The search was looking in the wrong place, and the search was right to (PROG-R4, unit U2)

There is a class of bug that never throws, never fails a test, and never produces a wrong number. It produces a correct number about the wrong population. This is one of those, and it is worth writing down because it is not specific to fluid dynamics: any pipeline that ranks candidates globally and then filters them by a property the ranking doesn't know about will hit it.

The setup

The programme is trying to recover published relative periodic orbits of 2-D Kolmogorov flow at Re = 60. Lucas & Kerswell 2015 list eight of them by name, with periods between T = 14.776 and T = 19.334. The standard method, which is theirs and Chandler & Kerswell's before them, has three steps:

1. run a long turbulent simulation;
2. find moments where the flow nearly repeats itself, near-recurrences, and record the time t and the interval T at which it nearly repeated;
3. hand each of those (t, T) guesses to Newton's method and see whether it converges onto an exact orbit.

Step 2 produces far more candidates than step 3 can afford, so you rank them by how close the near-repeat was and take the best few hundred. That is the obvious thing to do and it is what everyone does.

What came out

The simulation ran to T = 1e5: 400,000 snapshots, 3.44 hours. The recurrence scan looked at 95,542,640 pairs of times and found 913,301 genuine local minima of the recurrence measure. Plenty. The best one scored R = 0.016543, an order of magnitude better than the best this repo had ever found before (0.177, from a run 50× shorter). Everything looked healthy.

Then the top 400 were passed forward, and by the time they reached Newton's method there was one usable seed left.

Here is the funnel:

The plan called for ~100 attempts. There was one.

Why

Nothing was broken. The ranking did exactly what it was asked to do.

Near-recurrences at short intervals are easy. If you compare the flow to itself 1.5 time units later, it has barely had time to go anywhere, so the difference is small and the candidate scores well. If you compare it 19 time units later (long enough for the chaos to do real work) the difference is much larger, and the candidate scores badly even when it is a genuine near-miss of a real orbit.

There are also vastly more short intervals available than long ones.

So a global top-400 fills itself with T = 1.25 to T = 2.5 candidates. They are real near-recurrences. They score well because they are short. And every orbit anyone has ever published for this flow lives at T ≈ 15 to 19, which is exactly where the ranking is least willing to look.

The ranking was not wrong. It was answering "which pairs of times are most similar?", and the honest answer to that question is dominated by pairs of times that are close together. The question that was actually needed was "which pairs of times are most similar given how far apart they are?" Those are different questions, and the pipeline was quietly asking the first one.

The fix, and the part that needs saying out loud

Spend the candidate budget per named period instead of globally. Keep the whole global list, remove nothing, and additionally take the best few hundred local minima within ±1.0 of each published period. Everything added faces exactly the same tests as everything else.

One to 133.

Now the part that needs saying out loud: this change was made after seeing that the first attempt produced one seed. That is a researcher degree of freedom, and pretending otherwise would be the actual dishonesty here. It went into the pre-registration as a labelled amendment that says so in its first sentence.

Two things make it defensible rather than merely disclosed.

It cannot manufacture the answer we want. The gate this feeds asks whether at least one named published orbit converges to a residual of 1e-8, with the converged period and shift then matched back against the published values to within 0.05. Offering more starting guesses does not relax any of that. A bad guess simply fails to converge and is recorded as a failed attempt. The change alters which seeds are offered; it cannot alter what counts as a recovery.

The knob was closed, not left open. The rule written down says the per-band budget is raised once, in a single step, and if the pool still came up short the run would proceed on whatever it had and report the shortfall as a number. Otherwise "raise it until we get 100" becomes "raise it until we get the result", and those look identical from the outside once the run is over.

The other filter, which is not a bug at all

One row of that funnel deserves its own note: 345 perfectly good candidates were dropped because they involve a discrete shift in the cross-stream direction, and this programme's Newton solver carries only a continuous streamwise shift. Such a candidate cannot be expressed as a starting guess for it.

That is not a defect to be fixed in passing. It is a limit of the specific implementation, and the repo has a rule for exactly this (lesson 91: a negative result must name the implementation it was obtained in). It happens to cost nothing here (all eight published orbits have zero cross-stream shift, so nothing named is lost) but if this search comes up empty, "we did not find them" would have to be stated as "we did not find them with a solver that cannot represent one of the two shift classes". Those are very different sentences, and only one of them is true.

What this does not say

It does not say any orbit was found. That is a separate gate, running as this is written, and its answer is not in yet. It does not say the method works, only that it is now pointed at the right part of the search space. The best candidate residual of 0.016543 is a starting guess, not a converged solution, and the gap between those two numbers is the entire content of the question that comes next.

And it moves nothing toward the Millennium problem this repo is nominally aimed at. The odds stay where they were, at about 0.05%. This was a plumbing fix on an instrument, written up because the failure mode is general and silent: a global ranking will starve any band your filters care about, if the ranking doesn't know your filters exist.

Built and checked in a single session with no independent reviewer, and therefore labelled UNVERIFIED under this repo's own rules. Verification is a fresh pair of eyes or it is not verification.