A family that is a multiple

Remove one organ from a growing stem and the pattern above the hole sometimes never recovers. When it does not, the wrecked run still holds one hop rigid — the angle from an organ to the one some fixed number of places above it stays what it was — and that number is the surviving family.

Almost always it is one of the two the counter returns. On thirty cuts of the census it is a member of the counted pair every time, which is a strong enough regularity that the exceptions are worth naming individually.

Two exceptions are known and both came from cutting a band at every rise it holds rather than from the census. They are the whole of the evidence about what an off-pair survivor can be, and until the second one arrived the first looked like the beginning of a rule.

The first exception, and what it looked like #

The golden 8/13 band, cut at every one of its 126 rises, produced cuts that keep 4. Four is not 8 and it is not 13. It is half of 8.

That is the kind of coincidence that reads as a rule, and it read as one here. A hop of four on a lattice whose contact families are 8 and 13 is the hop of every other member of the 8-family, so a stem holding the 4-hop rigid is holding something the 8-family already contains. It is not an outside number arriving; it is a coarser reading of a number already there.

Nobody wrote that down as a rule, and the reason is worth stating: one case is not a rule, and there was no second case to test it on. The band it came from is the one whose cuts change their answer nineteen times, so the 4 was not an isolated reading — three of its six wrecking offsets keep it somewhere, at thirteen separate stretches of rise. A regularity across thirteen stretches of one band is still one band.

The alternative reading, which nobody preferred and nobody could rule out, was that 4 is simply the number that happened to be rigid there and its relation to 8 is arithmetic.

The second exception #

The golden 5/8 band, cut at every one of its 112 rises, produced cuts that keep 20.

Twenty is not 5 and it is not 8. It does not divide either of them. It is four times 5, which is the other direction entirely: the 20-hop is not a coarser reading of a family the lattice has, it is a finer one — every fourth member of the 5-family, or equivalently the 5-hop taken four times.

So whatever an off-pair survivor is, it is not “half of a counted number”. It is not “a divisor of a counted number” either.

Both bands are golden, both were cut at every rise, both were read by the same machinery with the same tolerance, and the two answers sit on opposite sides of the relation that looked like a rule. That is as clean a refutation as two cases can produce: the second case was not a near miss, and it is not explained by anything the first case’s reading offers.

What survives of the reading #

One sentence, and it is nearly a tautology: the off-pair family is commensurate with the pair. Four divides eight; twenty is a multiple of five. In both cases the surviving hop is an integer multiple or an integer divisor of a hop the lattice actually has.

That is weaker than it sounds only because the alternative is so unconstrained. A lattice at these rises has forty lags whose hop lengths are worth ranking, and the great majority are coprime to both counted numbers. Two exceptions out of forty candidates both landing commensurate is not nothing.

But it is two cases. The honest statement is that no off-pair survivor observed so far is coprime to the counted pair, over two observations.

Under a coin that puts each of forty lags equally likely, two commensurate results in two draws is about one chance in fifty — suggestive, and not a number anybody should quote as evidence, because the forty lags are not equally likely candidates for anything and the two draws are not independent draws from them.

Why a multiple is stranger than a divisor #

A hop held rigid is a distance the pattern refuses to change, and holding one is what separates a wrecked run from a destroyed one. If the 8-hop is held then so is the 16-hop and the 24-hop, because they are the 8-hop repeated — a fact about arithmetic rather than about the stem. Holding the 4-hop is stronger than holding the 8-hop: it implies the 8-hop and is not implied by it.

So the first exception is a stem holding more than the counted family requires. The pattern above the hole has kept the 8-family and kept a finer regularity inside it, and the 8 is reported only because the reading returns the smallest rigid lag and 4 is smaller.

The second is the reverse. Holding the 20-hop is implied by holding the 5-hop and does not imply it. A stem that holds only the 20-hop has let the 5-hop go while keeping every fourth step of it, which is a stranger object: the pattern has forgotten four fifths of a family and kept the rest.

How the survivor is read #

Two organs a fixed number of places apart on the control, and the same two on the cut run. If the angle between them is what it was to within a tolerance, for every such pair in the reading window, that lag is rigid. The survivor is the smallest rigid lag.

Smallest is doing the work, and the reading was built that way for a reason. Because a rigid lag implies its own multiples, the set of rigid lags is closed upwards under multiplication, and reporting the smallest is the only way to report it once. That is why 4 is reported rather than 8 on the first band: both are rigid there and 4 is smaller.

Which makes the second case genuinely different rather than a difference of convention. On the 5/8 band, 20 is rigid and 5 is not.

If the convention were the other way — report the largest rigid lag, or report the counted member if any is rigid — then the first band would report 8 and the second would still report 20, because there is nothing else to report. The two cases are asymmetric in the runs and not in the reading.

Which rules out one explanation immediately #

If the reading were merely coarse — if 20 came back because the tolerance was tight and the 5-hop failed it by a hair — then 10 would be rigid too, since 10 is a multiple of 5 and a divisor of 20. It is not. The smallest rigid lag is 20 and nothing between 5 and 20 is rigid.

So the pattern above that hole holds a distance of twenty places and holds none of its own factors. That is a positive statement about the wrecked run, not an artefact of where a line was drawn.

The tolerance itself is a gap rather than a threshold, which is the check that makes the statement safe. Across the census the rigid lags hold their angle to within a fraction of a degree and the non-rigid ones miss by tens, so any tolerance between about one degree and about ten returns the same table. Twenty rigid and ten not is a separation of that kind, not a decision.

The two rises it happens at #

Offset 5 on the golden 5/8 band, at rises 0.01625 and 0.01605. At every other rise it wrecks at, the same offset keeps 5.

Twenty-eight rises of the band have offset 5 wrecking and two of them keep 20. So this is not a stretch of the band with a different character; it is two rises inside a stretch that otherwise behaves.

Between them the offset recovers five times, which means the two are adjacent in the sequence of rises where anything is measured and not adjacent in the rise.

What is the same at those rises #

The counted pair is 5/8, as it is everywhere on the band. The settled divergence is within a twentieth of a degree of every other rise’s. The two contact hops are in the same order — the crossing is thirty rises further down — and their lengths differ from their neighbours’ by parts in a thousand.

Nothing available distinguishes those two rises from the twenty-six around them. That is the same finding the fine sweep of a rung produced for a different quantity, and it is the same uncomfortable shape: a discrete answer changing where every continuous quantity is smooth.

It is not a paradox. A rigid lag is a discrete reading of a continuous run, and a discrete reading can change under an arbitrarily small change in what it reads. What it does mean is that no summary of the lattice available here predicts which rises will do it, and a prediction is what would turn two cases into an account.

The candidate that is not available #

The obvious account of a family off the pair is that the counter is wrong there — that the lattice really has a 20 in its contact set and the counting radius missed it.

It is not available, and the reason is that the counter and the survivor are computed by machinery that shares nothing. The counter reads a point set and returns the two shortest steps across the surface; the survivor reads two runs of azimuths and returns the smallest lag whose angle held. A counting mistake would have to produce a 20 in one and a rigid 20 in the other, by different routes, at two rises and nowhere else.

It is also checked directly. The counted pair at both rises is 5/8, and the twenty most tightly ranked hops at those rises are the ordinary ones — 5 and 8 at the top, then 3, 13, 10 — with 20 a long way down. The counter is not returning anything unusual there.

That separation is the site’s own founding discipline rather than a convenience. A number computed twice by machinery that shares nothing is the only kind of agreement worth reporting, and here it is a disagreement that has to be explained rather than absorbed.

What a third case would be worth #

A great deal, because two cases can be described by anything. If a third off-pair survivor divides a counted number, the divisor reading is back with two of three. If it is a multiple, the multiple reading has two of three. If it is coprime to both, the one sentence that survives is gone.

The Lucas 4/7 band is the only band left uncut that wrecks at all, and it is 86 rises. It is the one place a third case could come from without going outside the design, and it would answer a second question at the same time — whether the branch is what decides that a band changes its answer at all.

Failing that, the fine end of either branch produces lattices the census does not hold, and one search of it has already turned up a lag nothing else carries. A search there is cheaper than a band and finds different things: it reaches rises the ladder’s rungs do not cover, where the counted pair is stranger and the settled divergence is off the branch entirely.

And a place it could not have come from #

The census cuts ten lattices at ten rises and every one of its thirty wrecked cuts keeps a counted number. So the census would never have found either exception, and neither appeared until a band was swept.

That is the argument for sweeping. A census samples the ladder at rises chosen for coverage, and a rise chosen for coverage is a rise chosen without knowing what is at the rises between. Two of the two off-pair survivors known sit at rises no census would have visited.

The reverse is also true and is worth keeping in view: the census’s ten lattices are read at every offset and to a depth no band sweep reaches, so the two designs find different things. What a band sweep cannot do is anything that needs a long run or a second reading, because it is already 1,120 stems.

What it does to the exchange #

Nothing, and that is worth checking rather than assuming. The exchange is quantified over wrecked cuts whose displacement profile has one balanced pair of exceptional chains, and it uses the surviving family as the period the profile is folded on.

A survivor of 20 folds a profile into twenty classes, and with a hundred and twenty organs read that is six samples a class — below the eight repeats the reading requires. So these two rows are outside the exchange by the same rule that puts a short window outside it, rather than by a judgement about them.

The refusal is the useful part. A profile folded onto a lag it does not have room to repeat returns twenty class means each read from six organs, and the spread within a class would be a statement about the sample rather than about the run. The window is an instrument setting that has been shown to decide answers, so a reading that needs it stretched is a reading that has to be declined.

The number twenty, said properly #

Twenty is four times five and it is also 5/8’s product minus 20 — which is the kind of arithmetic that can be made to work for any number and is worth naming so it can be dismissed. Five times eight is forty and twenty is half of it; 8 + 5 + 7 is twenty; and so on.

The only relation that means anything is the one with a mechanism behind it: a rigid hop of 20 is the 5-hop taken four times, and the pattern above the hole is holding that repetition without holding its unit. Every other relation is arithmetic on two small numbers.

Whether that is a mechanism is a further question and the answer here is no. It says what the reading means, not why the run does it. A stem holding every fourth step of a family and no smaller step of it is a description; the account of why four rather than three or six is not in any file on this site.

Why two cases is where this has to stop #

The ladder offers six bands and two of them wreck nothing at all, so the whole supply of objects that could produce an off-pair survivor is four. Three are cut and one is not.

That is not a resource limit. It is the size of the design, and it means the sentence the off-pair family is commensurate with the pair will rest on two cases, or three, and never on enough to be a rule.

What could enlarge it is a different ladder — a different falloff exponent, or a geometry other than a cylinder — and that changes the rungs, the pairs and the fronts together. A third case from there would be evidence about a different object, which is worth having and is not a replication.

What the reading would have to see to be wrong #

An off-pair survivor coprime to both counted numbers. On a lattice with a pair of 5 and 8 that would be a rigid lag of 3, 7, 9 or 11 — a hop the counter ranks nowhere near the top and a regularity the pattern has no obvious way to hold.

Nothing forbids it. The survivor is the smallest lag whose hop the wrecked run holds, and the run is under no obligation to hold a lag related to anything the counter returns. Two of two being commensurate is the whole of the evidence that it does.

Writing the refutation down is what makes the sentence a claim rather than a description. The next off-pair survivor either divides a counted number, or is a multiple of one, or is neither — and only the third outcome would need a new sentence.

What is claimed #

That two of the ladder’s bands produce cuts keeping a family off their own counted pair; that one of those families divides a counted number and the other is a multiple of one; and that this refutes the reading a single case suggested.

What is not claimed is a replacement. Commensurate with the pair fits both cases and would fit a great many others, and it has two observations behind it. It is written down here so that a third case can contradict it rather than be absorbed into it.