The Trinity Triangle Meets Euler: Geometric Structure of the φ, e, π Relationship within the ENSO Framework Cascade

This paper examines the geometric relationship between $\varphi$, $e$, and $\pi$ within the ENSO Framework cascade. It shows that $e$ occupies a unique weighted geometric-mean position between $\varphi$ and $\pi$, given by e=ϕaπ1−ae=\phi^a\pi^{1-a}e=ϕaπ1−a, with $a\approx 0.21813$. This position generates the Trinity Triangle, whose sides $\varphi ,e,\pi$ produce interior angles close to 30∘30^\circ30∘, 60∘60^\circ60∘, and 90∘90^\circ90∘. The paper also relates this geometry to the Trinity Tetrahedron result and to the wider ENSO residual family. A Python validation script is included to reproduce the numerical claims.

For further information about the ENSO Framework, please contact Eric Needham:ensotheory1@gmail.com