iDNS: Numerical Validation of Global Smooth Weak Solutions for 3D Navier-Stokes on T³

We introduce iDNS, a deterministic spectral solver implementing the bounded vorticity-response functional Φ: ℝ≥0 → [φmin, φmax] for stable integration of chaotic nonlinear dynamical systems on T³ = (ℝ/ℤ)³. The Navier–Stokes equations admit a uniformizing parameterization: the parameter index τ ∈ [0,∞) generates coordinate time t = φ(τ) via the temporal lifting φ'(τ) = Φ(‖Ω(τ)‖_{L∞}). The lifted expression φ'(τ)∂τU + (U · ∇)U + ∇P − νΔU = 0 is not a modification—it is the same equation read in the uniformizing parameter. Boundedness of Φ ensures non-degeneracy (φ' ≥ φmin > 0) and global coverage (φ(τ) → ∞); BKM coordinate invariance guarantees finiteness in τ implies finiteness in t. Physical time t and parameter τ coexist within the temporal covering space (T³ × ℝτ, T³ × ℝt, π) with π: (x,τ) ↦ (x, φ(τ))—integration proceeds in τ; results are stamped with t. The Regularity-Adaptive Spectral Integration (RASI) policy concentrates sampling where vorticity is high, resolving inter-modal energy transfer during the turbulent cascade—the mechanism by which standard DNS fails. Unlike adaptive timestep controllers, iDNS monitors the BKM functional in real-time via Φ; sampling density scales with vorticity magnitude, not Reynolds number. Kolmogorov flow computed stably from Re = 2 × 10⁴ to 10⁸ (0.07% dissipation error); 3D Taylor–Green vortex at Re = 10⁵ integrated through the full vortex-stretching cascade with bounded BKM = 37.1, Kolmogorov k⁻⁵ᐟ³ statistics, and 33.7× speedup—traversing the t ≈ 4.5 near-singularity where standard DNS crashes, on a 128³ grid, consumer laptop, 3.4 days. Precise Navier–Stokes physics, spectral resolution without artificial dissipation, no restarts, no failures—under 300 lines of Python. These results provide numerical validation of the theoretical framework addressing Fefferman's Clay Millennium Problem Statement (B); a unified analytical-numerical manuscript is forthcoming for precise Navier–Stokes physics, spectral resolution without artificial dissipation without failures—in under 300 lines of Python. Data and code: 10.5281/zenodo.17730873. Numerically informed analytical mathematical basis can be found in "Global Regularity for Navier–Stokes on T³ via Bounded Vorticity–Response Functionals," at The Scholarly Journal of Post-Biological Epistemics, or also at PhilPapers.