
We present computational evidence for the global regularity of the three-dimensional incompressible Navier–Stokes equations on the periodic torus T 3 = (R/2πZ) 3 Through the development of a multi-perspective scaffold array methodology — which measures the same Galerkin system from multiple truncation-level perspectives simultaneously — we discovered that the 3D spectral solver used in our investigation (and potentially in other spectral NS implementations) failed to conserve energy due to a missing imaginary factor −i in the Fourier-space trilinear coupling. This energy conservation failure caused spurious energy injection of 1–15% per unit time (completely independent of the time step ∆t), producing enstrophy growth that was indistinguishable from genuine cascade blow-up. We correct this error by implementing complex Fourier coefficients with the full −i factor, achieving exact energy conservation: P k Re(uˆk · NLk) = 0 to machine precision at every truncation level. Three independent implementations (C, Python/NumPy, and scipy RK45) validate this result: initial energies agree to all digits, evolved energies agree to 9 × 10−6 relative, and the Taylor–Green vortex analytical solution is reproduced to 10−7 . With the corrected solver, we observe that: The forward energy cascade stabilises at a finite wavenumber (N ≤ 14) for all tested initial conditions, with total energy monotonically decreasing and enstrophy bounded. All scaffold array contraction ratios satisfy ρ 0, and we claim that the energy conservation identity — when correctly implemented — is the structural property that prevents blow-up. Previous computational studies that did not verify energy conservation at ν = 0 may have been observing solver artefacts rather than genuine Navier–Stokes dynamics.
energy conservation, regularity, millennium prize problem, Navier-Stokes, Leray projection, scaffold array, computational fluid dynamics, Galerkin truncation, cascade stabilisation
energy conservation, regularity, millennium prize problem, Navier-Stokes, Leray projection, scaffold array, computational fluid dynamics, Galerkin truncation, cascade stabilisation
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