A band that follows the rise

A counter that cannot be slid slid bands of eighteen fixed widths up seven growing stems and found that the index-free chain counter reads the ladder only inside a band of widths — and that outside it, it returns a pair rather than a refusal. The two edges of that band have different causes and both were located.

The lower edge is coverage. A family is kept only when it links most of the band, so a band narrower than about the larger count divided by 0.45 drops the finer family and returns the rung below: a perfectly good pair, from the rung underneath the right one, with nothing anywhere to say so. That is the failure shape a sequence that was a reading found elsewhere in the same instrument, where a count threw away a determination it had already made. The upper edge is rise. A band spanning much more than a third of a rung holds a little of one lattice at its bottom and a little of the next at its top, and the pair it returns belongs to neither.

That second number is a statement about rise and a width is a number of organs, which is why the same width was right on one stem and wrong on another, and why on the fastest bijugate stem no width worked at all. So the fix suggests itself: a band whose width is set from the rise it is read at, rather than chosen once.

What the counter is allowed to know #

The positions of the organs in the band, and nothing else. Not the rise, not the rate, not the order of arrival, not the divergence angle. That restriction is the whole point of the index-free counter and it would be worthless to relax it here: a counter handed the rise could be handed the answer.

So the rise has to be measured from the points, and there is exactly one thing in the band that carries it. The organs sit at successive heights, and the gaps between those heights are the rise at each whorl. A stem whose rise is falling geometrically makes those gaps a geometric sequence, so a line fitted through their logarithms has the decay as its slope.

The one number a band learns about its stem #

That fit is the only addition to the counter, and it returns one number: how fast the rise is falling, per organ. Measured along a stem against the rate the stem’s own recorded rises actually fall at, the worst departure over the readings taken is under one per cent on every stem read.

Why the heights and not the turns #

A band carries two coordinates and only one of them is used. The turns around the cylinder are what the counter reads to find the families; the heights are what the fit reads to find the rise. That separation is worth noticing, because it means the rise estimate cannot be contaminated by the thing it is being used to count: a stem whose families were misread would still give the same gaps between its whorls.

It also means the estimate needs no ordering. Sorting the distinct heights of a band is not a claim that the organs arrived one at a time, only that they sit at heights — which is exactly the restriction counting without an index was built to respect, and the reason that counter exists at all.

From a decay to a width #

A rung of rise is a factor of $\varphi^{2}$ — the transitions of the ladder are that far apart — so a third of a rung is a factor of $\varphi^{2/3}$ in rise, which is 0.3208 in the logarithm. Divided by the decay per organ, that is a number of organs, rounded up to whole whorls so that a multijugate band holds whole whorls.

The widths that come out are 49 organs on a stem whose rise falls over 150, 97 on one over 300, and 193 on one over 600. A stem grown four times as slowly chooses a band 3.94 times as wide, which is the proportionality the upper edge predicted and is the reason no single width fits the seven stems.

Why the count is taken at the widest allowed width #

There is an order-of-operations decision here that matters more than it looks, and the wrong order has a fixed point in it.

The natural procedure is to count, read the coverage floor off the pair counted, and narrow to it. That settles, and it settles on the wrong answer. A band narrower than the coverage floor drops the finer family and returns the rung below, whose own coverage floor is smaller — so the counter arrives at a width that confirms its own undercount and reports it with nothing to mark it.

Counting once, at the widest width the rise permits, cannot do that. The floor is then computed from the finest pair the band was able to show, and it becomes a test rather than a target.

How well it reads #

Given no width at all, the counter reads 100 per cent of bands correctly on both stems falling over 300 organs, 92.8 and 92.2 per cent on the two falling over 600, 92.3 per cent on the trijugate stem, 83.7 per cent on the fast ordinary stem and 61.5 on the fast bijugate one.

Against the width a person could not have chosen #

The comparison that matters is against the best of the eighteen fixed widths for each stem, because that is the width a person could only have picked by already knowing the stem’s rate. The self-sizing counter is within eight points of it on six of the seven stems and beats it on two — 100.0 against 98.8 on the bijugate stem at 300, and 92.3 against 91.5 on the trijugate one.

And the one it falls short on has no good width either #

The exception is the fast bijugate stem, where the self-sizing counter reads 61.5 per cent and the best fixed width reads 84.6. That stem is the one the previous reading singled out: not one of its eighteen widths reads nine bands in ten, so its best width is a best among bad ones rather than a working setting. A counter that cannot find a width there is agreeing with the stem rather than failing on it.

What is left over, and which edge it comes from #

The failures divide cleanly. On the two fast stems the residue is the rung below — 16.3 per cent and 38.5 — which is the coverage edge: the rise allows only 49 organs and the counts have grown past what 49 organs resolve. On the three slowest stems the residue is pairs on no rung at all — 6.5, 6.5 and 7.7 per cent — which is the rise edge, narrowed by the sizing but not closed.

So the sizing fixes the edge it was built to fix and leaves the other where it was. That is the honest reading and it is not a disappointment: the two edges are independent, and only one of them was a width chosen wrongly.

A band that could not have seen what it is looking for #

The coverage residue can be tested for, and the test is available to the counter. A pair (m, n) read from a band of W organs sits on the ladder, and the rung above it is (n, m + n). If W is smaller than the coverage floor of m + n, the band could not have shown that rung even had the stem been on it — so the reading cannot be distinguished from an undercount, and the counter refuses rather than reporting it.

The check removes every undercount #

With it, not one band on any of the seven stems returns the rung below. The 16.3 per cent on the fast ordinary stem and the 38.5 on the fast bijugate one become refusals, and on the two stems that had no coverage failures to remove, nothing is refused at all — so the test costs nothing where there is nothing to catch.

And it refuses a great many correct readings #

It also removes 47.8 points of correct reading on the fast ordinary stem to remove 16.3 of wrong one, and 50.5 to remove 38.5 on the fast bijugate. On the stems with nothing to remove it costs 18.0, 5.3 and 10.3.

The reason is not subtle. A band too narrow to have shown the rung above is often looking at a stem the rung above has not reached, and the pair it read is right. The test cannot tell those apart, because the thing it is testing for is a family that is not there — and a family that is not there because the stem does not have it looks exactly like a family that is not there because the band is too narrow.

What a refusal is worth on a stem #

How many organs a pair needs made the argument that the counting window a pair needs is not a constant but the counter’s own arithmetic, and that a window one organ too narrow returns the rung below with no sign of it. The check here is that arithmetic applied forwards: instead of asking whether the window was wide enough for the pair reported, it asks whether the window was wide enough for the pair that would have been reported had the stem been one rung finer.

That is a stronger question, and it has to be, because the first one cannot fail. A counter that read (m, n) from a band wider than the floor of n has demonstrated nothing except that it read what it read.

Which of the two counters is the instrument #

Both, and they answer different questions. A survey asking what pair does this stem show at this height wants the reading without the check, because the reading is right about five times in six on the worst stem and nine times in ten on the rest. A survey asking what can be said with certainty wants the check, because everything it returns is right on four of the seven stems and better than nine in ten on the others.

What neither of them is, is a counter with a free parameter. The window nobody varied is the account of what an unvaried setting hides, and a width chosen once and reported nowhere is exactly that. Here the width is computed from the points and reported with the count.

What the width says about the stem #

A by-product worth keeping: the width the counter chooses is itself a measurement of the stem. It is a third of a rung expressed in organs, so it is the number of organs the stem takes to cross a third of a rung — which is a rate, and a rate is the quantity the rate decides the branch showed decides which ladder a stem keeps.

So a counter that has sized its own band has, incidentally, measured the thing that decides the stem’s fate, from the same points and at the same time as the count. That is not what it was built for and it is worth reporting alongside the pair, because a pair with no rate beside it leaves out the one number that says whether the pair is stable.

A third of a rung is not a tuned constant #

The share of a rung the band is allowed to span was taken from the fixed-width reading, so it is worth asking whether the result survives moving it. Swept from a sixth of a rung to a whole one, the reading is flat between a quarter and a third and falls away on both sides — refusals rising below, pairs on no rung rising above.

And what happens at a half is an edge #

At half a rung the share on no rung at all goes from under nine per cent to between 60 and 92 on five of the seven stems. That is not a slope; it is a threshold, and it is the clearest confirmation available that the upper edge is about rise rather than about organs. Two stems do better at a quarter or a sixth than at a third, so the edge tightens a little as the rate falls — which is worth recording rather than smoothing away.

Where each band’s reading sits on its stem #

The refusals are not scattered through a stem. On the fast stems they are the whole upper half, where the counts have outrun what a band spanning a third of a rung can resolve; below that the same stems read cleanly. The fast bijugate stem is the extreme case, and it is the one two at a time argued the index-free counter exists for: a pattern with no genetic spiral to index, which the counter can read and the index counter cannot.

That pattern is the two edges closing on each other, drawn along one stem instead of across a table of widths. The band the rise allows shrinks as the stem’s counts grow, and somewhere up every fast stem the two cross.

What this does not fix, and would have to be fixed elsewhere #

The rise edge stays. On the two slowest stems the self-chosen band is 193 organs and around seven per cent of its readings land on no rung at all, and narrowing the span to a quarter or a sixth recovers most of them. That is a hint rather than a result: the span that works best falls slowly as the rate falls, which suggests the upper edge is not exactly a third of a rung but something with a weak second term in it.

Finding that term would need the same sweep on more rates than three, and it is the sort of measurement that is worth doing only if a reader would ever be in a position to use it. A person counting a shoot does not choose a span; the counter does. So the honest version is the one drawn here — the span matters, a third is close to the best, and the two slowest stems say the best moves.

What is claimed, in one line #

A counter that fits the decay of the rise through the spacings of its own band’s whorls, and reads the band spanning a third of a rung, reads a growing stem as well as the best fixed width without being told which width that is — and can refuse the bands it cannot resolve, at the cost of refusing many it can.

What it does not establish #

That a person counting a real shoot estimates the rise this way, or at all. That a real stem’s rise falls geometrically: these do, by construction, and a stem whose rise falls some other way would give the fit a different thing to measure. The coverage constant 0.45 is this counter’s own threshold, and a counter that kept families on a different criterion would have a different floor.

Nor does any of it say that a stem read correctly has been read at a rung. A band spanning a third of a rung sits somewhere inside one, and the pair at its middle and the pairs at its two ends are all counted as right — which is the same convention the fixed-width reading used and is the right one for a stem whose rise never stops falling.

What would withdraw it #

A band whose measured decay differs from its stem’s own by more than a few per cent. A chosen width that does not scale with the stem’s rate. A stem on which the counter returns the rung below with the check on. A span sweep with no edge between a third of a rung and a half. A check that refuses bands on a stem with no coverage failures. Each is checked every time the measurement runs.

Still open: a stem whose rise does not fall geometrically #

Every stem here has a rise falling by a constant factor per organ, which is what makes the decay one number and the fit a straight line. A real shoot’s rise falls because its apex is changing size, and nothing says that change is geometric — a cone’s rise falls one way and an ogive’s another, which the shape of the apex already measures. The next test grows stems whose rise falls on those profiles and asks two things: whether a straight-line fit through the log gaps still recovers a usable local decay, and whether a band sized from a local decay on a curving profile still spans a third of a rung, or systematically more at one end of the stem than the other.