Navier–Stokes

How can speed grow without bound?

OpenAI’s proposed construction starts from rest under a smooth external force. Fluid spirals inward and exits along the axis. As the intense core shrinks, speed grows without bound while whole-flow kinetic energy remains bounded.

The teal, blue, and copper streamlines are inspired by OpenAI’s illustration. This is an artistic reconstruction, not a fluid solver or verification of the proof. Geometry and pulse rings are schematic. Four brighter highlight trails make the motion trackable. Rotation establishes itself for two seconds before contraction begins; its displayed angular rate is a compressed mapping of characteristic speed divided by radius. The moving highlights indicate direction only; they are not computed particle trajectories. Paired wave bands illustrate the location and role of the oscillatory corrections; their amplitudes, spacing, and timing are schematic. The vortex keeps a fixed axial height to isolate radial thinning. Below 12% of the reference radius, radial magnification keeps its structure visible; the zoom factor is labeled. The separate cross-section always shows the unmagnified radius ratio.

The readouts illustrate asymptotic scaling, not measured values of the constructed solution; t = 0 is a reference for these ratios, not the initial rest state. The time t is normalized so the proposed singularity occurs at t* = 1; it is not wall-clock seconds. Readouts follow normalized leading-core scaling laws with illustrative h = 0.008: speed ∝ τ^−½−h, radius ∝ τ^½, energy ∝ τ^½−3h. τ is time remaining. The timeline stops before zero.

The surrounding bands mark the annulus where oscillatory momentum transport compensates for the background’s imbalance. Further corrections are essential; this animation does not calculate that cancellation.