Synergetic Evolution of the Prime Existence-Entity: From the Minimal Existence-Entity to Complete Synergy via Prime-by-Prime Incremental Dynamics

This paper integrates the five-dimensional ontology framework with analytic number theory, introducing inter-dimensional synergy coefficients into the dynamic evolution analysis of prime distribution. We rigorously define the minimal prime existence-entity, the truncated prime existence-entity, and the ideal prime existence-entity, establishing an embryo--growth--maturity three-stage developmental model. Two quantitative synergy indicators are constructed: (1) the inter-entity synergy coefficient, measuring the developmental approximation of a truncated entity to the ideal template; and (2) the intra-entity self-synergy coefficient, characterizing internal coupling among the five dimensions. We prove that both coefficients are strictly monotonically increasing in N with limit 1, and derive the asymptotic decay law for self-synergy defect. Numerical experiments reveal a "form precedes spirit" developmental law: throughout the growth stage the inter-entity synergy coefficient is persistently larger than the self-synergy coefficient, with peak deviation at N ≈ 17. The Prime Number Theorem is reinterpreted as a strength characterization of complete self-synergy. Euclid's proof of the infinitude of primes is recast as an ontological necessity driven by synergy completeness, supplemented by a second argument from the direction--intensity matching perspective. Finally, we establish a connection between truncated entities and truncated Euler products, proposing the Spectral Upgrade Hypothesis that bridges the prime existence-entity and the Riemann-zero existence-entity.