Not the shorter of the two

A wrecked stem keeps one lattice hop rigid, and the hop it keeps is one of the two contact families at twenty-nine of thirty offsets. That is as far as the census had been pushed, and it leaves a question with only two possible answers, which is the kind of question that usually gets settled in a sentence.

The sentence everybody reaches for is this one: the rule keeps its shortest hop. It has everything a good guess needs. The rule places each organ at the minimum of a sum of inverse powers of distance, so it is a rule about being far from near things; the nearest neighbour is the shortest step; a family made of the shortest steps is the one most strongly held.

It is false. Across the census it is right at twelve of thirty offsets and wrong at the rest.

What “shortest” means here, and how close the two are #

A hop’s step is not measured up the stem. It is measured across the surface: the azimuthal difference and the height difference taken together, on a cylinder whose circumference is one turn. That is the distance the placement rule itself uses, which is why it is the one to rank by.

Ranked that way, the first two entries are always the contact pair. The gap between them is the number this essay turns on, and it is small. Across the ten lattices that produce a wrecked stem it runs from 1.0024 to 1.1072 — that is, the second-shortest step is between a quarter of a per cent and eleven per cent longer than the shortest.

So the two candidates are nearly the same length, and the question of which one the rule holds is not the question of which one is nearer by any comfortable margin. On a lattice where the difference is a quarter of a per cent it is not really a question about length at all.

Seventeen of twenty-nine keep the second #

Set aside the single offset whose survivor is not a contact family — the long four-hop at 8/13, which is neither the shortest nor the second — and twenty-nine offsets remain with a survivor that is one of the pair.

At twelve of them the survivor is the shortest step. At seventeen it is the second-shortest.

Some individual rows are worth reading, because the effect is not a scatter of near-ties. At the 5/8 lattice with a rise of 0.013 the shortest step belongs to the eight-family, and both wrecked offsets keep the five — whose step is 1.074 times as long. At 8/13 with a rise of 0.005 the shortest step is the thirteen and four of the five wrecked offsets keep the eight, at 1.088 times the length. On the Lucas branch at 4/7 and a rise of 0.020, the shortest is the seven and both wrecked offsets keep the four, at 1.082.

Those are three different lattices, on two different branches, all keeping the family with the longer of the two contact steps. And the reverse happens too: at 3/5 both wrecked offsets keep the five, which there is the shortest, and at the Lucas 4/7 with a rise of 0.013 four of five offsets keep the seven, which is the shortest there.

Neither reading survives. “The shortest hop is kept” holds at twelve of twenty-nine; “the longest of the two is kept” holds at seventeen. Both are descriptions of a subset dressed as accounts — which leaves the offset as the only ingredient not yet tried, and it does not finish the job either.

The ranking is not the same as the pair #

One clarification before the counting, because the ranking and the counted pair are two different measurements that agree, and the agreement is doing work.

The counted pair comes from the positions. A counter is shown the coordinates of the organs, follows chains of near neighbours, and reports how many chains run in each of the two directions. It never sees a divergence angle and never sees a lag.

The ranking comes from the angle. Given the settled divergence and the rise, the step of the lag-k family is a two-line calculation, and sorting those gives an order over every lag out to forty.

The two agree: at every lattice in the census, the first two entries of the ranking are the pair the counter returns. That is what “contact family” means, and having it come out of two independent computations is why the phrase can be used in a claim rather than as a definition dressed up as one.

Before the cut there is no asymmetry to appeal to #

It is worth ruling out the possibility that one family was already the weaker of the two, since that would make the whole question a matter of which was more fragile to begin with.

On an undisturbed stem both contact hops are exactly rigid. Measured over the same hundred and twenty organs the census uses, the divergence of a settled stem holds to a hundredth of a degree or better, and every hop derived from it holds to the same. There is no wobble in one family and steadiness in the other; there is a lattice, in which every lag is as fixed as every other.

So the asymmetry that decides which family survives is created by the removal. It is not a pre-existing weakness that the removal reveals, which is what a length-based account would have to be.

The lattice where the words stop meaning anything #

One row deserves separating out, because it is the case that shows why a claim about length was never going to work.

The Lucas lattice at a rise of 0.008 carries 7/11, and its two contact steps differ by a factor of 1.0024 — a quarter of one per cent. All four of its wrecked offsets keep the seven, which is the second-shortest. But on a lattice where the first and second differ by two parts in a thousand, saying that the rule “chose the longer one” is reporting a rounding.

The machinery flags rows like that rather than dropping them: a separation of one per cent is the threshold, two of the ten lattices are under it, and their rows are counted in both totals with the flag attached. Dropping them would have moved the score and hidden the reason.

Both families are inside the neighbourhood, and that is measurable #

The reason “shortest wins” fails is not mysterious once the rule’s own profile is looked at directly.

At the growing tip, the rule minimises a sum of terms, one per organ already placed, each falling off as a power of the distance. Rank those terms by size and read off which organs they belong to. The five largest belong to lags 13, 8, 5, 21 and 26 — and they are the same five at every falloff exponent from 1.5 to 6.

Both members of the contact pair are near the top of that list, and by amounts that are comparable rather than separated. The rule is not holding one family while barely noticing the other; it is holding both, and a cut has to decide between two things it is doing at once.

So the shape of the answer is: the rule holds two families, a single removal breaks the symmetry between them, and the length of the step is not what decides which way it breaks. A quantity that varies by four per cent across the census cannot be what settles a binary outcome that changes within a single lattice as the offset moves by one organ.

How much longer, exactly #

The gap between the two contact steps is the quantity every version of the guess depends on, so it is worth having the whole distribution rather than the range.

Across the ten lattices that produce a wrecked stem, the second-shortest step is longer than the shortest by 1.0024, 1.0131, 1.0738, 1.0750, 1.0760, 1.0817, 1.0867, 1.0883, 1.0917 and 1.1072. Eight of the ten lie between seven and eleven per cent; two lie under one and a half.

Those numbers have a reason and it is the ladder. A rung is the range of rises over which one pair is the shortest two, and the boundaries of a rung are exactly the rises at which the second-shortest overtakes the shortest — so the gap necessarily passes through zero at every transition and is largest in the middle. The two narrow lattices are the two nearest a boundary.

So “the shorter family survives” is a claim whose subject matter shrinks to nothing twice on every rung, and whose margin is eleven per cent at best. Even where it is right, it is right by an amount that a placement rule reading a hundred and eighty organs has no obvious way to notice.

The census is stacked in the guess’s favour and the guess still fails #

There is a way of reading the distribution above that strengthens the refutation rather than merely describing it, and it turns on how the rises were chosen.

The gap between the two contact steps is zero at every transition and largest in the middle of a rung. So a census assembled at rises near the rung boundaries would be a census where the ordering carries almost no information, and refuting “the shortest wins” there would prove nothing: a rule cannot be said to have ignored a distinction that is not there. This census is the opposite. Eight of its ten wrecking lattices sit between seven and eleven per cent, which is the top of the range the ladder makes available anywhere, because the rises were picked in the middle of their rungs for an unrelated reason — to keep the counted pair unambiguous.

So the sample is biased towards the cases most favourable to the hypothesis, and the hypothesis holds at twelve of thirty anyway. Had the score come out at twelve of thirty on a census of near-transition lattices, the honest reading would have been that the question was badly posed. Here the question is as well posed as the ladder permits: at eight of ten lattices one contact family is clearly nearer than the other, by a margin that is as large as any rise anywhere could offer, and the rule keeps the further one more often than not.

The two narrow lattices are then not the awkward part of the evidence but the least important part of it. Removing them entirely leaves eight lattices, a gap of seven per cent or better at every one, and a score that is still not a majority.

The rule’s own weighting makes the gap smaller still #

The margin is worth converting into the currency the rule actually works in, because a difference in distance is not a difference in influence.

The placement rule sums a falloff in the separation, and the site’s working exponent is an inverse cube. A step eleven per cent longer therefore contributes 1.1072³ = 1.357 times less — so at the widest gap anywhere on the ladder the shorter contact family is weighted about a third more heavily than the longer one. At the narrow lattices, 1.0024³ is 1.007: seven parts in a thousand.

That range is the whole of what a length-based account has to work with, and it holds across the exponent sweep as well. The five largest terms in the rule’s profile are the same lags at every falloff from 1.5 to 6, and at the extremes of that range the widest gap converts to 1.165 and 1.842. So on no exponent the site has ever used does the shorter family carry even twice the weight of the longer one, and on most lattices it carries a few per cent more.

A few per cent of weighting is being asked to decide an outcome that is binary, that changes within one lattice when the removal moves by a single organ, and that puts the stem twenty-seven degrees apart depending on how it goes. Quantities of that size do decide things in nonlinear systems — but they decide them by tipping a balance, which means the balance is what the account should be about, and the balance is a property of the cut rather than of the two lengths.

That is the sharpest form of the negative result. It is not that the shortest step fails to win often enough; it is that the shortest step never had enough of an advantage to be the reason for anything, and the twelve offsets where it does win need no more explanation than the seventeen where it does not.

Keeping the longer family makes the slip larger #

The choice is not cosmetic, and the size of what it decides is worth stating, because it is the reason a binary outcome is worth an essay.

A wrecked stem’s divergence lands at the one it was cut from plus a whole number of turns of the surviving family, divided by that family’s period. Keep the eight and the available destinations are 45° apart. Keep the five and they are 72° apart. The measured slips bear that out exactly: at the offsets that keep an eight the divergence moves by 44.97° to 45.03°, and at the offsets that keep a five it moves by 71.95° to 72.00°.

So a difference of four per cent in step length — or a quarter of a per cent, on the lattice above — is deciding a difference of twenty-seven degrees in where the stem ends up. That ratio is what makes “the shorter step wins” an unsatisfying account even where it happens to be right: the quantity it appeals to is far too small and far too smooth to be producing an outcome that large and that discrete.

The same refutation on the Lucas branch #

Everything above is stated over a census that mixes two branches, and it is worth separating them, because a result that held on one and not the other would be a result about seeds rather than about rules.

The golden branch contributes nineteen wrecked offsets and the Lucas branch eleven. On the golden branch the survivor is the second-shortest step at eleven of nineteen; on the Lucas branch at six of eleven. Both branches keep the shortest sometimes and the second-shortest more often, in about the same proportion, and neither is close to doing one or the other consistently.

That matters because the Lucas branch is where the two clearest individual cases sit. The 4/7 lattice at a rise of 0.020 keeps its four-family at both wrecked offsets, and four is the longer of its two contact steps by eight per cent. The 7/11 lattice at 0.008 keeps its seven at all four, and seven is longer than eleven by a quarter of a per cent. If the census had been golden-only, the second of those — the case that shows the ordering can be meaningless — would not be in it at all.

What a refuted guess is worth here #

Two things, and they are the reason this is an essay rather than a footnote.

The first is that it removes the account that would have stopped the search. “The rule keeps its nearest neighbours” is satisfying, sounds like mechanism, and would have been repeated. It is worth knowing that it is wrong before it is repeated, and the way to know is to count the cases rather than to check the case that suggested it.

The second is that it points at what the answer has to depend on. If neither member of the pair is favoured by length, then what distinguishes them must be something about the cut rather than about the lattice — where the removal lands relative to each family, rather than how long each family’s step is.

That is the direction the next essay takes, and it produces the best rule available over those two numbers — together with the pair of runs that shows no such rule can be the whole account.