This paper proposes a structural framework for the study of prime numbers based on the notion of admissible state spaces. Rather than treating primes as outcomes of probabilistic laws, statistical regularities, or algorithmic generation, their distribution is reinterpreted as the manifestation of hard structural constraints imposed on an underlying symbolic dynamical system. The starting point of this work is the observation that many classical and modern approaches to prime distributions fail not due to insufficient analytic power, but due to a mismatch between the objects being controlled and the structural conditions that govern their existence. A formal notion of structural admissibility is introduced, specifying which symbolic transitions are permitted and which are excluded a priori, independently of randomness, density arguments, or asymptotic heuristics. Within this framework, prime occurrences are characterized as admissible readouts of a constrained symbolic dynamics defined on logarithmic-phase coordinates. A central consequence of this perspective is the emergence of an intrinsic bound on admissible transitions, expressed as a frame constant that restricts the state space itself rather than individual trajectories. This bound is not imposed externally, nor derived from empirical gap statistics, but arises from the internal consistency of the admissibility structure. The framework clarifies why apparent irregularity and unpredictability coexist with strong structural rigidity in the distribution of primes, and why off-structure phenomena such as unbounded resonant transitions are categorically excluded. The present paper is purely structural in nature: no new estimates, conjectures, or numerical claims are introduced. Instead, a unified admissibility principle is provided, reorganizing several previously disconnected observations into a single, closed conceptual system. This structural viewpoint is compatible with, and motivated by, recent progress in mean-square stability arguments related to the Riemann Hypothesis, but is logically independent of any specific conjectural outcome. Its role is to identify the invariant constraints under which any viable theory of prime distributions must operate. AI Usage Declaration: The core concepts and mathematical intuitions of this paper were conceived by the author (Byeong-Young Oh). The technical formalization of the mathematical framework and the English drafting process were performed in collaboration with Artificial Intelligence (AI) models. The author takes full responsibility for the contents and logical integrity of this work. This work is part of an ongoing independent research program. Official research index :https://jingyu-papa.github.io/publications/ Files
Byeong-Young OH (Thu,) studied this question.