The angles left over

A basin is the set of starting angles from which a stem ends up at one settled divergence. The word has been used here since the first stretch of starting angle was given an extent, and every use of it has carried an assumption nobody wrote down: that the basins divide the circle between them, so that an angle not in one is in the next.

Twenty and a quarter degrees swept without a hole says otherwise. Between the last starting angle that reaches 101.5 degrees and the first that reaches 139.3 there are 1.75 degrees of starting angle, six sampled values, and not one of them settles anywhere at all.

What was swept #

Two ten-degree windows, joined. One sits over the neighbourhood between the basin at 101.5 degrees and the widest basin the settling table holds; the other sits over that basin’s own lower boundary. Together they run from 111.25 to 131.5 degrees at a quarter of a degree, which is 82 runs, and the join is checked rather than assumed: every consecutive pair of sampled angles in the merged scan is one step apart.

That check is the whole difference between this reading and an artefact. A scan assembled from two windows with a hole between them would produce a gap by construction, and the gap would be in the sampling.

The gap, located #

The last starting angle that reaches 101.5 degrees is 114.5. The first that reaches 139.3 is 116.25. Between them lie six sampled angles at 114.75, 115, 115.25, 115.5, 115.75 and 116 degrees, and none of the six settles onto anything.

Not onto a third destination, not onto an off-ladder divergence, not onto a rational angle. The stems grown from those six starting angles are still moving at the organ their run ends.

So the two basins do not abut. Between them is a stretch of starting angle that belongs to neither and to nothing else, and its width is 1.75 degrees against a sampling step of 0.25.

Why a gap at a quarter of a degree is a gap #

Because the alternative explanation has a size and this is larger than it. A stretch of starting angle that appears empty because nothing was grown in it is a miss; a stretch in which seven consecutive quarter-degree steps were each grown and each failed to settle is a property of the placement rule at those angles.

Seven steps is what separates the two settled samples. Six of them are the failures and the seventh closes the interval. A miss would need the sampling to have skipped all six, and the merged scan is checked for exactly that.

The other way a gap could be an artefact is the run length, and the settling times answer that below. What is left after both is a statement about the rule: from those six starting angles, at that rise and that falloff exponent, the stem does not arrive anywhere.

What reaching nothing means here #

A run has settled when its divergence stops moving, and the measure is the spread of the run’s own last two hundred organs against a threshold of 1.5 degrees.

That is not a marginal call anywhere in this sweep. Pooled over all 574 runs, everything that settles spreads by 0 to 0.442 degrees and everything that does not spreads by 30.828 to 39.712 — so the threshold sits in an empty band about seventy times wider than the whole settling population.

A run in the gap is therefore not a run that nearly settled. It is a run whose divergence is still wandering by thirty degrees at the end.

That separation is what makes a negative worth stating at all. A threshold sitting inside a continuous spread of values would turn every non-settling angle into an argument about where the threshold was put, and there is no such argument available here — the two populations do not overlap and are not close to overlapping.

And a wider void beside it #

The gap is not the largest thing of its kind in the neighbourhood. From 118 to 120.75 degrees there are twelve consecutive sampled angles reaching nothing, a run three degrees wide, and it sits above the gap rather than below it.

That matters for how the finding is stated. A gap of 1.75 degrees between two basins could be read as a boundary with an unusual amount of width to it — a transition zone rather than a crossing. A void of three degrees sitting just above it, with a degree and a half of settled angle between the two, cannot be read that way at all.

Between the gap and the void the strip is not empty: seven consecutive angles reach 139.3 degrees. So the neighbourhood alternates rather than grading, and neither of the two non-settling stretches can be absorbed into a boundary belonging to a basin on one side of it.

Five stretches in ten degrees #

Cut into runs of consecutive sampled angles that reach the same thing, the ten-degree neighbourhood window holds five stretches, of which two reach nothing. Twenty-three of its forty-one angles settle and eighteen do not.

Read as a partition that would be a strange one: two of five regions are not basins of anything and they hold nearly half the angles in the window.

The five stretches are not five equal things either. They run from a quarter of a degree to three and a quarter, and the two that reach nothing are the second and the fourth — so the window reads as basin, nothing, basin, nothing, basin, which is an alternation rather than a sequence of neighbours.

The island #

Between the gap and the void sits something the assumption has no room for at all: a detached stretch of 1.5 degrees, from 116.25 to 117.75 degrees, whose seven angles all reach 139.3 degrees — the widest basin’s own destination.

It is 6.6 degrees below that basin’s located lower boundary, and separated from it by 2.75 degrees of angle that reaches nothing.

So the set of starting angles reaching 139.3 degrees is not an interval. It is a long stretch with at least one piece of itself lying detached below it, and a partition of the circle into intervals was never going to describe that.

Which was visible before and unreadable #

An earlier reading of the settling table noticed exactly this shape and could not decide what it was. At one rise and one exponent three separate single angles reached one destination with other things between them, and the table could not say whether that was one basin cut into pieces by the sampling or three arrivals from three separated stretches.

At ten degrees between samples it could not be told. At a quarter of a degree the pieces have widths and the things between them have widths, and the answer is that both are real.

The fringe at the basin’s own edge #

The same shape appears again at the widest basin’s located lower boundary, which is why it has a name of its own. The boundary sits at 124.375 degrees, standing on twenty-nine consecutive angles that reach 139.3.

Beyond it, seven of twelve sampled angles still reach 139.3 degrees, in three separate islands, the furthest 2.875 degrees outside the crossing. Downwards from the edge those islands are 123.25 to 123.5 degrees, 122.75 on its own, and 121 to 122.25 — with angles reaching nothing at 123.75 to 124.25, at 123, and at 122.5.

So the border frays rather than ending #

That is what makes the boundary a fringe rather than an edge, and it is the same fact as the island six degrees further down. A basin here does not stop; it thins out over about three degrees into alternating stretches that do and do not reach it.

Whether a fringe is what an edge of a basin generally is is a question the six boundaries answer separately, and the answer there is that it is one of five things.

The claim, stated #

Settled destinations do not tile the starting angles. Between two basins there is starting angle that reaches neither of them and nothing else, at a sampling seven times finer than the gap it finds.

That is a claim about the placement rule rather than about the sweep. It says that for some initial conditions this rule has no attractor to fall into — not a slow one, not a distant one, none.

What would refute it #

A finer sweep of the same six angles finding a destination in them. The gap is bounded below by the sampling: a basin narrower than half a degree would sit inside it unseen, and this sweep cannot go finer because a quarter of a degree is where the placement rule stops distinguishing two starting angles at all.

So the claim is exactly as strong as its floor allows, which is: nothing wider than half a degree settles in that stretch. That is a bound rather than an absence, and the difference is the whole of what a sampled negative can say.

What it is not #

It is not a claim that the gap is empty of structure. Six runs that each wander by thirty degrees are six runs doing something, and nothing here says what.

It is also not a claim about the whole circle. Twenty and a quarter degrees were swept unbroken and the rest of the range was not, so starting angle is left over is established where it was looked for.

And it is not everywhere #

The narrow basin is the counter-example, and it is worth having. At its lower boundary forty-one of forty-one sampled angles settle: three destinations abut with no non-settling angle anywhere between them, 136.9 degrees over 3.75 degrees of starting angle, 101.8 over 0.75, and 151.0 over 5.25.

That window is a partition, over ten degrees, exactly as the assumption expected. So the finding is that the destinations tile the starting angles in some places and not in others, and a rule of the first kind was being read into every place.

Which makes the left-over angle local #

Two windows ten degrees apart, at two rises and two falloff exponents, disagree about whether there is angle between the basins. That is not a contradiction; it is the thing to measure next.

The natural reading is that the gap belongs to the pair of destinations rather than to the rule as a whole — that some neighbouring pairs meet and others do not. Nothing here tests it, because the six boundaries cut so far were chosen to compare kinds of border rather than to survey pairs.

The clock has nothing to say about it #

The obvious worry about a run that does not settle is that it was stopped too early, and the settling times refuse it outright. The slowest settling anywhere in the 574 runs is 89 organs, against the twelve hundred every run was given.

So nothing in this sweep is close to running out of length, and the median settled run gets there in thirty-three organs. A stem that has not settled by organ eighty-nine in this sweep is not a stem that needed longer.

Which the collection already knew twice over #

Tripling the run length changes not one row of the settling table, and the same statement survives a falloff exponent that nearly triples how long settling takes. This sweep makes it a third time and at a quarter of a degree, which is the spacing at which a budget rather than a wall would have shown.

That is worth one sentence rather than an essay, because the negative has now replicated at three samplings and is the least surprising thing here.

What a settling share was a share of #

Every share this collection has quoted is a share over a list of starting angles, and the list has been resampled three times without anybody asking what the non-settling angles are.

They are not scattered failures. In this neighbourhood they are two runs, three degrees and 1.25 degrees wide, holding eighteen of the window’s forty-one angles between them. A share of about three in seven is the same number either way, and the geometry underneath it is a region rather than a sprinkling.

Which changes what a share can be read as #

A share computed over angles sampled ten degrees apart is an estimate of how much of the circle settles. That reading survives.

What does not survive is the reading that has been sitting beside it — that the non-settling angles are the ones the sampling happened to catch between basins. They are stretches with extents, and at least one of them is wider than the basin gap it sits beside.

What the earlier samplings could have seen #

Almost none of this. The forty-angle table samples at 3.12 to 4.38 degrees, so the 1.75-degree gap falls between two of its angles more often than not, and the 1.5-degree island would have been a single filled cell if it were caught at all.

That is not a criticism of the table, which was refined twice for other reasons and confirmed the widest basin was a basin rather than an artefact of its own spacing. What a refinement of that kind cannot do is find anything narrower than the spacing it lands on. It is the ordinary relationship between a feature and a spacing, and it is why the ten-degree windows were swept at all.

The same shape has been resolved once before here #

A coarse sample of a band once showed one answer alternating with another, and the obvious reading was a period. At full resolution it turned out to be thirteen islands one to three rises wide, with gaps of every size, and a fitted period bought nothing.

This is the same thing one thread over: what a coarse sampling shows as alternation is islands when it is swept, and what it shows as adjacency is sometimes a gap.

Why the assumption was reasonable #

Because a rule with attractors usually does partition its initial conditions, and because the divergence this rule produces is an output rather than an input — which makes every starting angle a point that goes somewhere.

The word basin carries the partition with it, and it was borrowed for a good reason. What this sweep shows is that the borrowing has a limit, and where the limit falls.

What is genuinely open #

Whether the non-settling stretches are one thing or several. A run that wanders by thirty degrees could be circling between two destinations, drifting, or doing something with no name here, and the tail spread does not distinguish those.

Whether the gap closes at any rise or exponent. Two of the three basins cut here have a neighbouring destination immediately beside them and one does not, on three cells of a table with thirty-two.

And whether the island below the widest basin is one island or the visible member of a family, which needs the middle of that basin swept rather than its edges.

What this adds to the settling picture #

One structural fact, and it is the kind that changes how a word is read rather than adding a number. The destinations reached by this rule cover part of the starting angles and do not partition them: between two of them there is a stretch that reaches neither, and beside that stretch a wider one, and inside the wider neighbourhood a detached piece of one basin.

The destination list itself has only ever shrunk under refinement, so the left-over angle is not waiting on more destinations being found. It is waiting on nothing.

What a reader should carry #

That 1.75 degrees of starting angle between two basins reaches neither of them and nothing else, at a sampling of a quarter of a degree with no hole in it; that a three-degree void sits beside it; and that a detached island of one basin sits between the two.

And that the strength of all three is set by the sampling floor rather than by the sweep: what is established is that nothing wider than half a degree settles in those stretches, which is a bound and not an emptiness.

The one line #

Between the last starting angle reaching 101.5 degrees and the first reaching 139.3 there are 1.75 degrees in which nothing settles, sampled at a quarter of a degree without a gap in the sampling — so the settled destinations of this rule do not tile the angles a stem can start from.