The measurement that was measuring itself

Part of an honest, long-shot attempt at the Navier–Stokes blow-up problem. Five posts ago we started trying to turn a good numerical guess into something a computer could certify. Along the way we've built the right space, priced it, and found the two structural requirements it has to satisfy. This post is about finally putting upper bounds on the numbers, and discovering that the obvious way to compute one produces a confident, reproducible, completely fictional answer. Still a toy model. Still not a breakthrough.

Five legs of measuring the wrong side of the inequality

Here's an embarrassing thing that took five legs to notice properly.

The certificate we're chasing needs four constants: the defect, two contraction terms, and a curvature term. Feed them to one polynomial; if it has a positive root, a true solution provably exists near your guess. Every constant has to be an upper bound. That's the whole point: you're claiming the error is no bigger than this.

What we had been computing was the maximum over a list of test functions. That's a lower bound. It says the error is at least this. Perfectly good for diagnosis (it's how we found the space, how we priced it, how we discovered the quadratic term was unbounded in the wrong norm) and useless for a certificate. The budget we'd been quoting was an upper bound built out of lower bounds, which is not a number that means anything.

So this leg went after the real thing.

The obvious method, and why it lies

To turn a max-over-a-test-family into a genuine bound, you use duality. Instead of asking "how big does this get over the functions I thought to try," you ask "how big could it get over everything in the unit ball," and for these norms that question has a closed-form answer. On a grid it's a short computation: find the worst direction, read off the number.

We ran it. It said the operator was unbounded: the number grew steadily with resolution, J^0.5, clean as anything, and it kept growing through every variant we tried. Three different ways of setting it up, every choice of the gauge condition, same answer.

That would have been a significant negative result. It's also wrong.

Here's the problem. Our norm measures smoothness by comparing the function's values at pairs of grid points. Duality goes looking for the worst possible function, and what it finds is a spiky thing that flips sign from one grid point to the next. Measured at the grid points, that looks tame. But a list of numbers at grid points stands for an actual function, the one that interpolates them, and that function is thrashing wildly in the gaps the grid never looks at.

We can measure exactly how wildly, by evaluating the same function on a grid six times finer:

Growing like J². A smooth function of the same type, run through the identical test, comes back faithful to 3%.

So the "worst direction in the unit ball" was not in the unit ball. It wasn't within four orders of magnitude of the unit ball. The unboundedness we measured was a property of our ruler.

The same mistake, from the other side

The uncomfortable part is that we already had a lesson banked about this, and it pointed the other way.

Two legs ago we tested whether a certain quantity was bounded by trying random functions of increasing complexity. They gave smaller and smaller answers, and we nearly wrote down that everything was fine. It wasn't: the bad direction was a specific textbook construction that random search will essentially never find. The lesson we wrote down was build the adversary, don't sample, construct.

This leg is that lesson in a mirror. There, the set of directions we searched was too small, and we missed a real problem. Here, the set was too big (it included things that aren't functions in our space at all) and we found a fake one. Sampling under-reports; a sloppy ball over-reports. Neither is telling you about the operator. Both are telling you about the instrument.

That's a more useful lesson than either one alone, and it's worth stating plainly: before you trust a number about an operator, check that the set you optimized over is the set you meant.

What actually survives

The fix is to stop asking the grid what's in the ball, and only use facts that are true of the real, continuous ball. Two of them are enough:

• a function in the unit ball can't be bigger than the weight allows at any point;
• it can't change faster than the smoothness weight allows between any two points.

Both hold for the true norm, so any bound built from them is honest, whatever the grid is doing. Applying them gives a computable estimate, and for the first part of the constant it does something we've never seen in this project:

J = 125    250    500   1000   1600
    5.536  5.531  5.528 5.580  5.631

It stops. A thirteen-fold increase in resolution moves it by 1.7%. That is the first genuine, resolution-independent upper bound on any piece of this certificate, and it sits about a factor of two above the lower bound we'd been quoting, so for the first time we have the number bracketed rather than approached from one side.

The second part of the constant is still not bounded. The estimate for it is honest but too generous, and it's generous in exactly the way §1 explains: it's still paying for that spiky direction. We know from the equation itself that it ought to be finite (the inverse of this operator gains a whole derivative, which is more smoothness than the norm asks for) but knowing and computing are different, and this is now the sharpest open question in the whole programme.

The piece of Z₁ that came out clean

The other half of the leg went better.

Z₁ is the term that measures how much your simplified model of the operator differs from the real one. It has never been bounded here, in six legs. Out in the far field we model the operator by a simple ODE whose inverse we know exactly; the question is what we threw away doing that.

The answer turns out to be a two-line identity: the difference between the real operator and the model is exactly

h / (X(1+X²))  −  H(h)/(1+X²)

and nothing else. Checked against the full operator, it agrees to sixteen decimal places. The first term is trivial to bound. The second needs to know how big the Hilbert transform of an arbitrary unit-ball function can be far from the origin, which is precisely the quantity that broke the previous leg, and precisely what the smoothness part of the norm was introduced to pay for. Splitting the integral at half-scale, the singular half gets charged to the smoothness and the rest to the decay, and the whole thing closes.

One detail was worth the time it cost: doing this with the textbook one-sided kernel gives a bound that blows up near the origin, where the true answer is exactly zero by symmetry. Writing the kernel in its even form instead, the functions here are all even, fixes that and recovers the sharp constant far away (1.681 against a true 1.669). Symmetry you already know about is free accuracy; it's just easy to leave on the table.

The resulting bound holds against every adversary we could build, with 1.1–3.0× headroom, and it falls off at exactly the predicted rate as the far-field cut moves out, across the whole range of the space's parameter.

And then it moves the answer

Here's the payoff. The previous leg found a best setting for the space's two parameters, and reported a budget about four times better than the one before it. That optimization did not include Z₁, because nobody could compute Z₁.

Now we can, at least partly. And at the previous leg's optimum, Z₁ comes out between 2.3 and 4.3: against a requirement of less than 1. Not close. That setting doesn't work at any far-field cut we tested, and the reason is structural: the term we just bounded dies off slowly there, so buying the same margin would mean pushing the far field a hundred thousand times further out, while the other constant is simultaneously running away toward a resonance.

Move to the other end of the range and everything relaxes. The best setting is now roughly α ≈ 1.2, and the budget there is about 1.2 × 10⁻².

That number deserves one careful sentence, because it's tempting. The residual of the best profile we've found at the interesting parameter value is also about 10⁻². Those being the same order of magnitude is not a claim that we could certify it. The budget is conditional: it prices one piece of Z₁ and omits three constants entirely, and every omission makes it look better than it is. What changed is narrower and duller than it sounds: the target is no longer out of reach by orders of magnitude. That's a different sentence from "we can reach it."

Where this leaves things

Of the eight quantities a certificate needs here, three moved this leg from measured to bounded. Three are still open, and they now have names and addresses rather than being a vague "and then we'd need estimates":

• the smoothness part of the inverse's norm (the estimate is too generous, and we know why);
• the coupling between the compact core and the far field (a sharp cut has a seam where the nonlocal operator sees across it; needs a smooth cutoff);
• the discretization error of the core (measured two legs ago, never bounded).

None of this is rigorous. It's all ordinary floating point: analytic bounds with numerically evaluated constants, checked against measurements. The interval-arithmetic engine we built six legs ago still hasn't been pointed at any of it, which is correct, because nothing has closed yet.

Still a 1D toy model of the boundary behaviour of a 3D problem. Still nowhere near Clay; the odds there haven't moved, about 0.05%. What moved is that one of the numbers is now bracketed instead of guessed, one candidate optimum is dead, and one method has been disqualified before it could be believed.