The far copy's weight sets the step, whatever the cut's shape

A whorl feels the far side of its stem found that a folded bijugate stem is an ordinary stem whose repulsion counts each neighbour twice, once where it is and once a folded turn away, and that this far image is the whole of the one and a half per cent by which it keeps a long Lucas seed to a higher edge-to-seed ratio than an ordinary stem: 2.2155 against 2.1848. Cutting the far side in half put a gaussian range on the repulsion and found the step outlasting the far side — larger at three spacings than uncut, still 0.024 at two, gone at one and a half — and the ordinary stem’s exit from the seed changing from the whorled pair 2 and 4 to the Fibonacci pair 2 and 3 between four spacings and three.

The gaussian is the fastest-falling of the three cut-off shapes a neighbourhood is a hypothesis set out. That essay ended on the shape with a mechanism behind it. A substance made at each organ that diffuses and decays falls off exponentially with distance, and an exponential keeps far more of a copy several spacings off than a gaussian of the same range does. The question was whether the step collapses where the far copy’s weight falls to a few hundredths, whatever the shape, or at a range of the shape’s own — and whether the exit changes at one weight of the far copy too.

The same stems, a different falloff #

The design is the one the window was not carrying it used: Lucas seeds of sixty and eighty whorls, a folded rise falling from 0.12 to 0.004, the rule’s inverse-cube repulsion read in a window six spacings wide, and the rate edge found by the same scan, bisection and fine search. Every term is now multiplied by $e^{-d/(\lambda s)}$ instead of $e^{-(d/\lambda s)^2}$, with $s$ the local spacing and $\lambda$ the range in spacings, from six down to a half. Under a gaussian of the same form the grower reproduces the earlier gaussian stems at every node; under the exponential it does not.

Stems counting one image and two are grown; the third image, which the gaussian had shown going first and under a thousandth from two spacings down, is not. All 44 stems change from keeping the seed to losing it exactly once across their edges.

At a range of one spacing the exponential keeps 0.38 of the nearest neighbour, a spacing away, and 0.042 of its far copy, a little over three spacings round. A gaussian of the same range keeps nearly the same share of the near copy and almost nothing of the far: the two curves cross close to one spacing and part company beyond it. Across all the ranges grown, the far copy’s weight under the exponential runs from 0.58 at six spacings down to 0.0019 at a half.

The levels rise faster #

Every level rises as the range shortens, as it did under the gaussian: a rule that feels fewer neighbours has to fall more slowly to keep a seed at all. Under the exponential it rises from the start. At six spacings the one-image stem’s level is already 2.2146, against 2.1901 for a gaussian of six and 2.1848 uncut, because an exponential six spacings wide still trims the near neighbours by a sixth where a gaussian of six leaves them whole. By one spacing it is 2.4273, and by half a spacing 2.6968.

So the exponential is not a gaussian with a longer tail. It is weaker close in and stronger far out, and the two stems the step is the difference between both feel it.

The step under each shape #

The exponential’s step starts larger than uncut, 0.0341 at six spacings against 0.0306, and falls more steadily than the gaussian’s: 0.0286 at four, 0.0274 at three, 0.0205 at two, 0.0091 at one and a half. Then it does something the gaussian did not. At one and a quarter spacings it falls to 0.0038; at one spacing it is back to 0.0134; at seven eighths it is 0.0004, and from three quarters down it is gone, swinging between −0.0118 and −0.0033 as the one-image and two-image edges cross.

The dip and recovery are not noise between seeds. The step at one and a quarter is 0.0037 on the sixty-whorl seed and 0.0039 on the eighty; at one spacing it is 0.0134 on both. So the step is not a monotone function of the range under the exponential, and the range at which it is last alive, one spacing, is not where a smooth fall would have put it.

By range alone, the shapes disagree: the gaussian’s step is last alive at one and three quarters spacings and the exponential’s at one. That is the first half of the answer. The step does have a range of the shape’s own.

On the far copy’s weight, they agree #

Put each range at the weight its cut-off leaves on the far copy of the nearest neighbour — the organ a turn round from the one that matters most, at the first organ grown above an eighty-whorl seed — and the two shapes come together. The gaussian’s step is last alive at a weight of 0.035 (one and three quarters spacings) and gone at 0.011 (one and a half). The exponential’s is last alive at 0.042 (one spacing) and gone at 0.027 (seven eighths). Both lose the step between about four and three hundredths of the far copy.

So the question as asked has a clean answer: the step collapses where the far copy’s weight falls to about three hundredths, whatever the shape. What the shapes disagree about is only how many spacings it takes to get there — 1.6 or so for the gaussian, about 0.95 for the exponential, because the exponential reaches three hundredths of a copy three spacings off only when its range is under one.

The share does not line up #

The earlier essay’s suspicion was that the far side’s share of the repulsion is the wrong ruler, and the second shape confirms it. Where the exponential’s step goes, at seven eighths of a spacing, the far copies still carry 2.8 per cent of everything the organ feels; where the gaussian’s step still stood, at one and three quarters, they carried 2.4. On this axis the two curves cross each other rather than agreeing.

The reason is that a cut-off of one shape does two things a cut-off of another shape does in different proportions. It trims the far copies, which is what the step is about, and it trims the near neighbours, which is what the share divides by. The exponential trims the near neighbours harder, so its far share stays high while the far copy’s own weight falls. The weight is a property of the copy; the share is a property of everything else too.

The weight sets where, not how much #

The agreement is about the collapse and nothing else. At equal weights of the far copy the two shapes’ steps are not equal: with the copy weighted at about a fifth — the exponential at two spacings, 0.201, the gaussian at two and a half, 0.191 — the exponential’s step is 0.0205 and the gaussian’s 0.0301; at about a third, 0.0274 against 0.0364; at about three fortieths, 0.0038 against 0.0243. Above the collapse, the exponential’s step is smaller than the gaussian’s at every matched weight.

That fits what the levels showed. At a matched far-copy weight the exponential has trimmed the near neighbours far more — a near-copy weight of 0.62 at two spacings against the gaussian’s 0.86 at two and a half — and both of the stems the step is a difference between are then growing on a weaker neighbourhood and at a higher rate. The far copy decides whether the whorled stem’s extra term matters at all; how much it matters depends on everything else the rule still feels.

The exit, on the same ruler #

The exit changes on the same axis. Under the exponential a one-image stem grown one per cent past its edge leaves an eighty-whorl seed by 2 and 4 at six spacings — the whorled pair the Lucas ladder never visits — and by 2 and 3 at four spacings and every range below. Under the gaussian it changes between four spacings and three. In ranges the two shapes disagree; in the far copy’s weight, the exponential changes between 0.58 and 0.45 and the gaussian between 0.52 and 0.31, and the two intervals overlap between 0.45 and 0.52.

So an ordinary stem stops leaving its seed by a whorled pair once its rule keeps less than about half of the far copy, and its step over the bijugate stem goes once the rule keeps less than about three hundredths. Both are statements about one weight on one copy, not about a range or a shape. The window’s edge, which the window was not the neighbourhood had already found was not what a placement rule answers to, stays put throughout; the weight on the copy inside it is what moves.

Where the copy sits #

The ruler works because the copy does not move. Read at the first organ above the seed on a two-image stem at its own edge, the nearest neighbour sits 0.97 spacings away at every range of either shape, and its far copy 3.13 to 3.23 spacings round — the far copy of a neighbour a spacing off along the generative spiral, a folded turn away. The cut-offs change how much of it the rule feels, and almost nothing about where it is.

The near copy, by contrast, is weighted very differently by the two shapes at the ranges where the step goes: the gaussian keeps 0.73 of it at one and three quarters spacings, the exponential 0.33 at seven eighths. The step does not answer to that. It answers to the copy a turn away, which is the only thing a whorled stem’s neighbourhood has that an ordinary one’s does not.

A rule for any profile #

That turns into a prediction for a cut-off of any shape $f$, before any stem is grown: the step should go at the range at which $f$ leaves three hundredths of a copy 3.2 spacings off, and the exit should change where it leaves a half. For the gaussian, $e^{-(3.2/\lambda)^2} = 0.03$ puts the step’s collapse at 1.71 spacings — measured between 1.75 and 1.5 — and $e^{-(3.2/\lambda)^2} = 0.5$ puts the exit’s change at 3.84 — measured between four and three. For the exponential, $e^{-3.2/\lambda} = 0.03$ gives 0.91 spacings, measured between one and seven eighths, and a half gives 4.62, measured between six and four.

All four fall inside the brackets the stems measured. For these two shapes that is a restatement rather than a test: the three hundredths and the half were read off the same stems, and a weight is a fixed function of the range for each shape, so the brackets were bound to agree. What makes it worth writing as a rule is that it costs nothing to apply to a third profile, and a third profile would test it. The hard edge has the sharpest prediction of all — the exit and the step both changing exactly where the far copy crosses the edge, at 3.2 spacings.

Each seed on its own #

A step is a difference of two edges, each found to a fine rate on one seed, and the two seeds do not always agree on its size: at three spacings the exponential’s step is 0.0230 on the sixty-whorl seed and 0.0317 on the eighty, at two spacings 0.0239 and 0.0170. The seed-to-seed spread is up to nine thousandths — about a third of the step itself at long ranges. So the sizes of the step above two spacings are levels, not single readings, and the steady fall they describe is a fall in the mean.

What the seeds do agree on is the shape of the collapse. Both carry the step at one spacing (0.0134 and 0.0134), both carry next to none at seven eighths (0.0003 and 0.0004), and both carry the dip at one and a quarter (0.0037 and 0.0039). The places where the step goes and comes back are the same on two seeds of different length; the sizes in between wander by a few thousandths.

What the dip says #

The dip at one and a quarter spacings is the one place the clean story leaves something over. With the far copy still weighted at 0.078 — more than twice the three hundredths that ends the step — the step falls to under four thousandths, then recovers to thirteen thousandths at a weight of 0.042 before going for good.

Something other than the far copy is changing there. An exponential at one and a quarter spacings keeps under half the weight of any neighbour more than a spacing off, and the transitions a stem makes alone depend on which neighbours it ranks first. A rule losing its second ring of neighbours can change which transition carries it off the seed without any change in the far side. The step is the difference of two edges, and at that range both edges move for a reason of their own. That is named here and not traced.

A mechanism’s range, stated in copies #

The reason to care about the shape is that a plant’s inhibitor, if it acts through a diffusing substance, would have an exponential profile with a range set by how far the substance travels before it decays. This measurement says how to state the far side’s role for such a rule without knowing that range in spacings: the far image matters while the substance reaching a neighbour’s position a turn round is more than a few hundredths of what reaches the neighbour itself, and the stem leaves its seed by a whorled pair while it is more than about half.

Both are ratios of the profile at two distances, a spacing and three, and both can be read off any profile, of either shape or another. That is the practical use of finding the clock in the copy rather than in the range.

A shape that breaks the ruler would undo it #

A shape under which the step survives with the far copy weighted below two hundredths, or is gone with it above five. An exit that changes at a weight outside four to six tenths. A step that falls monotonically with the range under the exponential, which would say the dip at one and a quarter was an artefact. Each is checked against every stem grown above.

Two shapes, two images, one window #

Anything about a plant’s inhibitor, its shape or its range. The hard cut-off, the third shape, is not grown, and neither is the third image. The window stays six spacings wide at every range, so the exponential’s long tail is truncated there exactly as the uncut stem’s was; at a range of six spacings a neighbour at the window’s edge is still weighted at a third, and the longest ranges are partly the window’s. The far copy’s weight is read at one organ, the first above an eighty-whorl seed.

Still open: the hard edge on the same ruler #

Two smooth shapes agree on the far copy’s weight, as the ordinary and whorled stems agree on everything but it; the hard cut-off is not smooth, and a hard edge is not a falloff found that it changes which neighbours exist as the candidate moves. Under a hard edge the far copy’s weight is one or nothing, so the ruler found here has only two readings. The measurement is the same stems under a hard cut-off at ranges either side of the far copy’s distance, asking whether the step and the exit change exactly where the far copy crosses the edge — which the weight ruler predicts — or at some other range set by the near neighbours the edge removes first.

Measured, a hard edge has no step to read: under a hard cut-off the exit changes from 2/4 to 2/3 by four and a half spacings with the far copy counted in full, the step is 0.029, under 0.002 and 0.041 at six, four and a half and four spacings with the far copy inside at all three, and from four spacings down a stem’s keeping its seed stops depending on its rate in one direction. Below about 1.8 spacings — half a turn — the two stems are one stem, node for node. The rule held for the smooth shapes and fails for the hard edge.