A Diagnostic Scalar for Gravitational Time Dilation in Static and Stationary Spacetimes: Constraint Intensity, Tidal Deformability, and Compactness Inference

We introduce a scalar quantity S(x), defined as the inverse of the lapse function in static spacetimes, and derive from it the dimensionless constraint intensity σ(x) = S(x)² - 1. This construction is fully equivalent to standard general relativity and introduces no new dynamical degrees of freedom. The central result is the exact relation σ = z(2+z), where z is the gravitational redshift, together with its direct inversion χ = σ/(1+σ), where χ = 2GM/(rc²) is the compactness parameter. We further show that the dimensionless tidal deformability satisfies Λ = (64/3) k₂ [(1+σ)/σ]⁵ (geometric units G=c=1), and express standard TOV+tidal solutions for four representative equations of state (APR4, SLy, MS1, WFF1) in σ-space. The resulting Λ(σ) curves separate clearly by EOS stiffness and are consistent with the GW170817 tidal constraint. We extend the framework to Kerr spacetime, where σ diverges at the ergosphere. The scalar σ serves as a universal classifier across nine orders of magnitude in gravitational strength and defines a multimessenger null test of general relativity: σ_EM = σ_GW.