Deterministic Origin of Quantum Tunneling in the Quantized Space I11

This paper presents a deterministic geometric explanation of the quantum tunneling effect within the cyclic quantized number space I11, defined on the interval from 0 to 11.

The model introduces a mirror axis at position 5.5, a minimal structural offset of 0.5, and a universal drift constant of 1 divided by 1444.

At the fractal depth level k = 721, the accumulated drift reaches exactly half the mirror offset (0.5), enabling a deterministic coupling through the symmetry axis.

This replaces the probabilistic interpretation of tunneling with a geometric resonance condition based on spiral drift and mirror symmetry.

The study defines a recursive projection operator that combines cyclic closure with internal fractal subdivision sequences:

F1 = [2, 1, 2, 2, 4],

F2 = [1, 1, 0.5, 0.5, 1, 1, 1, 1, 2, 2],

and

F3 = [0.5, 0.5, 0.5, 0.5, 0.25, 0.25, 0.25, 0.25, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 1, 1, 1, 1].

These hierarchical sequences maintain the cyclic boundary while introducing quantized internal geometry.

Through three spiral turns, the model demonstrates that all external states above the mirror axis converge toward the fixed point at 5.5,

but true locking occurs only when the accumulated drift exceeds 0.5 at k = 721.

This threshold marks the onset of tunneling as a deterministic transition between mirrored domains.

The result provides a unified link between microscopic resonance and macroscopic quantum behavior, connecting tunneling, reflection, and spin coupling under a single quantized spatial framework.