundamentally resolves the classical topological paradox between the bounded metric constraint of compact complex manifolds and the unbounded asymptotic growth of prime sequences. It realizes the smooth homologous embedding and synchronous evolution of discrete arithmetic structures into high-dimensional holomorphic complex manifolds. Systematic verification confirms the well-posedness, strict order preservation and statistical ergodic equivalence of the proposed embedding system, establishing a rigorous intrinsic correspondence between the statistical distribution law of prime gaps and the topological phase evolution of $$CY_3$$ manifolds. This research eliminates the long-term disciplinary separation between discrete analytic number theory and pure Calabi-Yau complex geometry, constructs a unified self-consistent arithmetic-geometric theoretical framework based on standard complex manifold ontology, and provides an original fundamental paradigm for the geometric solution of classic number-theoretic problems and the topological normalization of string theory vacuum structures.