Constraint Softening as a Structural Criterion for Coherence in Nonlinear Systems

This paper introduces an operational diagnostic based on the lowest eigenvalue of the variational Hessian that determines whether nonlinear systems admit coherent, bounded dynamics. The diagnostic provides a structural criterion separating intrinsically radiative or turbulent regimes from softened constraint regimes in which coherence is possible. The framework is illustrated using nonlinear wave equations, gauge field theories, and applied plasma and material systems. The results are relevant to nonlinear dynamics, mathematical physics, and stability analysis.