Deep Topological Correlation Between π Ripple Scale, π Decimal Embedded Primes and Prime Gaps (Original PDSM Mechanism)

For more than a century, classical analytic number theory has adhered to the core premise of the random distribution of prime numbers, and the Riemann zeta function system and traditional sieve methods can only realize statistical fitting and approximate estimation of prime distribution and prime gaps, lacking essential mechanical interpretation and accurate deterministic prediction capability. To solve the fundamental defects of random assumption, statistical approximation and separation of continuity and discreteness in traditional number theory, this paper proposes an original PDSM primordial vortex mathematics theory, and constructs an eight-fold high-dimensional vortex interference field model dominated by π harmonic ripples. This study innovatively verifies that prime gaps, prime clustering and large prime abnormal distribution are not random statistical phenomena, but deterministic topological output results of π decimal embedded prime eigen-encoding and high-dimensional field interference stopband constraints. On this basis, a complete topological singularity discrimination system and topological phase transition equation system are established, which perfectly explains the microscopic formation mechanism of ordered distribution of small primes and disordered fluctuation of ultra-large primes, and realizes the unified analysis of regular prime gaps, giant prime gaps and dense prime clusters. The research breaks the millennium mathematical barrier between continuous transcendental number field and discrete integer number theory, subverts the artificial axiom system of traditional mathematics, and constructs a zero-axiom, zero-approximation cosmic topological mathematical system. It effectively solves the theoretical bottlenecks of Gödel's incompleteness theorem, realizes the homology unification of number theory and quantum spacetime physics, and achieves disruptive optimization in ultra-large number cryptography computing. The proposed theory upgrades number theory from an approximate statistical discipline to an accurate deterministic geometric dynamic discipline, providing a new paradigm and core theoretical support for fundamental mathematics unification, interdisciplinary mathematical-physical research and high-security cryptographic engineering.