The ruler that holds its ground.
Close the jaws, speed up the flow, and discover why the way you measure a field changes what you can control.
Reduced motion is enabled. Choose still frames with the controls.
The critical ruler stays unchanged as the jaws close.
What this instrument measures
The unforced whole-space Navier–Stokes rescaling is uλ(x,t)=λu(λx,λ²t). Length scales by1/λ, velocity byλ and time by1/λ². All readings are ratios to a normalized baseline, not equipment units. The caliper is an invented analogue for characteristic length; no fluid simulation is running.
Homogeneous Ḣˢ ratio: λ¹⁺ˢ⁻ᵈ⁄²
Energy is a squared L2 norm, while the other rulers display norms or seminorms. The critical order is s=d/2−1: half a derivative in3D and zero derivatives in2D. Energy falls like1/λ in3D, but stays constant in2D. The H1 option is the homogeneous gradient seminorm, not the full inhomogeneous H1 norm.
These exact identities require whole-space profiles with the indicated finite homogeneous norms. A fixed bounded domain or a full inhomogeneous Sobolev norm has different bookkeeping. The dimension selector changes the mathematical domain; the illustrated bench remains a three-dimensional object. Scaling alone proves neither singularity nor regularity, and the comparator does not test an existence theorem.
Tao: scaling and critical homogeneous spaces ↗
Fefferman: the official problem statement ↗