Deterministic Layered Exponential State Machine Model

In this study, it is demonstrated that the seemingly random gaps in the prime number sequence ($\Delta p$) and the exponential collapse steps ($k$-sequence) in the Collatz sequence are governed by a common deterministic state machine. The system consists of fractal shells with a fundamental core unit $S_1 = 6$, following the exponential growth law $S_n = 6 \cdot 2^{n-1}$. The shell rupture occurring as a result of internal energy accumulation ($\sum k = 12$) enables the sub-dimension to reach saturation and perform a quantum leap to a higher dimension.