## Abstract The Clay Millennium Problem on the global regularity of the Navier–Stokes equations asks whether smooth initial data can develop finite-time singularities. We argue that this question is ill-posed as a statement about physical fluids. The Navier–Stokes equations are a Newtonian approximation that permits arbitrarily fast transport of energy. We show that once flow is reconceptualized as bounded energy transport—expressed solely through continuity and a causal flux bound—finite-time blow-up is kinematically impossible. ## The Clay problem and its hidden assumption Implicit in the formulation of the Millennium Problem is a Newtonian assumption: the velocity field $u$ is unconstrained in magnitude, and transport may occur arbitrarily fast. Nothing in the equations forbids the instantaneous delivery of large amounts of energy into an arbitrarily small region. This assumption is mathematical, not physical. ## Flow as energy transport In physical fluids, we observe the transport of energy. We take as primitive an energy density $u(x,t) \ge 0$ and an energy flux $S(x,t)$, related by the continuity equation: $$ \partial_t u + \nabla\cdot S = 0 $$ ## The only empirical bound: causal transport All observed energy transport satisfies a causal bound: energy does not propagate faster than a maximal speed $c$. $$ |S(x,t)| \le c\,u(x,t) $$ Thus, physical flow is a transport process with bounded speed. ## Why continuity plus a speed bound forbids blow-up Finite-time blow-up would require the concentration of a nonzero amount of energy into an arbitrarily small region (radius $r \to 0$). Integrating the continuity equation over a ball $B_r$: $$ \frac{d}{dt}\int_{B_r} u\,dx \le \int_{\partial B_r} |S|\,d\sigma \le c\int_{\partial B_r} u\,d\sigma $$ The maximal inflow scales like $c \, r^2 \sup u$. For a point singularity to form, the energy density must scale like $u \sim r^{-3}$. However, the required inflow rate to sustain this density would scale like $r^{-1}$. The available inflow capacity (Surface Area) scales like $r^2$. Because $r^2$ vanishes faster than $r^{-1}$ diverges, the singularity creates a **Flux Bottleneck**. > Energy cannot be supplied to a point fast enough to produce a finite-time > singularity if transport speed is bounded. ## Statement relative to the Clay problem The original formulation of the Clay Millennium Problem is purely mathematical and permits flow regimes that are physically unrealizable. While mathematical regularity can be proved conditionally by imposing topological constraints (as shown in our companion technical paper), the physical resolution is simpler: the speed limit $c$ provides the ultimate constraint. ## Conclusion Water does not blow up because it flows. Flow is the continuous, bounded transport of energy. Once this is taken as fundamental, finite-time singularities are excluded by geometry alone. The Navier–Stokes blow-up problem is therefore not a question about fluids, but about the consequences of removing causality from a mathematical model.