The viscous scale as a test of blowup constructions
One necessary condition. Two constructions published a day apart, twenty thousand orders of magnitude between them. And a prior result of ours that measured a consequence of it, weeks early.
Justin Hill, Independent Research. Paper four, following the viscous response study. Two finite-time blowup constructions for incompressible fluids went public this week, both formalized in Lean. Terence Tao's exposition of the first noted the methods have “high likelihood of also extending to Navier–Stokes,” stated as an expectation. This puts a number under it. Every constant is read from the published Lean sources; every figure regenerates in seconds from the public repositories.
Paper three measured a consequence of this criterion, weeks early. This paper measures the premise that implies it.
k is ν independent ⇒ νk2 regular ⇒ linear response
and separately k is ν independent ⇒ not at the viscous scale
These are implications, not equivalences, and the direction matters. Regular perturbation only requires k = o(ν−1/2); a wavenumber going like ν−1/4 would still give linear response and would still not be ν independent. So the August paper, which measured linear response, does not on its own establish the Euler scale. It measured a consequence. The premise is measured here, from the same runs but a different diagnostic.
What is fair to say: two results from the same runs, obtained months apart by different methods, converge on one fact about the certified profile. The August measurement is the same concept reached empirically rather than deductively, narrower in scope, and carrying quantities the deductive version does not produce.
Leray's lower bound forces any Navier–Stokes blowup to the viscous scale, where the nonlinear and dissipative rates are both 1/(T−t). The energy identity caps the total enstrophy budget, and that cap survives forcing. Together they give a necessary condition on the wavenumber:
k(t) ∼ ( ν (T − t) )−1/2 ⇔ 2 log kn ≤ −log τn
This constrains the scaling data alone, so it can be checked without compiling a proof. The logic is deliberately one sided. Violated is a refutation the remaining lines cannot repair, because the energy identity holds for any solution. Satisfied proves nothing; it is the absence of this one refutation.
| Construction | What was measured | Verdict |
|---|---|---|
| Alpöge–Buckmaster ladder, stage 1 | exceeds the bound by e17551 | violated |
| OpenAI ChartScales, n = 10 to 106 | ratio 0.24 to 2.57 | satisfied |
| Certified Chen–Hou profile (our runs) | d log k / d log ν = −0.015 to −0.072, mean −0.037 | Euler scale, not asymptotic |
The first two checks are against constructions whose scaling was already worked through by hand, so neither could surprise the criterion. The third applies it to a certified Chen–Hou profile integrated at three viscosities, from runs produced weeks before either formalization was public.
Under the Leray rescaling a viscous scale flow collapses to spread ≈ 1. Measured spread is 7.3 to 9.4, against the ν1/2 = 10.0 predicted for a wavenumber set by the inviscid dynamics. It does not collapse. Tripling the viscosity moves the wavenumber by under three percent.
The slope steepens from −0.0145 to −0.0720 across the window and the runs stop at t/T* = 0.83, so this is not asymptotic. It would have to steepen roughly sevenfold to reach −1/2, and these runs cannot exclude that.
This does not verify or refute either proof. Neither has been compiled outside the groups that wrote them. It is a scaling analysis, the paper says so on page one, and Section 5 names the step most likely to be wrong. The Alpöge–Buckmaster theorem concerns the inviscid forced system and is untouched; the obstruction is to extending the method to Navier–Stokes, which was an expectation rather than a claim of theirs.