The previous run ended with a diagnosis rather than a result. We had gone looking for eight specific published orbits in a turbulent flow, found eight different ones, and worked out why: the pipeline that supplies starting guesses was quietly filtering on the very property that makes the targets unusual.
Each orbit has a shift: how far the flow pattern slides sideways over one period. The eight published orbits all sit between 0.295 and 0.707. Our candidate pool was overwhelmingly below 0.15, because the score used to admit candidates turns out to be correlated with shift. We were handing Newton's method a pool of small-shift guesses and then noting, with some puzzlement, that it kept returning small-shift orbits.
So this unit did the obvious thing: stop ranking candidates globally, and spend the budget in shift bands with quotas fixed in advance. Fill the band the targets live in, deliberately, and see what happens.
It worked, in the sense that the budget really is stratified now. And the answer is still zero.
That combination is the whole point of the exercise, so it's worth being precise about both halves.
The repair, and that it was a real repair
You cannot stratify at the stage where candidates are first scored, because the score is built from amplitude spectra and is shift-invariant by construction: shift doesn't exist yet at that stage. That is the same property that makes the pre-filter safe, and it's why the previous fix (in period, where the coordinate does exist) doesn't transfer.
So instead of stratifying the mine, we exhausted it: take every anchored near-repeat the Newton window cannot provably exclude, rank nothing, truncate nothing. That took the candidate pool from 2,014 to 75,873, 37.7× deeper, and then stratified downstream, at the one stage where the shift is known.
The results, against the previous run:
| before | after | |
|---|---|---|
| candidates in the published shift band | 35 | 72 |
| attempts seeded in that band | 31 | 60 |
| Newton iterations spent | 4,629 | 2,104 |
Note the last row. We did not buy the result with extra compute: the run spent less than half the iterations of the one it is being compared to, because a stall-exit rule ends attempts that have visibly stopped descending. That rule was validated first by replaying it against the previous run's stored iteration histories: it would have cut none of its 14 convergences, with a factor of 6.9 to spare. Everything else (tolerances, caps, the admission window, the matching rule) was held at the previous run's values by assertion at startup, not by hand-copying.
The result: the "we just weren't looking there" explanation is dead
Two explanations were written down in advance, before any of this ran, along with what would kill each.
"There aren't enough candidates in the band": plausible, and now refuted. Its premise was repaired: the supply doubled, the spend doubled. Its prediction was that more in-band seeds would produce a recovery. Sixty in-band attempts produced none.
"In-band seeds are genuinely harder": favoured, but honestly, not settled. Comparing seeds of matched quality across both runs pooled, in-band seeds converged 6 times out of 91 and out-of-band seeds 17 out of 109. That's a real-looking gap, and it fails significance (p = 0.073). It is a lean, and it is reported as a lean.
What we found instead: Newton walks out of the band
Here is the sharpest thing in the run, and it is not a null.
Of the nine attempts that converged, five were seeded inside the published band. Four of those five ended up outside it.
| seeded at shift | converged at shift |
|---|---|
| 0.619 | 0.587 |
| 0.397 | → 0.117 |
| 0.425 | → 0.100 |
| 0.311 | → 0.133 |
| 0.346 | → 0.101 |
Eight of the nine convergences finished below 0.15. Put a starting guess squarely in the band where the published orbits live, and the solver leaves, reliably, and lands on the same handful of low-shift solutions it always lands on.
That changes the diagnosis. The previous run's finding was about the seed supply: a filter that removed large-shift candidates. This run removed that filter and got the same destination anyway. So the bias is not only in what we feed the solver. It is in the basin structure: in this flow, at this resolution, the low-shift solutions have the large basins, and Newton falls into them from wherever you put it.
The honest caveat is that "the basins are bigger" and "our particular globalisation strategy drifts that way" are not distinguished by this data. Both are about the solver's destination rather than its menu, which is the part that matters.
This was not a story we constructed after seeing the numbers. The same pattern was recorded from the previous run's data before this one started (13 of its 14 convergences also finished below 0.15) and a separate note committed to the repository while this run was still going, before any of its numbers were visible, fixed in advance that this is what would decide the unit. This run is the replication, on an independent 60-attempt arm.
The one thing it did find
Nine convergences, but not nine orbits: they collapse to five distinct solutions. Four of those five were already found by the previous run. One was not:
Period 20.418, shift 0.587. New. And it is the only solution either run has produced inside the published band: the one in-band convergence that didn't run away.
So the stratification bought exactly one orbit that a globally-ranked budget had not reached, for about 57 core-hours. That is a narrow, real, unglamorous result, and it should be read narrowly: it is not one of the eight named orbits, the count against those is still zero, and the gate about them remains under-resourced rather than answered. Nothing here reopens it.
It also says something uncomfortable about the method. Four of five re-finds, across runs, not just within one: Newton spends most of a fresh hundred-attempt budget rediscovering what the last hundred already found. The obvious next move is deflation: subtract the known solutions from the problem so the search cannot land on them again.
Two smaller things worth writing down
A number that looks like a bug and isn't. The deeper mine's best candidate scores worse than the shallow one's, 0.0685 against 0.0165. That reads like a regression until you look: all 81 of the old pool's better-scoring candidates have periods between 1.75 and 2.5, i.e. they are trivial short-time near-repeats that no published orbit is anywhere near. The new rule requires candidates to be anchored to a real target period, so it never had them. Every one of the old pool's anchored admissible candidates is inside the new one.
An option got priced, and the price was informative. One tempting next step is to let the solver handle candidates with a discrete vertical shift instead of discarding them, and it would unlock a lot, 334 of the 575 in-window candidates, 58% of the pool. But of those 334, exactly one sits in the published shift band. So it is not a fix for the band problem, whatever else it is. That is the kind of thing worth measuring before spending ten hours on it.
What this doesn't say
It doesn't say the eight published orbits aren't there. A hundred attempts is not the scale that question is posed at, and a silence at the wrong scale means nothing.
It doesn't say the solver is broken. Three planted controls fired exactly as designed: a positive that had to converge and did, a negative that had to fail and did, a conditional positive that had to converge and did. A null from an instrument with a dead control would be uninterpretable; this one isn't.
And it moves nothing toward the Millennium problem this repository is aimed at. Those odds stay where they were, around 0.05%. What changed is smaller and duller: an instrument that was pointed wrong has been pointed correctly, and now reports that the thing we were blaming was not the only thing wrong.
Produced and checked in a single session with no independent reviewer, and therefore labelled UNVERIFIED under this project's own rules. Verification is a fresh pair of eyes or it is not verification.