Geometric Derivation of the Scalar Constant ψ

This article formally defines Newman’s Omega Operator (\Omega) and the associated scalar constants within the Crown Omega Framework. The operator is introduced as an affine contraction on \mathbb{R}, governed by a coupling parameter \lambda derived explicitly from the Golden Ratio (\phi) and Archimedes’ constant (\pi). While \Omega itself is contractive, an explicit inverse mapping (\Xi) is defined, enabling reversible reconstruction of pre-image states within the framework. A principal contribution of this work is the geometric reinterpretation of the scalar constant \psi = \phi^{1/\phi}. Although \psi is classically known as the limiting value associated with infinite power towers, it is shown here to arise naturally as an emergent geometric quantity under the action of the Omega Operator. Through inverse geometric reconstruction using \Xi, we demonstrate that \psi corresponds to a unique, non-canonical geometric configuration characterized by an aspect ratio r_{\psi} \approx 1.629, distinct from the standard Golden Ellipse (r = \phi). This establishes a spatial interpretation for \psi that is independent of its traditional algebraic formulation. All numerical values are reported using a consistent 15-decimal truncation policy to promote deterministic reproducibility across computational platforms.