The rule comes back

The rows the exchange sets aside were added up a round ago. Each one’s exceptional chains have a displacement, and the sum of them says whether the rearrangement closes — whether the chains end where chains began.

Seven closed and six did not, and the lag the cut kept sorted them thirteen for thirteen: every row that closes kept 7 or 8, every row that does not kept 4 or 5. Then a lattice from outside the census added three rows and one of them broke it, at fifteen of sixteen.

The objection that came with it #

Written down in the round’s own file. The sum is over the chains that sit away from the common level, and the level is a median over the classes — so on a profile with three of four chains exceptional the median is itself an exception, and the offsets are offsets from an anomaly.

That is a candidate artefact with a direction: it would corrupt exactly the rows with many exceptions, which are exactly the rows the rule was fitted on.

Recomputing the level as the densest cluster moves it on eight of the thirty-six rows, and every one of the eight is in the set-aside group.

What the sums become #

The set-aside group is smaller under the new level, because one row gains a balanced pair and leaves it. Fifteen rows rather than sixteen.

Of the fifteen, ten close and five do not. Under the old level nine closed and seven did not, so two rows change: one leaves the group entirely, and one crosses the line.

The one that crosses is l006 at offset 7, which is the out-of-sample row that broke the rule. It summed to −22.6 degrees and now sums to +6.4.

Which restores the rule #

Every row that closes kept a lag of 7 or more; every row that does not kept 4 or 5. Fifteen of fifteen, including the three rows from the lattice the rule was never fitted on.

So the objection was real, the artefact was where it was predicted to be, and correcting it puts the rule back where the in-sample score had it.

That is the least likely of the three outcomes. A candidate artefact usually turns out to be absent, or to be present and to weaken the thing it was raised against. This one is present and restores it.

Why that is uncomfortable #

Because it is the outcome most easily produced by wanting it. A statistic changed, a rule improved; the change was proposed by the people who had the rule; and the change was made after the rule failed.

Three things keep it honest, and they are worth stating rather than assuming.

The objection was written down before the recomputation, in the file that reported the failure, with the mechanism named. The new estimator has no free parameter — its cluster width is the exception tolerance already in use. And it was scored on something else: how much of each profile sits at the level, which neither estimator defines and on which it wins 145 to 138.

And the price #

The gap the line is drawn in halves. At the median level the closing rows sit 0.2 to 10.5 degrees from nothing and the open ones 16.9 to 173.6, a gap of 6.3 degrees. At the recomputed level they sit 0.1 to 8.9 and 12.1 to 160.0, a gap of 3.2.

The threshold is twelve degrees, and the nearest open row is now at 12.1. Under the old level the nearest was at 16.9.

So the better score is read on a thinner separation, and one row sits a tenth of a degree outside the line it is called open by. That is the honest cost and it is reported everywhere the score is.

What a thinner gap means #

That the classification is closer to being a decision. A line in an empty gap is a way of writing down a separation; a line through a continuum is a decision dressed as one, and the distance between the two populations is what says which it is.

At 6.3 degrees the line could have moved anywhere between about 11 and 17 without changing a verdict. At 3.2 it can move between about 9 and 12, and at 12.1 the nearest row is inside the range a different tolerance would put it on the other side of.

Neither reading is a continuum. Both have a real gap and the second’s is half the first’s, which is worth knowing and is not the same as the rule being in doubt.

Which reading is right #

The recomputed one, and the argument is not about the score. A level that stands on a class sitting by itself is a level nothing is measured against; the sums taken from it are sums of offsets from an anomaly.

That is a defect in the reading independent of what it does to any rule. It would be a defect if the rule got worse.

So the right order is: the level was wrong on eight rows, it has been corrected, and the consequences are reported. The rule improving and the gap narrowing are both consequences and neither is a reason for the correction.

The row that crossed #

l006 at offset 7, on the Lucas branch at a rise of 0.006, keeping a lag of 7. Its seven class means are 79.97, −125.0, −23.0, 81.1, −127.2, −23.5 and −23.9.

The median lands at 34.69, which is not near any class — the nearest is 79.97, forty-five degrees away. The densest cluster is the three at about −23.5.

Against the median it has five exceptions summing to −22.6 degrees; against the cluster it has four summing to +6.4. So the change is not a marginal shift: the level moves forty-seven degrees and the row loses an exception.

What the rule is #

A rearrangement closes when the cut kept a lag of seven or more. Three rivals were written down first and all three lose.

Four or more exceptional chains puts the one clean three-cycle on the wrong side, since it has three and closes. It scores 14 of 15. The golden branch scores 5 of 15, which is worse than a coin. The exceptions forming one consecutive run scores lower still.

So the winner beat rivals rather than being the only rule tried, which is the site’s standing requirement for reporting a rule at all.

What it does not explain #

Why the lag should decide. A rearrangement closing means the exceptional chains’ displacements sum to nothing after folding — the chains end where chains began. The surviving lag is the period the profile was folded on.

There is no account here connecting the two. The rule is a correct prediction over fifteen rows with no mechanism behind it, which is the same position the branch account is in on the bands and is worth stating in the same words.

What would be an account is something that predicts the size of an open row’s sum, and nothing does. The open rows sum to 12.1, 13.0, 41.0, 127.5 and 160.0 degrees, which is not a pattern anybody has fitted.

The two largest of those are l020’s two cuts, which are the pair of rows that carry exactly two exceptions displaced the same way — a pair that is not equal and opposite and is therefore not a pair. They are the furthest from closing in the table and they are the rows whose exclusion was miscounted when the set was first described.

The out-of-sample rows #

Three of the fifteen are from l006, the lattice a search for a fifth surviving lag found for reasons that have nothing to do with this rule. They are the only out-of-sample test available and they arrived by accident.

Under the median level the rule was right on two of the three. Under the recomputed level it is right on all three.

That is the part of the fifteen that carries the weight. Twelve rows fitted the rule when it was written and three did not exist yet, and the rule is right on all three.

What a fourth out-of-sample row would need #

Another lattice with unpaired rows at a lag the rule can be wrong about, which means a lag of 4 or 5 that closes or a lag of 7 or more that does not.

The census’s lags are 4, 5, 7, 8 and 11, so the material exists. What does not exist is a supply of new lattices: the fine-end search covered twenty rises across both branches and found one lattice the census does not hold.

So the rule’s evidence is fifteen rows from eleven lattices, three of them out of sample, and the next test would need a search of a stretch nobody has cut. The stretch between two rises the search stepped over has now been cut and produced no new lattice with unpaired rows, so the cheapest place is already spent.

What is still true of the median #

That it agrees with the recomputed level on every row the exchange keeps, worst 3.13 degrees. So every number in that thread stands.

And that it was never wrong on a row where its own condition held. Its condition is that the exceptions are a minority, and that fails on exactly the rows the exchange sets aside — which are exactly the rows this rule is scored on.

So the two threads have been using two readings all along and neither knew: the exchange’s rows had a valid level and the set-aside rows did not.

That is the tidiest form of the whole round’s finding. No published number moves; a reading that was undefined on sixteen rows returned a value anyway; and the rule scored on those sixteen rows was the only thing that could have noticed.

How the sums are taken #

Each exceptional chain’s offset from the level is a number, and the sum is over the exceptional chains only. Every class at the level contributes nothing by construction, which is checked: adding every class instead agrees to within a degree and a half everywhere.

The sum is then folded into a half turn either way, and the reason is the census’s one clean three-cycle. Three chains rotating into one another’s places each move about a third of a turn the same way round, so the raw sum is −359.6 degrees — the largest in the table by a factor of two — and folded it is 0.4 degrees from nothing.

Read raw, that row is the worst anomaly in the census. Read folded, it is the one row that unambiguously answers the question the rows were set aside with. The unfolded value is carried beside every folded one so the distinction stays visible.

What the control is #

The rows the exchange keeps. A balanced pair sums to nothing by construction — two displacements equal and opposite — so every one of them closes, and they close within 4.6 degrees.

That is not a confirmation of anything; it is the check that the addition is not an instrument that says yes to everything. A reading that called every row closed would be measuring its own arithmetic.

Twenty-one rows closing by construction and ten of fifteen closing by measurement, with five not, is a reading that separates.

The rule tested on one lattice’s own cuts #

The complaint any rule scored over rows invites is that the rows sit on fewer lattices than it counts. Fifteen rows sit on seven lattices, so a rule about rows might be a rule about lattices.

It is answered by a lattice whose own cuts fall on both sides. g005 cut at offset 6 keeps a lag of 4 and is open; offsets 7 and 8 keep 8 and close. One rise, one seed angle, one history below the hole, and the split follows the lag.

That is the check that separates the rule from the lattice, and it survives the recomputation: the same lattice still falls both ways, with the same lags on the same sides.

What changed and what did not #

Two rows moved. One left the set — gaining a balanced pair and joining the exchange — and one crossed the line from open to closed.

Thirteen rows kept their verdict. Their sums moved by between nothing and a few degrees, which is what a level moving by a fraction of a degree does, and none of them came near the threshold.

So the restoration is two rows out of sixteen, and the rest of the table is the same table. That is worth stating because a rule that went from 15/16 to 15/15 sounds like a wholesale re-reading and it is two rows.

Why the threshold is twelve #

Because the measured sums are 0.2 to 10.5 degrees on one side and 16.9 to 173.6 on the other, and twelve sits in the gap. It was stated rather than fitted, with the two nearest values reported beside every count taken with it — which is what makes it possible to say now that the gap has narrowed rather than to discover it later.

Under the recomputed level the same twelve sits between 8.9 and 12.1. So the line has not moved and the population has moved towards it on the open side, by 4.8 degrees.

Leaving the line where it was is the right choice and it is worth saying why: moving it to the middle of the new gap, at about 10.5, would be fitting a threshold to a re-reading, and the verdicts it would produce are the same. A line that could be moved without changing an answer should not be moved.

What the rule is standing on #

Fifteen rows from eleven lattices, three of them from a lattice the rule was never fitted on, with one lattice’s own cuts falling on both sides of it.

That last is the load-bearing part. Thirteen of the rows sit on seven lattices, so a rule scored over rows could be a rule about lattices — and g005 answers it: one rise, one seed angle, one history below the hole, and its three set-aside cuts fall both ways in step with the lags they kept. Whatever the lag is doing, it is not standing in for which lattice was cut.

What the rule still lacks is any account of why a lag of seven should close a rearrangement and a lag of four should not. It is a correct prediction with no mechanism, which is the same position the branch account is in on the bands, and it is worth naming in the same words rather than dressing either of them up.

The shape of a correction that improves a result #

Three things make this one reportable rather than convenient, and they are worth listing because the shape recurs.

The objection came first, in the file that reported the failure, with the mechanism named — a median standing on an exception — so the prediction existed before the test.

The change had no free parameter. The cluster’s width is the exception tolerance already in use, so there was nothing to tune towards an outcome.

And the estimator was scored on something else: how much of each profile sits at the level, which neither estimator defines and on which the new one wins 145 to 138 while never returning a level with one class at it.

Take any of the three away and the same arithmetic would be worth very little.

What is claimed #

That at the recomputed common level the lag rule sorts every one of the fifteen set-aside rows correctly, including the three from a lattice it was never fitted on, where at the median level it was wrong on one of sixteen.

That the row it had been wrong on sums to +6.4 degrees rather than −22.6, because its median level sat forty-five degrees from any of its classes.

And that the gap the classification is read in falls from 6.3 degrees to 3.2, with the nearest open row at 12.1 against a threshold of 12 — so the improved score is read on a thinner separation, and both numbers are quoted wherever either is.