What a quarter degree cannot see

Six boundaries of three basins have been located to an eighth of a degree, five names were needed where the question offered two, and 1.75 degrees of starting angle was found reaching nothing at all. All of it was read off a grid of a quarter of a degree, and everything the grid cannot resolve is the subject here.

Two things sit under all of it. Nothing in the sweep bounds a basin narrower than half a degree, and the 41.75 degrees in the middle of the widest basin were never swept at that grid — they are still sampled only at the forty-angle table’s own spacing.

Why the grid is a quarter of a degree #

Not because it was cheap. The placement rule puts each organ at the minimum of a sum over its neighbours across a fixed set of candidate azimuths, and the grid the stems are placed on has 1,536 of them. A quarter of a degree is one step of that grid.

So two starting angles closer together than a quarter of a degree are two the placement rule cannot tell apart. Sampling finer would not measure a finer structure; it would measure the same structure twice, because the rule rounds both angles to the same candidate.

Which makes half a degree the floor #

A feature is bounded when a sample lands inside it and a sample lands outside it. At a grid of a quarter of a degree that needs the feature to be at least two steps wide, so a basin narrower than half a degree cannot be given a boundary by this sweep at all.

That is not a budget. It is where the instrument and the rule stop being distinguishable from one another, and no amount of processor time moves it.

It is also the first floor in this collection that is a property of the model rather than of the sampling. The six-degree floor under the twenty-angle basins came from how many stems anybody chose to grow, and halving the spacing halved it. This one does not halve.

What the sweep did bound #

Six boundaries, each located between two sampled angles one step apart, and therefore placed to ± 0.125 degrees. A width formed from two of them carries ± 0.25.

Those are the error bars on every number the three basins produced, and they are the whole of what the grid buys. The 16.75-degree width of the narrow basin is 16.75 ± 0.25; the fringe at 124.375 degrees is 124.375 ± 0.125.

An eighth of a degree is small enough that none of the readings in this thread turns on it. The two lower boundaries that sit closest together — the widest basin’s at 124.375 and the same-rise basin’s at 124.875 — are two steps apart, which is four times the uncertainty on either, so the two are genuinely at different angles rather than one boundary read twice.

The narrowest thing anybody found #

0.75 degrees. At a rise of 0.030 and a falloff exponent of 3, a wedge of starting angle at 124.25 to 124.75 degrees settles onto 101.4 degrees with 139.1 on both sides of it — three sampled angles, a neighbouring destination driven into a basin.

It is the smallest feature this sweep resolved anywhere, and it is three steps of the grid. One step narrower and it would have been two samples; two steps narrower and it would not have been seen at all.

Which is exactly what could be hiding #

That is the reason the wedge matters more as a measurement of the instrument than as a finding. Something 0.75 degrees wide exists in this rule; it was found only because the sweep happened to put a ten-degree window over the place it sits.

Every stretch of starting angle that was not swept at a quarter of a degree could hold one, and the largest such stretch is the middle of the widest basin.

A wedge is not the only shape available at that size either. The widest basin’s upper crossing is a hole of non-settling angle 0.75 degrees wide, and the fringe below that basin opens with a hole of the same width. Three quarters of a degree is simply what the small features in this rule have measured so far.

The middle nobody swept #

The widest basin runs from its located lower boundary at 124.375 degrees to the reflection point at 180, and two ten-degree windows were swept at its two ends. Between them lies 41.75 degrees, from 131.5 to 173.25, sampled only at the forty-angle table’s own spacing of 3.12 to 4.38 degrees.

Forty-one and three quarters degrees is more than three quarters of the basin. Every claim about that basin being one connected stretch of starting angle rests on twelve of the table’s angles inside that stretch and nothing finer.

Twelve samples over forty-one degrees is the same density the whole settling table was read at, and it is the density that produced the readings this thread was written to check. So the middle of the widest basin is not less well measured than the rest of the collection; it is measured exactly as well, which is the point.

How often a coarse sampling misses a thing that size #

The figure derives it twice and prints both, because the two answers differ and the difference is the table’s own unevenness.

Against the table’s actual spacings inside that stretch — 3.125 to 4.379 degrees, uneven because the forty angles are the midpoints of twenty and inherit the golden angle’s awkwardness — a 0.75-degree feature falls between two sampled angles 79 per cent of the time. Against an even three-degree step it falls between two 75 per cent of the time.

Four times in five is the table reading. Three times in four is the even-three-degree reading.

Why the two numbers are not the same #

Because a miss rate depends on the spacing and the table’s spacing is not one number. Where its angles sit 3.125 degrees apart a 0.75-degree feature is caught rather more often than where they sit 4.379 apart, and the pooled figure over the twelve angles in that stretch comes out above the even-step answer rather than below it.

The four-in-five reading is therefore the one to quote about this basin, and the three-in-four reading is the one to quote about a hypothetical sweep at a round step. Neither is the other’s approximation.

What that licenses and what it does not #

It licenses the sentence an unmeasured sliver or puncture could sit anywhere in the middle of the widest basin and nothing here would have seen it. It does not license there is one.

The distinction is the same one the collection has had to keep before: a bound on what a sampling could detect is not evidence about what is there. What the miss rate adds is that the bound is not a formality — four times in five is not a small chance of having missed something.

The reason to be careful about it is that the sentence is easy to slide into a stronger one. A basin drawn as a single unbroken bar looks like a claim that it is unbroken, and it is a claim about its two ends with a straight line between them.

The wedge under a three-degree grid #

The clearest way to see it is to put the coarse grid over the one place a feature of that size is known to be. Drawn over the same-rise basin’s lower window, a three-degree sampling takes four of the forty-one angles, and whether any of the four lands on the wedge is a matter of where the grid starts.

The unswept middle is not a constant #

It is computed from the two windows rather than assumed, so it moves with the basin it belongs to. The widest basin’s middle is 41.75 degrees; the same-rise basin’s is 30.50; the narrow basin’s is 8.00.

That ordering follows from the basins’ own extents. A basin whose two boundaries are sixteen and three quarter degrees apart has most of itself inside the twenty degrees of window that bracket it, and a basin fifty-five degrees across does not.

So the narrow basin is the one this sweep knows nearly all of, which is a reason to prefer its numbers and is not a reason to prefer its shape.

What closing it would cost #

Four more windows of the kind already swept, over the widest basin’s middle, and about the same again for the other two. Each window is forty-one runs at a quarter of a degree, and the whole sweep so far is 574 stems.

So the middle is not unswept because it was expensive. It is unswept because the windows were placed where the boundaries were expected to be, which was the right design for locating boundaries and is the wrong one for finding features inside a basin.

That is a design decision with a shape the collection has met before. Sampling one value of a parameter per region answers the question the sampling was built for and leaves every other question about that region unasked, and the second question does not announce itself until somebody asks it.

The other setting, and why it buys nothing #

Every sampled angle in every window was grown twice: once to 1,200 organs and once to 3,200. That is 287 pairs.

Of the 287, none changes whether the stem settled. None changes the organ it settled at. None of the six boundaries moves by any amount, and none changes kind.

Which is the claim, not a redundancy #

A drawing made from the 3,200-organ sweep is identical to the same drawing made from the 1,200-organ one, and that identity is the result rather than a duplication to be tidied away. Every one of the six located boundaries reads the same to the last digit at both lengths.

An edge located by the length of the run it was grown for would not be an edge. It would be a statement about the budget, and it would move the moment anybody spent more.

Why a run length could have mattered here #

Because a boundary is the one place it should. A stem started well inside a basin has an obvious place to go; a stem started a quarter of a degree from the edge is the case where a slow arrival is most plausible, and where a sweep that stopped too early would report a boundary further in than the true one.

The same statement has been made twice before, over runs three to five degrees apart, which is nowhere near a boundary. This sweep makes it at the one spacing where a budget would have shown, and it still does not show.

And the clock agrees from the other side #

The slowest settling anywhere in the 574 runs is 89 organs, against the twelve hundred every run was given, and the median is 33. Nothing in this sweep is within an order of magnitude of running out of length.

That is two independent readings of the same negative — one from doubling the budget and finding no change, one from measuring how much of the budget was ever used. A falloff exponent that nearly triples settling times does not change it either.

What the run-length negative does not cover #

It covers whether a stem that failed to settle would have settled with more organs. It does not cover whether the settling test itself is the right test.

Nor does it cover the seed. Every run here starts from one organ at one starting angle, and a run length is a statement about how long the stem is grown rather than about what it is grown from.

The tolerance — how close later divergences must stay to count as settled — is the setting nobody has moved, and it has been named as such since the settling table was built. Here it sits in an empty band about seventy times wider than the whole settling population, so a different threshold inside that band would classify every one of the 574 runs identically. A threshold outside it would not be the same measurement.

What the table understated #

Worth recording, because it is the size of the correction a finer grid bought. The forty-angle table bracketed the widest basin between 128.75 and 176.25 degrees. The quarter-degree sweep puts its lower boundary at 124.375 and its upper crossing at 179.625.

So the table was 4.375 degrees short below and 3.75 short above. Both errors are of the order of the table’s own spacing, which is what a bracket read off a coarse sampling should be — and both go the same way, because a coarse sampling can only ever find a basin smaller than it is.

Which is the general shape of the limit #

Every sampling here reports a lower bound on a width and an upper bound on a gap. A basin measured between two samples is at least as wide as the samples say; a stretch of non-settling angle is at most as wide as the samples say, because anything narrower than the grid could be hiding a settled angle inside it.

That asymmetry is why the widths this collection quoted from twenty starting angles were all lower bounds and why the destination list could only ever shrink under refinement. It is arithmetic about sampling rather than anything about phyllotaxis.

What is still unbounded #

The width of anything narrower than half a degree, anywhere. Whether the widest basin’s middle is one connected stretch. Whether the two non-settling stretches found between basins hold a narrow basin apiece. Whether the wedge inside the same-rise basin is the only one of its kind or the only one that was looked at.

None of those is answered by growing more stems at this grid, and the first is not answered by growing them at any grid, because the rule does not distinguish the angles a finer grid would sample.

The second and third are answered by moving the windows rather than by refining them, which is the cheapest open question in this thread: the same forty-one runs a window, placed where nothing has looked instead of where a boundary was expected.

What would change the floor #

Raising the number of candidate azimuths. The floor is one step of the placement grid, so a finer placement grid gives a finer floor — and it also gives a different rule, whose basins would have to be relocated before they could be compared.

Tripling that count has been tried once here and changed nothing that was published, which makes it a reasonable thing to try and not a free one. A basin located under one placement grid and a basin located under another are two measurements of two rules.

What a reader should carry #

That every boundary in this sweep is located to an eighth of a degree, that no basin narrower than half a degree can be bounded at all, and that three quarters of the widest basin has never been looked at more finely than three to four and a half degrees.

And that a feature of the one size this sweep is known to produce would be missed by that spacing about four times in five. The middle of the widest basin is not evidence of smoothness; it is unexamined.

The one line #

The grid is a quarter of a degree because that is one step of the grid the stems are placed on, so the sweep bounds nothing narrower than half a degree — and the 41.75 degrees in the middle of the widest basin remain sampled at the forty-angle table’s own spacing, which would miss a 0.75-degree feature 79 per cent of the time.