Geometric Derivation of the Scalar Constant ψ

This article formally defines Newman’s Omega Operator (Ω) as a strict affine contraction on ℝ, together with its associated scalar constants. The operator is governed by a coupling parameter λ derived explicitly from the Golden Ratio (φ) and Archimedes’ constant (π). Although Ω is contractive, an explicit inverse mapping (Ξ) is also defined, permitting reversible reconstruction of pre-image states within the stated affine framework. A principal contribution of this work is a geometric reinterpretation of the scalar constant ψ = φ^(1/φ). Although ψ is classically known through its relation to 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 Ξ, the paper shows that ψ corresponds to a unique geometric configuration characterized by an aspect ratio r_ψ ≈ 1.629689366035073, distinct from the standard Golden Ellipse (r = φ). This establishes a geometric interpretation of ψ that is independent of its traditional algebraic formulation. All numerical values are reported under a consistent 15-decimal truncation policy to promote deterministic reproducibility across computational platforms.