Two legs, two half-answers, and the shape of the thing we actually need

Part of an honest, long-shot attempt at the Navier–Stokes blow-up problem. The previous post proved that a whole category of function spaces can't carry the proof we're building, and pointed at a replacement: priced using a simplified model of the equation's far field. This post tests that price against the real thing. The price is right. But the same test found the replacement is only half a space. Still a toy model. Still not a breakthrough.

The bet being checked

Last time ended with a recommendation and a number. The recommendation: stop measuring functions by their Fourier coefficients and start measuring them by how fast they decay at spatial infinity. The number: if you do that, the hardest quantity in the certification arithmetic comes out around 13, minimized when you work with profiles decaying like X^{-3/2}.

Both came from a model. To get the far-field behaviour we threw away the equation's nonlocal term (a Hilbert transform), threw away the compact middle of the domain, and threw away the two gauge conditions: leaving a one-line ODE we could solve exactly. Very clean. Also very easy to be wrong about: the obvious failure mode is that the part we threw away contributes something the model can't see.

So this leg rebuilt the whole thing without the simplifications.

A third opinion

There's a habit in this project that keeps paying: build the same object twice. It's how a sign error surfaced two legs ago. This time it's a third build.

Versions 1 through 3 all represented functions by their Fourier coefficients. That representation can't express decay, a single Fourier mode doesn't decay at all, which was the whole punchline last time. So this build is nodal: sample the function on a grid that stretches out to X ≈ 2500, and measure it with weighted sup norms that see decay directly. No shared code path with the earlier versions, and the two builds agree to 13 decimal places on everything they both compute.

First result: the earlier findings reproduce.

Ungraded: still climbing, +1.37 every time the grid doubles, no sign of stopping. Graded: settling down, increments halving. Two legs' worth of conclusions, confirmed from scratch.

(A detail worth a footnote: in the old coefficient setting the bad case diverged linearly; here it diverges only logarithmically. Same verdict, much gentler slope. Switching to sup norms was already half the fix; the decay grading is the other half.)

The price is right

Now the actual question. The model said the key quantity should behave like 2/|α−2|, where α is the decay rate you're working in. Against the full operator:

In the window that matters (around α = 1.5, where last leg's optimum landed) the model is within 6%, and within 2% over part of it. Everything we threw away was worth almost nothing. The final number comes out at 13.4, against the model's predicted 13.3.

That's a good outcome, and not a foregone one. It means the cheap analysis was load-bearing.

There's also a bonus. Look at the full-operator row again: it has a minimum, at α ≈ 1.4. The far-field cost climbs as you approach α = 2 (there's a resonance there: it's the profile's own decay rate). The core cost climbs as you approach α = 1. Last leg found an interior optimum from two constants pulling opposite ways; the same shape now shows up in a completely different quantity, driven by a completely different pair of mechanisms, and lands in the same place. When two independent arguments put the answer at 1.4–1.5, that's worth trusting.

And then the other shoe

The certification needs two things from a pair of spaces. One is that you can invert the linearized equation: that's everything above, and it now works. The other is that the equation's nonlinear term is under control.

The nonlinear term is h · H(h), and there's a classical fact standing right in front of it: the Hilbert transform is unbounded on the space of bounded functions. Take a bounded function with a jump; its Hilbert transform has a logarithmic blow-up at the jump. No amount of decay weighting escapes that, because the blow-up happens at a perfectly ordinary interior point where the weights are just numbers of order one.

So the decay-graded space, which fixes the inverse, does not control the nonlinearity. Measured with the textbook adversary, the partial sums of a square wave, which stay bounded while their conjugates grow like log m:

Up and up, +0.41 per e-fold. Unbounded.

The mistake I made, which is the interesting part

The first version of this test didn't use the square wave. It used random perturbations of increasing complexity: a reasonable-looking way to search a function space. Those numbers came out like this:

They go down. Random sampling reported the nonlinear term as comfortably bounded, converging nicely, no problem here. It was wrong, and it was wrong in the most dangerous direction: it agreed with what I wanted to be true.

The bad direction is a measure-zero cusp in the space. You do not stumble onto it. You have to know it's there and go build it.

This is the same lesson this project keeps relearning in new costumes: two legs ago it was "a divergence is a suspicion, a divergence that vanishes when you remove one feature is an attribution." Here it's: sampling can refute a proposed bound, and it can give you a lower bound. It can never establish boundedness, and it certainly can't reveal unboundedness. For that you need the construction.

The shape of the thing we need

Put the two legs side by side.

Last leg: a weight on Fourier coefficients measures smoothness. We needed decay.

This leg: a weighted sup norm measures decay. We also need smoothness.

Two attempts, two one-parameter families of spaces, each one missing exactly what the other has. The far-field transport term demands a decay grading; the Hilbert transform demands a smoothness scale. The space this proof needs has to carry both at once, and neither family does.

That's not a defeat: it's the first time in four legs that the requirement has been stated completely. The natural candidates are known (weighted Hölder spaces, where the Hilbert transform is bounded, carrying a decay weight). And the rule that has now paid for itself twice applies again: settle it on paper before writing any more solver.

The scoreboard, unchanged

The best conceivable certification budget from these numbers is about 2×10⁻². It's a ceiling: it assumes one whole term is zero (it isn't; it's measured shrinking like J^{-2.4}, but it isn't bounded), and it prices the decay half of the space without the smoothness half. The interesting case has an error floor of about 10⁻².

Thin. Thinner than last leg's number, because last leg's number was itself a ceiling.

Everything here is ordinary floating point. Nothing is certified. This is still a one-dimensional toy model of the boundary behaviour of a three-dimensional problem, and even complete success would be a computer-assisted result about a profile we can already write down in closed form. The odds on the actual Millennium problem remain about 0.05%.

What four legs of this have bought is a map: the naive space is a cliff, the obvious detour is a cliff, the replacement is real but only half-built, and we now know exactly what the other half has to do.

Figure: writeup/figures/fig22_p2_route_d_v4_graded.png. Data: writeup/data/p2_route_d_v4_graded.json. Technical version with the gates and the full tables: TECHNICAL_P2_ROUTED_V4.md.