The Navier–Stokes equation is normally introduced as a continuum law: one postulates a smooth fluid, writes conservation of mass and momentum, and closes the stress by the Newtonian constitutive rule. This paper gives the QTT derivation in a stronger form. Conditional on Quantum Traction Theory (QTT), the smooth fluid is not primitive. It is a coarse-grained projection of a finite-capacity address ledger. Bundle conservation gives mass continuity. Tick-wise momentum exchange gives Cauchy momentum balance. A5 isotropy, A6 finite capacity, A7 bundle closure, frame indifference, and near-equilibrium KMS structure force the first-gradient stress closure to the Newtonian form. The classical Navier–Stokes equation is recovered as the hydrodynamic shadow of the substrate. The paper supplies the mathematical spine. It states explicit substrate hypotheses, proves coarse-graining lemmas (bounded-averages, discrete-flux-balance, ledger-Laplacian), writes a capacity-preserving discrete substrate analog of the Clay equation (a lattice-BGK dynamics with QTT admissibility limiter), and proves global existence for that lattice dynamics on finite periodic address lattices. The proof is not a proof of the Clay Millennium Problem for the continuum PDE on idealized R³. It is a theorem about the QTT physical substrate: for every finite substrate time, all physical address observables remain bounded, all resolved coarse fields remain finite, and no state requiring support below VSQ = 4π ℓ̃³ exists. The continuum singular configuration requested by the Clay scenario is therefore not a physical state in QTT. The paper also gives a quantitative saturation bound. With ecap = ℏc/ (4π ℓ̃⁴) for energy density, τcap = ℏc/ (cA ℓ̃⁴) for stress, and Gcap = τcap/ (2μ) for strain rate, a continuum Navier–Stokes extrapolation must be abandoned no later than the first scale at which the local energy density, stress, or strain approaches the A6 capacity ceiling. A putative blow-up fit |D| ~ A (T* − T) ^ (−γ) is cut off before T* − T ≲ (A/Gcap) ^ (1/γ). This is a parameter-free structural bound once the QTT substrate scale and the measured fluid viscosity are fixed; it is not a fitted regularizer. Finally, the paper connects the substrate result to the Onsager regularity threshold. The filtered energy flux scales as Πᵣ ~ ρ (δᵣ v) ³/r; hence the classical 1/3 exponent separates vanishing flux from persistent ultraviolet flux. In QTT, the cascade cannot proceed below ℓ̃ and cannot exceed Πcap = ℏc²/ (4π ℓ̃⁵). The physical meaning of anomalous dissipation is therefore ledger-theoretic: visible coarse energy is not destroyed; it is transferred into anchored modular charge and hidden substrate sectors at the capacity-bounded ultraviolet end of the cascade. The standard equation remains empirically secure. The QTT-novel results are: (i) the substrate theorem for the form of Navier–Stokes from A1, A3, A5, A6, A7; (ii) the address-lattice BGK construction with global existence on finite periodic lattices; (iii) the no-blowup theorem for the QTT substrate; (iv) the explicit capacity cutoff and the saturation propositions; (v) the substrate reading of the Onsager 1/3 threshold and the visible-energy-flux ceiling. The claim that Clay-type physical blow-up belongs to the continuum idealization, not to the geometry of the physical fluid, is now a conditional theorem rather than a verbal argument. The paper does not claim that QTT axioms are established physics; the derivation is conditional on them. It does not claim to solve the Clay Millennium Problem for the abstract continuum PDE on R³; it works on a discrete substrate. It does not claim a new ordinary-fluid deviation from textbook Navier–Stokes. It does not claim to predict species-specific viscosities parameter-free; transport coefficients remain Green–Kubo correlators on the address lattice. It does claim that classical fluid mechanics is a derived continuum projection of a discrete substrate with finite local capacity, that this is the same substrate that already underwrites quantum mechanics, gravity, and the Standard Model charge ledger in the broader QTT corpus, and that the physical-fluid version of the Clay scenario is structurally forbidden on that substrate. Status: conditional substrate theorem inside the QTT axiomatic framework, not a claim that the axioms are independently established physics. The paper claims that, conditional on A1, A3, A5, A6, A7, the form of Navier–Stokes is forced as a continuum projection, the discrete substrate analog admits a global forward solution, the Clay blow-up configuration is not in the physical state space, and the Onsager 1/3 threshold has a parameter-free substrate interpretation.
Ali Attar (Mon,) studied this question.