Navier–Stokes Through Coherence: A Residual Growth Criterion for Structural Persistence

This work introduces a coherence-based perspective on the Navier–Stokes stability problem, reframing structural persistence as a question of residual growth under admissible transformations. Rather than analysing solutions of the governing equations directly, the framework considers how observable structure behaves under a constrained class of symmetry-preserving operators. Velocity fields are treated as elements of a scaling family within a Hilbert-space representation, and structural compatibility is assessed through an observable invariant map. A minimal residual is defined between the original and transformed representations, and its growth behaviour across scales is used as a diagnostic criterion. The central hypothesis is that coherent flow regimes correspond to polynomially bounded residual growth, while accelerated growth indicates structural incompatibility and potential breakdown. Illustrative results demonstrate that distinct structural regimes exhibit qualitatively different residual growth signatures. These are presented using raw, semi-log, and log–log representations to highlight the separation between polynomial and accelerated scaling behaviour. This approach does not attempt to solve the Navier–Stokes equations directly. Instead, it provides a model-agnostic structural stability criterion that may be applied in settings where governing equations are unknown, partially known, or computationally intractable. The framework is positioned relative to classical PDE-based methods and modern data-driven approaches, including Koopman operator analysis and equation discovery, but differs in that it does not seek to reconstruct underlying dynamics. Instead, it evaluates structural compatibility directly. The results presented are illustrative and do not constitute empirical validation. Future work will focus on applying the framework to simulated and experimental flow data, and on assessing robustness across different observable representations and admissible transformation classes.