Deriving Golden Ratio Divided by Pi

One of the most intriguing observations emerging from the study of recognition-based physics is the recurrent appearance of the dimensionless ratio φ/π ≈0.5149, where φ≈1.618 is the golden ratio and π≈3.14159 is the well-known constant from circular and spherical geometry. While earlier works within the Recognition Physics framework have empirically identified this ratio as the optimal parameter for minimal informational overhead in a three-dimensional (3D) setting, a rigorous derivation from first principles is still lacking. In traditional treatments, such constants are often viewed as numerological coincidences; however, a formal derivation would establish that this ratio is an inevitable consequence of the underlying geometry and the requirements for dual recognition in physical systems.