A sequence that was a reading

Sorting every counted pair in the settling table into the additive sequences they sit on produced ten sequences, and one of the ten was odd. The destination at 79.2° returns three pairs — 4/5, 4/9 and 5/9 — and two of them sit on 1, 4, 5, 9, 14 while the third sits on 4, 9, 13, 22, because 4 and 9 are two terms apart in the first sequence rather than adjacent.

The reading offered was that a count had been taken over the wrong patch and had stepped over a term. The essay that offered it named the test that would settle it: count the same runs over a hundred organs and over four hundred, and see whether 4/9 survives.

It survives. It survives at every window from twenty-five organs to eleven hundred, and so does every other pair in the table. The anomaly is real, the conclusion drawn from it was right, and the mechanism proposed for it was wrong.

What the counter computes and does not return #

The counter picks two index offsets whose hops are shortest and straightest in the band it is given, and it works out one further thing about them: whether the two families wind in opposite directions around the stem. That is not decoration. A parastichy pair is a pair of families that cross, and two families of one handedness are two sets of parallel lines that never meet.

The counter has always computed it. What happens next is that the count is taken through a shared call which returns the pair and drops everything else, and every count in this collection goes through that call.

Five of 384 #

Read the flag rather than the pair and the table separates. 379 of 384 settled runs return two families that cross. Five return two families of one handedness, which is 1.3 per cent.

Those five are exactly the readings that produce a pair skipping a term. Three of them are at 79.2° and count 4/9, one is at 65.1° and counts 5/11, and one is at 47.9° and counts 7/15. Nothing else in the table returns a pair that is not two adjacent terms of its own sequence, and nothing else in the table returns two families that wind the same way.

What a same-handed reading is #

Two rows of one handedness on a cylinder are two families whose offsets are two terms apart in an additive sequence rather than one. That is not a coincidence about these five runs; it is what parallel families on a cylindrical lattice are, and it is why the skip and the handedness are the same fact seen twice.

So a reading of 4 and 9 is not a badly-taken reading of 4 and 5. It is a correctly-taken reading of two families that happen to be the shortest two the counter could find, and it is not a pair at all.

The term that was stepped over #

Put the missing term back and each reading names a pair that does cross. 4/9 becomes 4/5 on 1, 4, 5, 9, 14. 5/11 becomes 5/6 on 1, 5, 6, 11, 17. 7/15 becomes 7/8 on 1, 7, 8, 15, 23.

In each of the three cases the sequence named after the repair is a sequence that destination’s own crossing runs are already on. The runs at 79.2° that return two crossing families return 4/5 and 5/9, both on 1, 4, 5, 9, 14. So the repaired reading does not introduce a new object anywhere; it merges a spurious one back into a real one.

A pair and one of its own sums #

There is a second way to see that a same-handed reading is not a pair, and it does not mention handedness at all. A lattice’s contact families are closed under addition, and exactly two of them are not sums of the others — the two smallest, which are the counted pair.

Read 4 and 9 against that. Nine is four plus five, so 9 is a sum of two families the same stem carries, and a set whose two smallest members are 4 and 9 would have to have no 5 in it. The stem plainly has one: other runs at the same destination count it. So the reading names a generator and a sum rather than two generators, which is the same defect the handedness flag reports and is checkable from the integers alone.

That the two agree on all five readings is not a coincidence, because the family that is skipped is exactly the one that would make the larger number a sum. It is worth having both because one of them is computed by the instrument and one of them is arithmetic anybody can do to a published pair.

Three stems, not one stem read three ways #

The other half of the correction is that 4/5, 4/9 and 5/9 were never three readings of one object. Which pair a run at 79.2° returns is decided by the rise and the falloff exponent it was grown at, and the cells are clean: 4/5 comes from the two coarsest rises at every exponent, 5/9 from a rise of 0.013 at exponents 3, 4 and 5, and 4/9 from a rise of 0.013 at exponent 2 alone — the shallowest rule in the sweep.

At forty starting angles the destination holds forty-three runs and the same three cells, with three same-handed readings instead of one. Widening the sample adds runs to the odd cell and does not spread it into the others.

The test that was named, and what it returned #

The essay that raised the anomaly did the thing this collection asks of every claim: it said what would settle it. A hundred organs against four hundred, on runs already grown.

That test was run, and it was run at seventeen widths from twenty organs to eight hundred and then at thirty-nine widths from thirty to eleven hundred. Not one of the 384 runs returns a different pair anywhere above the width its own pair requires. The window decides nothing on this table, and the test would have reported that 4/9 is a real pair.

A named test returning the wrong verdict is the most useful kind of failure, because it says the quantity being varied was not the one that mattered.

So the conclusion stands and the reason does not #

The published essay concluded that the table holds nine sequences rather than ten. That conclusion is now confirmed by direct measurement: dropping every same-handed reading takes nine starting angles from ten sequences to nine.

Its mechanism does not survive. The skip is not a count taken at the wrong radius, not a count over the wrong patch, and not a matter of degree at all. It is a boolean the instrument computes on every run and the reading discards, and it is right or wrong in a way that has nothing to do with how much stem was read.

What the limit angle could and could not do #

The route the published essay took was arithmetic on limits: the sequence 1, 4, 5, 9, 14 converges on a divergence of 77.96° and 4, 9, 13, 22 on 82.2°, and a destination at 79.2° is 1.2° from the first and 3.0° from the second, so it belongs to the first.

That is a good argument and it reaches the right destination. What it is not is a mechanism. It compares two candidate ladders after a person has noticed an odd pair, and it says nothing about how the odd pair came to be returned; the same comparison run on a genuinely new ladder whose limit happened to sit nearby would endorse the wrong one. The flag is not a comparison at all. It is computed on all 384 runs, it identifies the same one the limit angles identified, and it identifies two more nobody had looked at.

Two routes to one number #

Nine sequences at nine starting angles has now been reached twice, from arithmetic and from measurement, and the two share nothing but the table they are about. One computes a limit divergence for each candidate ladder from its first two terms and asks which limit the destination sits nearer. The other asks the counter, on every run, whether the two families it picked wind opposite ways.

Neither route is a check on the other in the ordinary sense, because neither could have been derived from the other. What they have in common is that both are cheap and both were available before the anomaly was noticed — the limit angles need only the seeds, and the flag was already being computed and thrown away.

The other two are the check #

That is the part that turns a repair into a result. 4/9 was known to be odd and was explained twice; 5/11 at 65.1° and 7/15 at 47.9° were not known to be anything, because at nine starting angles neither is in the table.

Both are flagged by the same rule, both are two terms apart on their own sequence, and both repair onto a sequence their destination’s crossing runs already occupy. Three independent instances of one mechanism, of which one was suspected and two were not.

Each is its own destination’s sequence, a term skipped #

The three destinations are unlike each other in every way the table records. 79.2° is reached at a rise of 0.013 by the shallowest falloff; 65.1° at 0.008 by the same; 47.9° at 0.0045 by an exponent of 3, and it is one of the destinations only a steep rule reaches.

They agree on the one thing the mechanism predicts. Each same-handed reading names its own destination’s sequence with the intervening term stepped over, and no two of the three name the same sequence.

The one place the window does touch it #

Exactly one run in the table changes its handedness with the window, and it does so below the width its own pair requires: the 7/15 run reads two crossing families at twenty organs and its own same-handed pair from twenty-five.

That is not an exception to the null; it is the other essay’s floor showing through. A window narrower than the larger count plus six organs returns the rung below, and the rung below happens to be a crossing pair. Above the floor the handedness of all 384 runs is constant at every width.

What the census loses #

Drop the five readings and three things move. At forty starting angles the table goes from 384 settled runs to 379, from 31 arrangements to 28, and from 15 additive sequences to 12.

The three sequences that go are the three the same-handed readings were the only support for: 4, 9, 13, 22 · 5, 11, 16, 27 · 7, 15, 22, 37. No other sequence in the table loses a single run.

Nine, twelve and twelve #

The same arithmetic at the three samplings the table has been grown at. Nine starting angles: 117 runs to 116, twenty arrangements to nineteen, ten sequences to nine. Twenty angles: 207 to 205, 24 to 23, thirteen sequences to twelve. Forty angles: fifteen sequences to twelve.

The count of destinations sitting on one of the collection’s two ladder sequences is five throughout — at every window, at every sampling, and with or without the same-handed readings. Nothing about the ladder moves. What moves is the number of things the table was said to have found off it, which is the number both the twenty-angle sweep and the forty-angle one were raising.

A correction to a census, not a re-reading of one #

Five readings out of 384 is 1.3 per cent, and the honest description of the change is that a census is smaller rather than that a measurement was wrong. Every one of the 379 runs that remains is untouched, every destination is still a destination, and every angle the table reports is the angle it reported.

That distinction matters for anything built on the table. A result quoted as the table reaches fifteen sequences has to become twelve; a result quoted about any particular destination’s divergence does not change at all.

What has not been counted, rather than what is not there #

The three arrangements those readings contributed are arrangements this collection has not counted. They are not arrangements that do not exist.

A run whose two shortest families wind the same way is a real stem that settled to a real divergence, and something is there to be counted; what is there is a lattice whose counted pair is the adjacent one, and the collection has no reading of it because the reading it took was refused after the fact. Calling that a deletion would be a second error on top of the first.

Why the flag was dropped in the first place #

Not carelessness, and worth saying because the fix is a line. The shared call was written to return a pair, one thing rather than two, and a pair is what every caller wanted. Every count on a disc in this collection deals with families that cross by construction, and the cylindrical counter was the one that had to drop a rule the disc counter used — so an extra return value about handedness looked like a fact about the cylinder rather than a check on the answer.

It is a check on the answer. A pair whose families do not cross is not a pair, and an instrument that computes that and does not say it is an instrument with a refusal it never makes.

Where else it is dropped #

Eight of this collection’s measurement libraries take their counts through the same shared call, and every one of them discards the same flag for the same reason.

The rate measured here — five in 384 — is the rate to expect in each of them until each is checked, and it is not a rate anyone can assume is smaller elsewhere. It is a property of the counter and of what it is pointed at, and several of those libraries point it at patches that have been cut, blocked or shortened.

What one per cent is worth #

Very little as a fraction and a great deal at a tail. A 1.3 per cent error rate that fell on arbitrary runs would be invisible and would matter to nothing; this one falls entirely on readings whose pair is unusual, which is precisely where a census of unusual pairs is counted.

So the same five runs are 1.3 per cent of the table and 100 per cent of the support for three of its fifteen sequences. That is the shape a rare failure has when the thing being counted is rare.

What this does not settle #

It does not say what a stem whose two shortest families run parallel actually looks like, or whether that configuration is a stable arrangement or a transient the settling criterion let through. Five runs is not enough to ask.

It does not touch the destinations. Every one of the nineteen is where it was, reached by the runs that reached it, and the census of destinations is unchanged. And it says nothing about whether the twelve remaining sequences are the right number, which is a question about how densely the starting angles were sampled and not about how the counting was done.

The refusal that would have caught it #

The reader assembling a pair now refuses four things: a pair claimed from one window, a window too narrow for the family it claims, two windows that disagree, and two families that wind the same way. It answers the fifth case, which is a pair two wide-enough windows agree on whose families cross.

A machine that refused everything would satisfy those assertions and prove nothing, so the refusals are checked against a case that must be answered as well as against four that must not. That is the difference between a check and a shrug.

What would refute this #

A settled run returning two same-handed families whose offsets are adjacent terms of an additive sequence. That would break the identity between the skip and the handedness, and it would mean the flag and the anomaly are two things that happen to coincide on five runs.

Three hundred and eighty-four runs have been asked and none does it. The other refutation is cheaper: a same-handed reading whose repaired sequence is one its own destination’s crossing runs are not on. None of the three is.

What carries forward #

Nine sequences at nine starting angles, twelve at forty, and a census that shrank without a single measurement changing. A test that was named honestly, run, and returned the wrong verdict because it varied the wrong quantity. And a flag that has been computed on every count this collection has ever taken on a stem, discarded eight libraries deep, and is worth one line to read.