Structural Reduction and Necessary Conditions for Coherent Triadic Accumulation in the Three-Dimensional Navier–Stokes Equations

We investigate the continuation problem for the three-dimensional incompressible Navier–Stokes equations from a structural, assumption-free perspective. Using the exact Fourier–helical representation and a dyadic shell decomposition, the nonlinear term is reformulated in terms of triadic interactions, allowing a scale-resolved analysis of energy transfer. Within this framework, we establish a complete structural reduction of the nonlinear dynamics. All cross-scale and non-coherent interactions are shown to be perturbatively controlled on every finite time interval and cannot produce non-integrable accumulation in weighted Sobolev norms on compact subintervals. As a result, any potential finite-time blow-up must be supported by a sharply restricted class of residual mechanisms. More precisely, we show that non-integrable accumulation of positive Sobolev-weighted transfer can occur only through either large-transfer same-scale interactions or endpoint accumulation of perturbative remainder contributions. All other interaction channels are excluded as possible sources of divergence by structural and energetic arguments. The analysis is entirely assumption-free and does not rely on any phase closure, temporal localization, or statistical modeling. It therefore provides a complete obstruction formulation of the continuation problem: blow-up is reduced to the viability of a minimal set of explicitly identified mechanisms. We further show that these residual mechanisms persist because the incompressible Navier–Stokes equations do not constitute a thermodynamically complete system. Interpreting the incompressible equations as a singular limit of the compressible formulation, we identify the loss of entropy-based dissipation as the structural origin of the missing control on positive nonlinear transfer. Motivated by this observation, we introduce a minimal ε-retained thermodynamic extension that restores a remnant of the free-energy dissipation mechanism. Under this extension, we show that the positive transfer becomes integrable and that both residual blow-up mechanisms are eliminated under the stated closure condition. This yields a precise conditional closure of the continuation problem. The results clarify the exact scope and limitation of Navier–Stokes-based analysis and reduce the global regularity problem to the question of whether a thermodynamic-type dissipation principle can be rigorously derived within, or as a limit of, the governing equations.