A Geometric State Variable for Early Turbulence Intensity-Geometry Separation, Vorticity-Weighted 3D Eigengeometry, and Falsifiable Tests of a Geometric-Embryo Framework

Early turbulence is usually described through amplitude-, energy-, spectrum-, instability-, or coherent-structurebased quantities. These descriptions are fundamental, but total vorticity intensity does not uniquely determinethe three-dimensional spatial organization of vorticity. Here we introduce a complementary state variable: anormalized vorticity-weighted 3D geometry tensor, C_omega, whose eigenvalues and eigenvectors describe theglobal spatial organization of |omega|^2 independently of a uniform rescaling of vorticity magnitude. Theprincipal novelty is therefore not the proposal of another vortex-identification criterion, nor a universalturbulence-onset predictor, but an explicit separation of early vortical state into intensity and geometrycoordinates that can be measured, transformed, compared, and falsified independently. In controlled tests, thedescriptor was invariant to rigid coordinate changes to numerical precision and tracked imposed 3Dmorphology. Two fields normalized to identical total enstrophy nevertheless possessed clearly different eigengeometries, directly demonstrating that equal intensity does not imply equal 3D state. Moderate-resolutionNavier-Stokes tests further showed that early geometry can retain dynamical information absent from intensity insome comparisons, while a stricter matched-family test showed that geometry is not universally superior tointensity as a predictor. We therefore propose the 'geometric embryo' as a state-description framework: earlynonlinear flow should be represented in a joint state space containing intensity, 3D geometry, geometricdynamics, and topology. A targeted literature comparison found extensive work on vortex identification, coherentstructures, transition mechanisms, vortex reconnection, pancake-like vorticity structures, and turbulence onset,but did not identify the same normalized vorticity-weighted covariance/eigenstate formulation explicitlydeveloped as an intensity-independent coordinate for early turbulence. This novelty claim is intentionallymethodological and remains subject to broader literature review and external validation.