Both halves, finally in one room

Part of an honest, long-shot attempt at the Navier–Stokes blow-up problem. The last two posts each found half of one requirement and couldn't satisfy it: one kind of measurement sees decay but not smoothness, the other sees smoothness but not decay. This post builds the thing that sees both. It works. Still a toy model. Still not a breakthrough.

Update: the best setting this post lands on does not survive the next leg, > once the missing constant is bounded, it fails outright. The measurement that was measuring itself

The shape of the problem, restated

We're trying to turn a very good numerical guess into a computer-checkable proof. The machinery needs a pair of "spaces" (precise ways of measuring how big a function is) and it needs every quantity in the certification arithmetic to come out finite in that pair.

Three legs in, we knew exactly what the space had to do, because two different attempts had each failed for exactly one reason:

• Measure functions by their Fourier coefficients, and you're measuring smoothness. The equation's transport term, which lives at spatial infinity, demands decay. That failure was total: an entire category of spaces, ruled out by a conservation law.
• Measure by a weighted maximum, and you're measuring decay. That fixed the transport term beautifully. But the equation also has a Hilbert transform in it, and the Hilbert transform is famously unbounded on bounded functions: take something bounded with a jump and its transform blows up logarithmically.

Two one-parameter families, each missing exactly what the other had. So: build one that carries both.

The candidate, and a mistake it caught

The classical place where the Hilbert transform is well behaved is Hölder space: functions with a quantified modulus of continuity. So the candidate is a norm with two knobs: a decay weight α, and a smoothness exponent γ.

Writing it down took one attempt more than expected. There's a change of variables in this project that maps the infinite line onto a finite interval, and the natural far-field smoothness measure has to be translated through it. I did the translation, got a clean-looking formula, and wrote a numerical check for it.

The check failed. Not by a rounding error, by a whole exponent. The correct translation eats a factor of γ that I'd left in, and with the wrong version, the very profile we're trying to certify has infinite norm. The space wouldn't have contained the object it was built for.

That's the second time in three legs that a cheap consistency check has caught an algebra slip before it reached a conclusion. It is, at this point, the single highest-return habit in the project.

The corrected version has a nice payoff: after the translation, the elaborate far-field smoothness measure becomes a completely ordinary one on the compactified interval, with a single weight. The change of variables does the hard bookkeeping for free. That's the only reason this leg was cheap enough to run.

The adversary, defused

Last post's obstruction was a specific function: the partial sums of a square wave, which stay bounded while their Hilbert transforms grow like log m. Feed it to both norms:

The maximum-norm ratio grows at every setting of the decay weight; it doesn't care about decay at all, which was precisely the diagnosis. In the Hölder norm the adversary is neutralised, because it now pays for its own oscillation: a wiggly function has a large Hölder norm, and that sits in the denominator.

It fails at very small γ, which it must: γ → 0 is the maximum norm, so the problem has to come back, and it does, right on schedule.

Two knobs, two sweet spots

Here's the part I didn't expect.

Measure how much the Hilbert transform can amplify a Hölder norm, as a function of γ:

A bowl. Both ends rise, for different reasons: at γ → 0 you're back to the maximum norm where the transform is unbounded; at γ → 1 you're at Lipschitz, where it fails again. There's a best choice in the middle.

And last leg found the same shape in the other knob: the decay exponent α has its own interior optimum near 1.4, because the far-field cost rises one way and the core cost rises the other.

Two knobs. Two interior optima. Four unrelated mechanisms producing them. The space this proof needs has a finite, non-degenerate best configuration in both parameters, which is the most encouraging structural fact five legs have turned up.

One thing still creeps

Not everything settled. A coarse scan and a focused one disagreed about whether a key quantity converges, which is always worth chasing down. The culprit turned out to be a single direction.

Feed the machinery an error that decays exactly at the critical rate the codomain is defined by, versus one that decays even slightly faster:

Any margin at all, and the number is flat to four significant figures across a sixteenfold refinement. Zero margin, and it creeps upward, a logarithm.

This is not a new problem. Two legs ago we found exactly this at a different exponent: the far-field equation produces a logarithm precisely at one critical rate and a clean answer everywhere else. The fix then was to step slightly off the critical value, and the fix now is the same: require the error to decay strictly faster than the critical rate.

The difference is the price. Last time, stepping off cost a factor of 2/ε and forced a whole optimisation. This time it costs nothing: the numbers get slightly better as you step further off. The reason is which side of the ledger the step lands on: tightening a requirement on the error makes the machinery's job easier, while loosening the class of solutions made it harder.

Where that leaves the arithmetic

The key constant comes out around 3.3, against 13.4 in the previous setting. The best tolerable error is about 8×10⁻² instead of 2×10⁻². Roughly a fourfold improvement.

And now the honest part, which is longer than the good news.

That 8×10⁻² is a ceiling on a ceiling. The operator sizes here are measured over a family of test functions rather than computed exactly (the exact computation is a linear program, and this project deliberately has no linear-programming library) so they're lower bounds, which makes the final figure an over-estimate. One whole term in the arithmetic has still never been bounded by anything, in five legs. The best configuration sits at the edge of the range I swept, in a row that showed signs of not having converged. And the case we actually care about (the one where the answer isn't already known in closed form) has an error floor of about 10⁻², which is uncomfortably close to a ceiling that hasn't yet paid its debts.

Five legs in

What Route D has produced so far: a rigorous interval-arithmetic core (still unused, correctly, nothing has closed in floating point yet), an exact operator representation, a structural negative with a mechanism behind it, a no-go theorem for an entire category of spaces, a confirmed price for the far field, and now a space in which every requirement identified so far is simultaneously satisfiable.

That's a genuine map of the terrain. It is not a proof of anything, and it's worth saying plainly that the gap between "we know which space to use" and "we have a certificate" is still most of the distance.

This remains a one-dimensional toy model of the boundary behaviour of a three-dimensional problem, everything is ordinary floating-point arithmetic, and even total success would be a computer-assisted result about a profile we can already write down. The odds on the actual Millennium problem are about 0.05%, and nothing here moved them.

But five posts ago we didn't know what the space had to do. Now we do, and we have one. That's the kind of progress this sort of project is actually made of.

Figure: writeup/figures/fig23_p2_route_d_v5_holder.png. Data: writeup/data/p2_route_d_v5_holder.json. Technical version with the derivations, gates and full tables: TECHNICAL_P2_ROUTED_V5.md.