A counter built on a falling rise

A counter that cannot be slid found that the index-free chain counter reads a growing stem only inside a band of widths. Too narrow and it drops the finer family and returns the pair below; too wide, more than about a third of the step of rise between one transition and the next, and it holds a little of two lattices and returns a pair on no transition. A band that follows the rise then made the counter choose its own width: it fits the decay of the rise through the logarithms of its own band’s spacings, and takes the band that spans a third of that step. A band that follows a curving rise moved that onto stems whose decay changed sixfold along their length, and found it reading them all, with a straight-line fit recovering the local decay to within half a per cent.

All of those stems have a rise that falls, the way a cone’s rise falls. A shoot that is starting to flower does not stay like that. Its apex swells and the rise falls; then, as the apex is used up by the organs it is making, the rise climbs again. The earlier essay ended by naming that case and a specific worry about it: near the minimum the rate of fall passes through nought, a third of that step becomes enormous in organs, and the counter’s widest allowed band would become the limit rather than the rise. It asked whether the counter reads the pair through the turn, where the stem climbs back down the transitions it came up, and whether its widest band, rather than the rise, sets what it can see there.

The answer to the question as asked is that the widest band is no limit at all. The turn itself is read correctly by a band of any size. What fails is something else, it fails on the way back, and it comes from a choice of sign that had never mattered.

Stems that turn round #

The stems are grown by the same free rule as every stem in this line of work, each organ placed against the organs already present on an azimuth grid, with six whorls of ideal seed to start. Only the rise is new. Here its logarithm is a parabola in organ number. Its decay per organ starts at a stated rate and falls linearly through nought at the turn, then carries on so that the rise climbs just as fast as it fell.

Three single-organ stems were grown. Two start at the fastest rate read before, 1/150 per organ, falling either thirtyfold to the turn at organ 1,020 or a hundredfold to a turn at 1,382. The third falls thirtyfold at half that rate, its turn at organ 2,041, so it has twice as many organs at every pair. A fourth, bijugate, falls thirtyfold at the base rate with two organs a whorl, and is treated separately below. Every one is then read at a band top every eight organs by the counter exactly as it stood, and by three fixed widths of 49, 97 and 193 organs, the ones it had chosen for itself on geometric stems at rates of 150, 300 and 600.

One stem, band by band #

The strip tells the story in one look. The top row is the counter as built. Before the turn it is nearly solid blue, the colour of a right answer, broken by small patches of refusal. Past the turn the blue stops. About three hundred organs on from the turn it runs into a stretch of orange, the colour of a pair on no transition, and the rest of the stem is a mix of that and the green of refusal.

The numbers agree with the look, and the three stems tell one story. Bands more than a fifth of the turn’s organ number before it are right 92, 84 and 88 per cent of the time, the rest being refusals. That is the counter’s normal record. On the same side nothing has changed. Bands an equal distance past the turn are right 12, 12 and 28 per cent of the time. On those bands it refuses 60, 67 and 42 per cent, and on 28, 21 and 30 per cent it returns a pair that sits on no transition at all. It returned 13/67 at a rise where the pair was 13/21, and 26/114 where it was also 13/21.

A failure of this kind is not a refusal. Silent wrong answers were the fault this counter was built to remove. The counter it replaced returned them when a band was too wide, and a fifth of the climbing side has brought them back.

Why it fails #

The width it chooses says what is going wrong. Early in the stem it chooses a band of about 68 organs, and the band widens as the rise slows, exactly as it should, reaching its largest allowed value of 320 organs at the turn. From there it never comes back. Far out along the climb, where the rise is moving so fast that a band of 50 organs would cover a third of a step, it is still reading 320. By then that band covers one and three quarters of a step of rise. A band that wide holds several lattices at once, and it is precisely the failure the counter was introduced to prevent, brought back in full by holding the width still.

The reason it does so is a single inequality. The counter takes its width as a third of a step over the decay when the decay is positive, and treats anything else as infinite. Which is correct on a stem whose rise cannot climb. A negative decay had never come up, and a test of that branch had never been possible, because every stem read before had a decay of one sign.

The fit was fine #

That places the fault very precisely, because the part that looks as though it should have been hard has no fault in it. The counter’s fitted decay, a straight line through the log spacings of the whorls in its band, goes smoothly through nought at the turn and comes out as a negative rate, and it is nearly right at every point. Read at organ 1,502 it gives minus 0.00288 per organ, and the true decay forty organs back, at the middle of the band it was fitted over, is minus 0.00289. Everything the counter needed was there.

So the repair is one word. Size the band from the magnitude of the decay, so a stem whose rise is climbing at a given rate gets the width that a stem falling at that rate would get. A third of a step of rise is a third of a step, in whichever direction it is crossed.

The climb, read #

On the falling side that repair changes nothing, since a positive number has the magnitude it had: the readings are identical, band for band. On the climbing side it transforms the result. The right share goes from 12, 12 and 28 per cent to 94 per cent on all three stems. Its width comes back from 320 organs down towards 50, tracking the climb, and every band it chooses from then on spans about a third of a step. A pair on no transition drops from a quarter of the climb to nothing on two of the stems and 6 per cent on the slowest, most of whose few such bands fall in the last stretch of a long climb, where the rise is moving fastest.

The bijugate stem agrees on all of this. Its falling side reads 90 per cent right, and its climb reads 14 per cent as built and 96 per cent repaired, at twice the organs a whorl, so nothing about the failure or its repair is specific to one organ at a time. In no band on any stem does the repaired counter return the pair below, which was the first guarantee the self-sized band gave and which is exactly as true of a climb as it was of a fall.

There is a real lesson in how small the fault was and how long it could have stayed hidden. All the workings of these counters was tested at length. Each band was scored against the transition it sat in, each stem against a static calculation, on geometric rates from 150 to 900 organs a fold and on stems whose decay changed sixfold. Every reading held. They all had a rise of one sign. A counter’s weakest point is usually a case that its inputs have never had a chance to contain, and it does not show until somebody asks a question for which the answer has to come from that branch.

What happens at the turn itself #

That leaves the question the earlier essay actually asked. Near the minimum, does the widest band become the limit? Bands within a fifth of the turn are right only 47 per cent of the time on the first stem and 67 on the second, and no repair of sign could change that, since they are mostly on the falling side. But nearly all the remainder are refusals. Not one of them returns the wrong pair, and on the first stem only 4 per cent return a pair on no transition. The fixed widths give the answer: at 49, 97 and 193 organs all three of them read 100 per cent of those same bands correctly on the first stem. So the bands were readable, and the rise there does not make them hard.

What the counter refuses them with is a reason it reports: “unsettled”. The search for a width starts at 80 organs, fits a decay and asks for the band that decay requires. Near the turn a decay read from a narrow band is steep, so the next width is much wider, and that wider band takes in slower spacings, so it asks for wider again. On a stem heading towards a stop, every widening reaches back into faster-falling organs, which then shortens the next request. Traced at organ 942, the sequence is 80, 320, 206, 270, 230, 253, 239, 247, 243, 245, 244, 244. It swings around its limit and takes eleven revisions to settle, and the counter allows itself six. At organ 900 it takes nine revisions and at organ 830, seven.

So near the turn the counter is refusing bands it would have read. That is an honest failure, it is the right direction to fail in, and it costs a stretch of about 200 organs on the fastest stems. The slowest stem has no trouble: over its gentler approach the search settles within six on every band near the turn, and it reads 100 per cent of them. More revisions, or a damped update, would very likely deal with the rest. It has not been done here, because a repair of that kind wants testing against all the banks of falling stems, whose refusal counts it would change.

Where the band reaches its widest, at about the turn itself, it reads right nearly throughout, two bands at its very start being the exception. The third of a step it wants is more organs than 320, the cap applies, and a cap applied to a rise that hardly changes costs nothing. The worry that the widest band would limit what the counter could see there turns out to be misplaced. On a hundredfold fall the pair at the turn is 21/34, whose coverage floor is 76 organs, far below the cap. A much finer pair, on a far larger stem, could in principle outrun it, and that has not been grown.

What three fixed widths would have done #

The fixed widths make a useful control across the whole of each stem, since they were never given any way to go wrong in the direction of the decay. At 49 organs on the thirtyfold stem, every band reads right. Across that stem the pair never rises above 13/21, whose floor is 47 organs, so 49 is enough throughout, and a width that was only ever a little too narrow for the deepest part never met it. On the hundredfold stem the same width fails on the whole stretch near the turn, returning the pair below on all of it, since 21/34 needs more than 49 organs. On that stem’s climb it is right on only 76 per cent.

A width of 193 reads the turn and little else: on the thirtyfold stem it is right on 5 per cent of the falling side and 25 per cent of the climb, and it returns pairs on no transition on nearly all the rest. A person who has chosen one width because it reads the quiet middle of a stem well would be reading its ends very badly. That is the fixed-width problem again, the reason for a self-sized band in the first place. It also places the failure of the counter as built: on the climbing side, it did about as badly as the widest fixed width, for the excellent reason that it had become one.

The stem comes back down #

All of that is about the instrument, and the shoot has its own question. As the rise falls a stem climbs a sequence of transitions; when the rise climbs back, does it simply retrace them, or does it hold on to the fine pair it had reached? A rule with a memory might do either. Hysteresis is a familiar thing in systems with a settled state, and the early organs of a stem do shape what follows.

Measured by the pair the rule’s own divergence gives at a band’s own rise, as the settling of a stem’s angle would lead one to do, the stem retraces them. The change between 5/8 and 8/13 happens at a rise of 0.0067 going down and 0.0068 coming back. The change between 8/13 and 13/21 happens at 0.0026 and 0.0027, and between 13/21 and 21/34 at 0.0010 both ways. On the slower stem the changes are at 0.0177 and 0.0181, 0.0068 and 0.0070, and 0.0026 and 0.0027. Every pair of those is equal to about the spacing of the bands they were read from.

The divergence is less tidy, and it is the one place a memory shows. Near the turn it barely moves, holding 137.4° on both sides of it, but it does not repeat itself at high rises: at a rise near 0.007 the stem’s mean divergence is 137.89° going down and 136.79° coming back, more than a degree apart, and by the end of the climb it has left the range altogether. The band of angles a pair allows at that rise is wide enough to hold both, which is what a count is worth, so the pair retraces itself while the angle underneath it does not. A stem whose rise climbs does not keep the count it reached at the turn. It gives it back at the rise at which it was gained. That sets a limit on one kind of reading of a flowering shoot: a count taken late in its development says what the rise is now, and not what it has been.

What this does not establish #

The rate here is modest. A decay of a hundred and fiftieth per organ takes more than a hundred organs to cross one step of rise, and a stem may retrace its transitions at this rate for a reason that fails at a faster one, where a turn takes fewer organs than a step. The profile is symmetric and smooth, a parabola, and real apical growth has no reason to be either. The bank does not include any stem climbing from a rise so low that the turn’s pair needs a band near the cap. And the counter’s near-turn refusals have been traced to its search and not cured: whether a damped search keeps every other property is the kind of change that has to be run against the banks of falling stems before anyone relies on it.

Measurements that would overturn it #

A repaired counter returning the pair below on a climb, on a stem of any rate, would mean the guarantee does not carry to a climb. A band refused as unsettled which, given more revisions, settled on a wrong pair would mean a refusal near the turn was doing useful work and not merely costing a reading. A stem’s pair changing back at a rise visibly different from the one it changed at on the way down, at this rate, would be the memory these stems did not show.

Still open: a turn quicker than a step #

The stems here spend a hundred organs or more on every pair, so the rule has time to settle on each. The measurement that would find a limit is the same parabola made steeper, the rate raised from 1/150 to a thirtieth and beyond, until the climb takes fewer organs than the deepest pair’s own count. It asks at what rate the rise of each change of pair begins to differ between the fall and the climb, whether the first such difference appears at the fine end or the coarse, and whether a stem hurried through its turn retraces the transitions at all or finds itself on another branch at the far side. If it does the second, a rise that turned quickly would be a route to a different family of pairs, and the only one here in which nothing was cut.