PDSM Vortex Field Conjecture on Geometric Emergence of Primes: Impossibility in Infinite Domain and Well-Posedness in Finite Domain

Abstract

Abstract The long-standing dichotomy between discrete number theory and continuous differential geometry constitutes the core academic bottleneck restricting the research on prime distribution mechanisms. To break through this theoretical dilemma, this paper constructs an original Primitive Vortex Mathematics (PDSM) eight-fold symmetric conch topological system, systematically proposing and fully proving the topological structure, limit constraints, and existential boundaries of the Shu-type geometric emergence conjecture for primes. By establishing an eight-fold symmetric steady-state vortex ripple field, constructing ridge manifold geometric criteria, and completing the vortex emission source decomposition mechanism, this paper builds a rigorous bidirectional mapping relationship between discrete prime sets and topological extremal structures of continuous partial differential equations (PDEs). Two core conclusions are drawn in this study: first, the infinite global steady-state eigen system contains inherent geometric asymptotic contradictions of PDEs, which fail to match the density distribution and gap evolution laws of the prime number theorem; the global uniform potential prime emergence conjecture in the infinite domain is strictly invalid; second, for any finite radial interval, this geometric number-theoretic inverse problem possesses complete well-posedness. A steady-state vortex field can be constructed and interval-customized truncated potential functions can be solved via adjoint gradient optimization algorithms, enabling accurate geometric emergence of primes and active suppression of composite magnitudes in finite domains. This work strictly equates the irreducible structure of discrete primes with topological extremal structures in continuous differential geometry for the first time, breaking the two-thousand-year disciplinary barrier between discrete number theory and continuous geometry and pioneering a new first-level research paradigm of geometric number theory. Relying on the original Shu-type topological entropy theory, this paper resolves the millennium paradox of primes—apparent randomness yet essential order—from the perspective of topological thermodynamics, providing a novel fundamental theoretical support for solving analytical number-theoretic problems, constructing new computing architectures, and unifying mathematical philosophy systems.