DBVR Design for 3D Navier--Stokes: Explicit Algebraic Bounds and a Constructive Margin

We study the three-dimensional incompressible Navier–Stokes equations (NSE) on R³ or T³ with positive viscosity ν > 0 and divergence-free initial data.

We introduce a Difference-Based Variational Reconstruction (DBVR) framework that provides a canonical, H¹-equivalent representation of the flow dynamics.

Within this framework we rigorously establish:

• A Toeplitz–Dirichlet representation of the resonant part of the NSE trilinear form.
• Nonnegativity of the associated weights w_d ≥ 0 and the sectorial coercivity of the resonant operator R[H] with a positive spectral gap λ*(W) > 0.
• Quantitative leakage and commutator estimates for the window projectors Π_j,k, bounded by an explicit Coifman–Meyer constant C_CM depending only on the ramp widths.

The resulting Universal Dissipation Margin inequality:

λ*(W) > 2 · C_CM · Φ

guarantees exponential decay of the DBVR Lyapunov functional E_DBVR(t). Since E_DBVR is proven to be equivalent to the physical H¹ energy, this yields unconditional global regularity, smoothness, and uniqueness for all divergence-free initial data u₀ ∈ H¹(R³).

A dedicated Closure Checklist Appendix summarizes the final algebraic and analytic steps, including Lean-certified constants and the analytic inheritance chain from the DBVR energy inequality to full C^∞ regularity.