A wall or a fade

Where a basin ends, one of two things happens. Either the destination changes — the angles just beyond settle somewhere else — or the angles beyond stop settling at all. Those are different claims about the placement rule, and the collection has never been able to tell them apart, because a table of destinations records where a run finished and not whether the runs beside it finished anywhere.

Naming that gap was the last thing the widest basin’s confirmation did. Cutting three basins at a quarter of a degree closed it, and it needed an instrument rather than a picture: six boundaries named by eye is six judgements, and what a name is worth depends on the same rule having been applied to all six.

Two instruments were tried. The obvious one fails, and the one nobody had read works by a factor of three hundred.

Why an instrument and not a look #

Because a window drawn sample by sample answers the question for one boundary and settles nothing about the next. Six windows read by eye produce six decisions, and the interesting result — that one basin is walled on one side and fades on the other — is exactly the kind of claim an eye produces whether it is there or not.

An instrument is a number computed the same way at every boundary, with a threshold stated before the six are read. If it separates the kinds it separates them everywhere, and if it does not, the failure is visible instead of absorbed.

That is the same requirement a settling test has to meet before any share computed from it means anything. A rule applied by hand to six cases is six cases.

The clock, and why it looked right #

The proposal was the settling time. A stem grown from a starting angle inside a basin settles after some number of organs; a stem grown from an angle just outside a fade never settles at all, so the time is infinite. Approaching a fade, then, the time should climb — a run nearly at the edge should take nearly forever — while approaching a wall it should stay flat, because the angles beyond settle perfectly well, only somewhere else.

Stated that way it is not merely plausible. It is the mechanism: a fade is the place where settling stops working, and settling time is what measures settling working.

The reading taken is the ratio of the slowest settling in the last degree inside the basin to the middle of the rest of the window. A ratio rather than a time, so that a fast basin and a slow one are comparable, and a rise rather than an absolute figure, so that the threshold is a number and not a calibration.

What the clock reads #

At the one clean wall, 1.31. At the two clean fades, 1.00 and 1.57.

The wall lies between the fades #

That is the whole of the failure and it needs no statistics. The wall’s rise is larger than one fade’s and smaller than the other’s, so any threshold set to call 1.57 a fade also calls the wall one, and any threshold set to spare the wall loses the fade at 1.00 as well.

There is no ordering to rescue either. The six ratios are 1.00, 1.00, 1.31, 1.54, 1.57 and 1.69, and the two kinds the question was about sit at 1.00, 1.31 and 1.57 — first, third and fifth of the six. The instrument does not separate them badly; it does not separate them at all.

A negative of this shape is worth more than a weak positive. It is not that the clock needs tuning, or more boundaries, or a better statistic over the same quantity. Six readings, three of them the kinds in question, arranged in the one order that cannot be cut.

And nothing diverges anywhere #

The second half is worse for the proposal. The largest rise anywhere in the sweep is 1.69, and a rise of 1.69 is not a divergence — it is a run taking two thirds again as long as its neighbours.

The slowest settling in all 574 runs is 89 organs, against the 1,200 every run was given. The median is 33. So nothing in this sweep is anywhere near running out of stem, and the quantity that was supposed to climb towards infinity at a fade climbs by two thirds at most and by nothing at two of the six.

That is consistent with what timing settling for the first time found: a run that settles does it early or not at all, and the long tail everybody expected is not there.

Why the clock could not have worked #

The failure is not bad luck, and this is the part worth carrying. A settling time exists only for a run that settles. On the far side of a fade nothing settles, so the clock has no reading there — and the far side of the boundary is the side the question is about.

So the instrument is defined exactly where the answer is already known and undefined exactly where it is wanted. Everything it can say about a fade it says from inside the basin, by extrapolation, and the extrapolation is the assumption under test.

That is a fault in the design rather than in the numbers, and it was available before a single stem was grown. It was not seen because the mechanism story was so good: settling time measures settling, a fade is where settling fails, therefore settling time detects a fade. Every step is true and the conclusion does not follow, because the detection has to happen on the side where the quantity does not exist.

Drawing the failure #

The clock gets two of this essay’s figures and the reading it fails at gets a third, which is more space than the instrument that works. That is deliberate.

An instrument that looked obviously right and is not is the more useful half of a comparison, because the next question of this shape will attract the same proposal. A reader who has seen the six ratios laid out will not need to be argued out of it, and a reader told only that the tail spread works has been given no reason the clock was ever a candidate.

It is also the more honest half. The clock was the instrument the sweep was designed around, and reporting only the one that worked would describe a study nobody ran.

The instrument that works #

Every run in this collection is tested for settling by asking whether its divergence stops changing, and the quantity that test computes is how far the run’s own last two hundred organs wander. It is the spread of the tail, in degrees, and it is produced on every run whether that run settles or not.

Nothing had ever read it. It sat inside a boolean: a run whose tail spread is under the threshold is recorded as settled and a run whose spread is over it is recorded as not, and the number itself was discarded at that point.

Read directly, at the first angle past each boundary, it does the separation the clock could not.

What it reads #

Past the wall, 0.101 degrees — as steady as the basin the angle just left. Past the two fades, 33.042 and 39.712 degrees. Factors of 327 and 393.

Three hundred is not a close call #

Two populations three hundred apart do not need a carefully placed threshold, and the practical consequence is that the reading is not a judgement. The angle past the wall wanders by a tenth of a degree; the angles past the fades wander by tens of degrees.

The second number is worth pausing on. A tail spread of forty degrees is not a stem slowly converging, and it is not a stem converging on something the tolerance was too tight to accept. It is a stem whose divergence is still moving across tens of degrees at organ 1,200, which is a different behaviour from settling rather than a slower version of it.

Pooled over every run #

The boundary-by-boundary reading could be six coincidences, so the same quantity is read over all 574 runs at once. Everything that settles spreads by 0 to 0.442 degrees. Everything that does not spreads by 30.828 to 39.712.

Which says something about the settling test as well #

The threshold that decides whether a run has settled is 1.5 degrees, and it sits in an empty band seventy times wider than the entire settling population. Nothing in this sweep lands between 0.442 and 30.828.

So whether a stem settles is never a marginal call here, and that is a check on a setting nobody had varied rather than a result about basins. A threshold sitting in a gap seventy times its own width is a threshold whose exact value cannot matter; a threshold with runs stacked either side of it would have made every settling share in the collection a function of where it was put.

It does not follow that the threshold is right everywhere. This sweep is 574 runs at three rises, aimed at the insides and edges of known basins, and the fine end of the rise range is where runs stop settling for reasons of their own. The gap is a statement about this sweep.

Why one instrument works and the other cannot #

The tail spread is defined on both sides of every boundary. The settling time is defined on one.

That single difference is the whole comparison, and it is a property of the two quantities rather than of the six boundaries they were read at. Any instrument for this question has to say something about angles that reach nothing, because reaching nothing is one of the two answers — and a quantity computed from where a run finished cannot, because those runs do not finish.

The tail spread survives that requirement because it is a statement about the run’s behaviour rather than about its result. A run that never settles still has a last two hundred organs and they still have a spread.

What the instrument does not do #

It does not name a boundary. It separates the angles beyond reach something from the angles beyond reach nothing, which is the wall-or-fade question and is not the whole of what the six boundaries turned out to be.

Three of them are neither. A fringe frays into islands of the basin’s own destination; a sliver is a wedge of a neighbouring destination driven into the basin; a puncture is a hole inside one. Telling those apart needs the neighbourhood — how many angles beyond settle, where, and to what — and the spread contributes one column of that reading rather than the verdict.

Six boundaries, five names, three edges #

The question offered two possibilities and the six boundaries needed five names for themselves. Only three of the six are edges in the sense the question meant: one wall and two fades.

That is not a complaint about the question. A question that turns out to have offered too few answers has done its job, and the three that are not edges are why one basin has no width to quote rather than an untidiness in the naming.

The kind belongs to the rule #

Here is the finding the second basin was cut for. Two basins sit at the same rise of 0.030, reach the same destination to a fifth of a degree — 139.3 against 139.1 — and are read over the same ten degrees of starting angle. They differ in one thing: the exponent of the falloff.

At an exponent of 2 the lower border is a fringe at 124.375 degrees, with three of the eight angles beyond it settling and all three back on the basin’s own destination. At an exponent of 3 it is a sliver at 124.875, with all eight settling and three of them on 101.4 degrees.

What that comparison controls for #

Everything except the one parameter. The rise is the same number, the destination is the same to finer than the tolerance that calls two destinations one, the window of starting angle is identical sample for sample, and the two located borders sit half a degree apart — two sampling steps, which is as close as this grid can put two things that are not the same thing.

So one parameter of the placement rule changes what the border is, at a place where it changes neither the destination nor, to half a degree, where the border sits. That is a stronger statement than a correlation across a table, because there is nothing else left to carry it.

It also puts the falloff exponent somewhere new. The exponent has been shown to change which destinations are reachable at all while leaving the share that reaches one alone. This adds the texture of a border to that short list, and it is the least expected entry on it: the exponent decides what a border is at a place it barely moves the border to.

And two sides of one basin need not agree #

The narrow basin is walled below and fades above. Same basin, same rise, same exponent, same destination — and its two boundaries are different kinds of object.

That disposes of the way the question was originally framed. An edge was being asked about as though it were a property a basin has, the way it has a width or a destination, and the phrase what an edge looks like carries that assumption inside it.

An edge is a property of the placement rule at a place. Two places on the border of one region are two places, and a rule that behaves one way at one of them is under no obligation at the other. That is the same lesson a hard cut-off and a smooth one at the same range produced from the other direction: what the rule does at a boundary is decided by the rule’s shape there, not by the region the boundary encloses.

What the run length did not change #

The clock is a time, so a longer run is the one change that could plausibly have rescued it. A run given 3,200 organs instead of 1,200 has more time to settle late, and a late settling is exactly what a rising clock predicts near a fade.

Every window was swept at both lengths — 287 pairs of runs. Not one changed whether the stem settled, not one changed the organ it settled at, no boundary moved, and no kind changed.

So the clock’s failure is not a budget. It is the instrument, and the earlier finding that a run either settles early or not at all is what it looks like seen from a boundary.

What this does not support #

Six boundaries on three basins at two rises is a small sample, and the tail spread has been validated on it and nowhere else. That it separates two populations by a factor of three hundred here says nothing about a rise at the fine end of the table, where almost nothing settles from any starting angle.

Nor does the reading test what happens at a boundary between two stretches that both fail to settle, because no such boundary was cut. Every window here was aimed at the side of a known basin, so every one has a settling population inside it, and an instrument read against a population that is always present has not been asked the hard question.

And the naming is a reading rather than a measurement. Fringe, sliver and puncture are descriptions of a neighbourhood, chosen because the three of them appeared, and a fourth neighbourhood would need a fourth word. What is measured is the located angle, the destinations either side, and the two instruments’ numbers; the names sit on top of that and could be drawn differently without any number moving.

What the leftover angles say #

One thing the spread does answer, which nothing else could. Between two basins there are stretches of starting angle that reach neither of them and nothing else, and until the spread was read those stretches were an absence in a table — a run recorded as not settling, indistinguishable from a run whose destination the sampling missed.

A tail spread of thirty to forty degrees says which it is. Those runs are not near anything; they are still moving at organ 1,200, and the angle they occupy is left over rather than assigned to a destination too narrow to see.

That is the reading the wall-or-fade question was really after, and it arrives as a by-product of answering it.

What is claimed #

That the settling clock cannot separate a wall from a fade: the one clean wall’s rise of 1.31 sits between the two clean fades’ 1.00 and 1.57, so no threshold cuts them apart, and nothing in the sweep diverges — the slowest settling in all 574 runs is 89 organs of the 1,200 each was given, against a median of 33.

That the clock could not have worked in principle, because a settling time is undefined on the far side of a fade, which is the side the question is about.

That the tail spread does the separation and was already being computed on every run: 0.101 degrees past the wall against 33.042 and 39.712 past the fades, and pooled over all 574 runs everything that settles spreads by 0 to 0.442 degrees and everything that does not by 30.828 to 39.712 — an empty band seventy times wider than the settling population.

And that the kind of a boundary belongs to the placement rule rather than to the basin: one exponent of the falloff turns a fringe into a sliver at a place where it changes neither the destination nor, to half a degree, where the border sits, and one basin is walled on one side and fades on the other.