Exact Coupling of π and φ in the Quantized Space I11

This publication presents the first successful derivation of an exact algebraic and geometric coupling between the circular constant pi and the golden ratio phi within the finite, mirror-symmetric number space I11 (interval 0 to 11).

Building on the author’s earlier discovery, A New Perspective on Infinite Number Spaces through Prime Number Distance and Quantization (Zenodo DOI: 10.5281/zenodo.14853309), the work introduces a quantized mirror correction that closes the classical open relation between pi and phi and establishes a provable equality inside I11.

For the first time, irrational constants are represented exactly and without distortion within a finite numerical topology.

This is made possible through the mirror operator M(x)=11-x and the quantized inversion map Iq(x, alpha).

The paper proves the Exact Coupling Theorem, provides full formal proofs of existence, uniqueness, and stability, and includes a geometric visualization linking the pi-circle and phi-spiral around the symmetry axis at 5.5.

This result demonstrates that transcendental and irrational constants can coexist in a closed, quantized, and mirror-symmetric space, marking a foundational step toward a new finite representation theory of number and geometry.