This research investigates the quasi-crystalline renormalization of integers using Fibonacci intervals and CMZ (Convergent Mathematical Zones) structures. We define a deterministic “elevator” operator that maps integers between CMZ levels, revealing a discrete self-similar structure governed by Fibonacci numbers and the golden ratio. Numerical examples illustrate the renormalization and climbing dynamics across CMZ layers. This work is open for collaboration with researchers interested in discrete mathematics, quasi-crystals, and integer sequence structures.