A Layered Structural Theory of Prime Growth: Integer Layers, Transition Dynamics, and Boundary Chaos

This paper introduces a layered structural framework for studying prime growth through the integer part of the normalized prime ratio floor (pₙ / n). The central idea is that the prime-index sequence can be decomposed into stable integer layers, revealing a previously unexplored level of structural organization within prime numbers. The framework introduces and studies several structural quantities, including: • Integer layers Tₖ• Layer lengths Lₖ• Transition points τₖ• Boundary-chaos measures Bₖ• Oscillation widths W_ (k, j) • Sign-change dynamics Nₛign An empirical law is proposed for the integer part of the prime ratio, achieving approximately 99. 687% accuracy on the first 10, 000, 000 prime indices. Additional empirical laws are developed for layer lengths and transition points, providing a coherent geometric description of the layered hierarchy. The work further develops a multi-scale structural chain linking local prime-gap fluctuations to large-scale layer organization: Prime gaps → Threshold probabilities → Sign changes → Oscillation widths → Boundary chaos → Transition points → Layer lengths → Integer layers Numerical experiments support the existence of stable integer layers, systematic layer growth, predictable transition behavior, and asymptotically localized boundary-chaos regions. The framework is presented as a structural complement to existing approaches in prime-number theory, including the Prime Number Theorem, Cipolla-type asymptotic expansions, Rosser–Schoenfeld bounds, and probabilistic prime-gap models. Rather than replacing these theories, the framework seeks to describe an additional level of organization that emerges within the prime-index sequence. Several components of the framework remain empirical and exploratory, and the paper identifies multiple open problems requiring future theoretical investigation.