A list that can only shrink

The settling table records where each run ends up, and two runs whose settled values are within half a degree are called one destination. Some destinations are reached only by runs at a steep falloff exponent — 4 or 5 — and never by a shallow one.

That list was three at nine starting angles, five at twenty, and everybody read the growth as a list filling in. At forty it is four.

What can happen to such a list #

A destination joins it when a steep run finds a place no run has been before. A destination leaves it when a shallow run turns up at a place only steep runs had reached.

Those are not symmetric. The list is defined by an absence — no shallow run reaches here — and every new run is a chance to violate the absence and never a chance to confirm it.

So a member of the list is a claim that becomes weaker with every angle sampled, and the list as a whole is an upper bound on the real one rather than a set. The round that established the list said as much and had no case to show for it.

What happened #

Two left and one arrived. 47.9 degrees and 55.2 degrees turned out to be reachable from a shallow exponent once an angle between two of the twenty was tried; 147.3 degrees is a destination nobody had reached before and is reached only by a steep run.

So the count goes five to four by two subtractions and one addition, which is not the same as one member falling off.

Both of the departures had survived a doubling already. 47.9 was on the list at nine angles and at twenty; 55.2 arrived at twenty. Neither was a marginal member.

Why this is the first time #

Because the first doubling added angles and the second added twice as many. Eleven new angles at twenty gave eleven chances to violate an absence and none did; twenty new angles at forty gave twenty and two did.

That is roughly what a rate of one violation per fifteen new angles would produce, which is a number with two observations behind it and is written down only so a third doubling can contradict it.

The more useful way to say it: the list has now been shown to be wrong in the direction it can be wrong, which is the difference between a bound whose looseness is asserted and one whose looseness is measured.

What survives of the reading #

What the list is like. Both departures and the arrival are countable — a stem at any of them has a parastichy pair a counter returns — and all three are two consecutive terms of an additive sequence.

That was the substantive claim: the destinations a steep rule reaches are not exotic, they are ordinary lattices from sequences other than the two ladders. It survives, because it is a statement about members rather than about membership.

How long the list is was always a property of the sampling and is now known to be so in both directions.

The whole destination list #

Fifteen at nine angles, eighteen at twenty, nineteen at forty. So the second doubling adds one where the first added three.

Every one of the four found by refining is off both ladders: 55.2 counting 6/7, 132.3 counting 8/11, 162.1 counting 9/11, and 147.3 counting 2/5.

And the count of destinations that sit on the golden or Lucas sequence is five at nine angles, five at twenty and five at forty. Over a fourfold refinement of the starting angle, not one on-ladder destination has been found that the original nine runs missed.

Which is the more interesting half #

The on-ladder list is closed and the off-ladder list is not. That says something about what the placement rule does: its Fibonacci and Lucas arrangements are few and are found immediately from almost any starting angle, and its other arrangements are many and are found only from particular ones.

The five on-ladder destinations are the golden angle at 137.8, 136.8 and 139.1 either side of it, and 99.5 and 101.6 on the Lucas branch. Those are the arrangements the whole subject is about.

The fourteen off-ladder ones count pairs like 6/7, 4/9, 5/7, 9/11 and 2/5, and each is a real lattice that a real counter would report. They are the part of the rule’s range that nothing outside this table has ever asked about.

What a destination is worth #

Not much on its own, and the number of runs reaching it is the honest weight. The most-visited destination is 101.6 degrees with 74 of the table’s 384 settled runs; the least is 147.3 with one.

So the arrival on the steep-only list is a destination reached by a single run, which is the weakest kind of member. If it turns out to be reachable from a shallow exponent at eighty angles, the list goes to three and the arithmetic becomes three departures and one arrival.

Reporting the run count beside each destination is what makes that visible. A list of nineteen values with no weights would put 147.3 and 101.6 on the same footing.

The arrangements and the sequences #

A destination’s arrangement is the counted pair a stem at it has, and one destination can have several because the pair depends on where up the stem it is counted.

Arrangements go 20, 24, 31 across the three samplings. Additive sequences — the families the pairs come from — go 10, 13, 15.

Both are still growing and neither is converging, because a destination reached from more angles is counted at more rises and more exponents. So those counts are measurements of the sampling as much as of the rule, and they are reported as such.

What the steep-only list is for #

It is the one place the falloff exponent visibly changes what the rule can do. The wall does not separate the exponents and the clock does, and the destination list is the third quantity — and the only one that is a list rather than a number.

So a shrinking steep-only list weakens the strongest visible effect the exponent has. Four destinations rather than five is still four places a shallow rule has never reached across 1,280 runs.

But it is a bound, and the bound has been shown to be loose by two.

The symmetric list #

There is one, and it is small. Destinations only a shallow exponent reaches were two at twenty angles, and the same argument applies to them in reverse: they can only lose members as steep runs are added.

The two lists together are the honest form of the exponent changes what the rule reaches, and both are upper bounds that shrink. What does not shrink is the union — every destination in the table is reached by something — so the total count grows while each exclusive list can only fall.

That is worth stating because a reader could take a growing destination list and a growing steep-only list as the same trend. They are opposite kinds of quantity.

What the two departures were #

47.9 degrees counts a 7/15 or 8/15 pair and sits on the 1,7 and 7,15 sequences. 55.2 degrees counts 6/7 or 7/13 and sits on 1,6.

Neither is near anything special. They are ordinary off-ladder lattices in the middle of the range, and the shallow runs that reached them started at angles between two of the twenty rather than at the ends or in some pathological corner.

So there is no story about where the violations came from. They came from the middle of the sampling, which is where twenty new angles mostly are.

What would settle it #

Eighty angles, which is another 1,280 runs and about three quarters of an hour. The prediction is one or two more departures and perhaps one arrival, on the rate the two doublings so far give.

The stronger version of the same test is cheaper: take the four current members and grow shallow runs at a dense sampling of angles near the ones that already reach them. That targets the violation rather than waiting for it, and it is a few hundred runs.

Neither has been run, and the reason for preferring the second is that a list defined by an absence is best tested by trying to violate the absence rather than by sampling more of everything.

How a destination is defined #

A run has settled when every later divergence stays within a tolerance of the mean over its last stretch, and its destination is that mean. Two settled values within half a degree are called one destination, which is two steps of the 1,536-sample azimuth grid.

The tolerance is why the list has nineteen entries rather than several dozen: the settled values cluster tightly and the gaps between clusters are large, so half a degree sits in a gap.

It is also a second way the count can move. Reading the same table at a tenth of a degree splits some destinations and gives a longer list, and the steep-only count is reported at both tolerances wherever it matters.

What “reached” means for the steep-only list #

That at least one run at exponent 4 or 5 settles there and no run at exponent 2 or 3 does, over the whole table of eight rises and forty angles.

So membership is decided by 640 shallow runs failing to land within half a degree of a value, which is a lot of runs and is still an absence. Each new shallow run is an independent chance to end it.

The asymmetry is worth spelling out because it is easy to read no shallow run reaches here as a property of the destination. It is a property of the destination and of how many shallow runs have been grown.

What the exponent is #

The power in the sum each organ is placed at the minimum of: a new organ sits where the total influence of its neighbours, falling off as one over distance to that power, is smallest.

Three is the value every other measurement on this site uses, so the sweep from two to five puts it in the middle of a range rather than at an end. Below two the neighbourhood stops being local at these run lengths and above five the sum is decided by one organ, which is a different rule rather than a steeper one.

So steep means 4 or 5 and shallow means 2 or 3, and the split is at the working value rather than around it. That is a choice and it is the one that makes the two halves the same size.

The one thing that has never moved #

Five on-ladder destinations, at nine angles, at twenty and at forty. That is the most stable number in this whole table and it has survived a fourfold refinement.

They are 136.8, 137.8 and 139.1 on the golden branch and 99.5 and 101.6 on the Lucas one — the angles the entire subject is about, found by the rule from almost any starting angle at almost any rise.

So the picture is: a small closed set of arrangements the rule falls into easily, and a long open set it reaches from particular angles. Refining the sampling adds only to the second.

What this does to the exponent’s own story #

The falloff exponent is the rule’s one free parameter and three quantities have been asked whether it decides anything. The wall does not, and it turns out never to have been measured well enough to say. The clock does. The destination list does, and it is the effect a reader would find most convincing because it is a list of places rather than a number.

This reading does not remove that effect and it does put a bound on it. Four destinations a shallow rule has never reached across 640 shallow runs is a real asymmetry; five was a real asymmetry too, and two of its members turned out not to be members.

So the exponent’s clearest visible effect is the one most exposed to further sampling, which is an awkward place for a headline to sit and is where it sits.

Which lists on this site are of the same kind #

Several, and it is worth naming them because the argument is general rather than about this table.

The families a wrecking cut has been seen to keep is a list of what has been observed, so it can only grow. The rungs a cut never wrecks at is a list of absences, so it can only shrink. Destinations reached from a single starting angle is a list of absences of neighbours, and it grew when the sampling doubled, which is the interesting case.

The rule of thumb is simple and worth writing down: a list defined by what was found grows with the sampling, and a list defined by what was not found shrinks. A list that moves the other way is a list worth looking at twice.

What the exponent sweep was for #

Two to five, with three the value every other measurement on this site uses. The point of sweeping it was to find something it decides, and the answer after three rounds is: the clock, the destinations, and not the wall.

That is a modest return and it is an honest one. A rule with one free parameter that changes almost nothing when the parameter is moved is a rule whose behaviour is robust, which is worth knowing and is not what anybody hoped to find.

What the destination list adds is the one visible, countable effect. Four destinations no shallow run reaches across 640 shallow runs is a real asymmetry between a steep rule and a shallow one, and it is the sort of thing a reader can check by looking at a table.

Why the count keeps moving in both directions #

Because the two halves of the reading behave oppositely, and it is worth separating them once more.

The destination list is a record of what was found. Every run added can only add to it, so it grows: 15, 18, 19. It converges when the sampling is dense enough that a new angle finds nowhere new, and the on-ladder half has already converged.

The steep-only list is a record of what was not found. Every shallow run added can only take from it, so it shrinks — and it grew from three to five only because the destination list grew underneath it. The two effects run against each other and the net count is the difference.

Untangling them is why this round reports the departures and the arrival separately rather than the net. Five to four is one number; two left and one arrived is what happened.

What a shallow rule and a steep one are #

Worth a sentence, because steep and shallow are doing a lot of work above. The placement rule puts each organ where the sum of its neighbours’ influence is least, and the influence falls off as one over the distance to a power. That power is the exponent.

At two the sum reaches a long way and many organs contribute; at five it is decided by the nearest one or two. So a steep rule is a local rule and a shallow one is not, and the question the sweep asks is whether that changes where a stem ends up.

The answer, over three quantities and three rounds, is that it changes how long a stem takes and which unusual arrangements it can reach, and not the rise at which it stops settling at all.

What is claimed #

That the list of destinations reached only by a steep falloff exponent is three at nine starting angles, five at twenty and four at forty; and that the fall is two departures and one arrival rather than one member dropping off.

That 47.9 and 55.2 degrees left it because a shallow run reached them from an angle between two of the twenty, and 147.3 joined it as a destination nobody had reached before.

And that a list defined by no member of this group reaches here can only lose members as the sampling improves, so every such list on this site is an upper bound — which had been asserted and is now measured.