Newman’s Omega Operator and the Definition of the Newman Constants

Abstract This work introduces Newman’s Omega Operator (Ω), a formally defined affine contraction operator designed for direct arithmetic evaluation of geometric scaling behavior. Unlike algorithmic stabilization methods or iterative numerical solvers, the Omega Operator is defined as a closed-form arithmetic transformation acting on real values, with a unique fixed point and exact analytic inverse. From this operator, a structured family of constants—collectively termed Newman’s Scalar Constants—is derived. These include the Omega Naught constant (Ω₀), the contraction factor λ, the logarithmic scaling constant κ, and the derived geometric scalar ψ. Importantly, ψ is not introduced as a primitive constant but is shown to emerge as a consequence of the Omega contraction structure. The framework enforces a strict 15-decimal truncation policy to ensure deterministic reproducibility across platforms. A Prime Boundary Seal (P = 23) is defined to quantify the precision horizon of the contraction sequence, establishing a mathematically bounded limit for stable geometric evaluation. This work presents a minimal, arithmetic-first formulation intended for geometric parameterization, not physical simulation or algorithmic optimization. The emphasis is on derivational clarity, numerical determinism, and formal attribution of constants arising from the operator itself. Key Contributions:• Formal definition of the Omega Operator as a reversible affine contraction• Introduction of Omega Naught (Ω₀ = φ + π) as a structural constant• Definition of Lambda (λ = φ / (φ + π)) as the intrinsic contraction ratio• Derivation of Kappa (κ = ln(1/λ)) as a geometric scaling exponent• Formal classification of ψ = φ^(1/φ) as a derived constant, not a primitive• Introduction of the Prime Boundary Seal (P = 23) as a precision bound• Strict truncation-based numeric policy to ensure reproducibility• Clear separation between arithmetic derivation and later application