Why Meaningful General Solutions for Prime Numbers Do Not Exist

This work investigates a long-standing and often implicit assumption in number theory: that prime numbers might admit a general solution in the form of a closed formula, explicit generator, or constructive expression. Rather than proposing a new candidate formula, the paper examines the problem at a structural level by clarifying what should count as a \emph{meaningful} general solution. A minimal axiomatic framework is introduced, formalizing natural requirements such as constructivity, finite descriptive complexity, non-circularity, locality, and the absence of embedded primality testing. Within this framework, a structural obstruction theorem is established, showing that no function satisfying all these conditions can generate prime numbers. Any purported solution must necessarily violate at least one axiom by reintroducing hidden decision procedures, global dependence, or unbounded descriptive complexity. The analysis reconciles a wide range of existing results. Diophantine representations, analytic formulas, sieve-based constructions, and closed-form expressions are shown not to be failures or incomplete attempts, but precise illustrations of the same underlying obstruction. This perspective explains why prime numbers consistently resist pointwise generative descriptions while remaining amenable to rich statistical and asymptotic analysis. The implications extend naturally to prime distribution and the Riemann Hypothesis. From the present viewpoint, the success of analytic approaches and the absence of explicit formulas for individual primes reflect not a gap in technique, but a structural feature of primality itself. The paper thus delineates a principled boundary between meaningful generality and formal artifices in representations of prime numbers.