This work introduces a prime-induced simplicial complex, referred to as theAdrian-Structure, and studies its global organization using topologicalinvariants. Rather than approaching prime numbers through classical analytic quantitiessuch as densities or asymptotic estimates, the focus lies on the structuralproperties of a finite complex generated under multiplicative constraints.As the prime bound increases, the Euler characteristic of the associatedsimplicial complex exhibits a non-monotonic behavior. This behavior is interpreted in homological terms as reflecting changes in thenumber of independent one-dimensional cycles of the complex. In this sense,prime numbers act as structural generators that reorganize a space of arithmeticreachability, which is more naturally described by topological quantities thanby growth-based measures. The paper does not claim new analytic results in prime number theory. Itspurpose is to provide a structurally consistent interpretation of numericalobservations supported by separately archived data and software.