Cutting the far side in half keeps its step

A whorl feels the far side of its stem found the whole difference between an ordinary stem and a whorled one in a single term of a sum. Folded to one node a whorl, a bijugate stem is an ordinary stem whose repulsion counts every neighbour twice: once where it is, and once a whole folded turn away, because on the unfolded stem those are two real organs half a turn apart. Grown that way from a long Lucas seed, the stem keeps the seed to an edge-to-seed ratio of 2.2155 where the ordinary stem keeps it to 2.1848, and the second copy — the far image — is all of the one and a half per cent between them.

Every repulsion in that measurement was an inverse cube with no range of its own. An organ felt each neighbour in its window at whatever strength the power law gave it, and a copy a turn away was simply a copy three times as far off. That essay ended on the obvious objection: nothing in a plant is known to reach a whole turn round a stem. An inhibitor with a range of a few spacings would feel the far side only where few organs fit round the stem, and perhaps not at all. The prediction was that the step would shrink with the range and vanish where the far copy falls outside the cut-off at the rise where the seed is lost.

It does vanish, and roughly where predicted. It does not shrink on the way there.

The cut-off, and why the far copy is still a copy #

The repulsion is the same inverse cube, with every term multiplied by a gaussian of the distance: $e^{-(d/\lambda s)^{2}}$, where $s$ is the local spacing, the square root of the rise, and $\lambda$ is the range counted in spacings. This is the gaussian of the three cut-off shapes a neighbourhood is a hypothesis set out, in the unit it set out, and it is smooth, which matters: a hard edge at the same distance changes which neighbours exist as the candidate moves, and makes a different rule rather than a shorter one.

The identity that made the far image an instrument survives the cut. Folding a k-jugate stem multiplies every distance by k and the rise by k squared, so the spacing by k as well: a distance counted in spacings is the same number on the folded and the unfolded stem, and so is a range. A bijugate stem grown unfolded under a cut-off of two spacings, then folded, coincides with the two-image ordinary stem under the same cut at all 443 nodes of a sixty-whorl seed’s stem; the one-image stem, at the same cut, differs at 383 of them. So under any range, the difference between an ordinary and a bijugate stem is still exactly the far image, and turning the range is a clean question about it.

Nothing else moves. The window an organ is placed against is left at six spacings, so with no cut-off the stems are the published ones to the digit, and the seed, the rate, the counting window and the grid of 512 azimuths are the design the seed’s edge was measured on.

Where the far copy sits when the seed is decided #

The far side matters where the seed is decided, which the far-side essay located at the first transition the stem has to make on its own, just after the seed ends, at a folded rise near 0.077. The spacing there is about 0.28 of a folded turn. At the first organ grown above an eighty-whorl seed, on a two-image stem grown at its own edge, the organ’s nearest neighbour is 0.97 spacings away by its near copy and 3.23 spacings away by its far copy, round the other side of the stem.

That is the number the cut-off has to be compared with. A range of six spacings keeps three quarters of the far copy’s weight; four keeps half; three keeps 0.31 of it; two keeps 0.076; one and a half keeps a hundredth. The near copy, a spacing away, is touched far less — 0.90 of its weight at three spacings and 0.79 at two — so the cut thins the far side much faster than it thins the near.

A cut-off a spacing wide, drawn #

Summed over the whole neighbourhood rather than one neighbour, the far copies’ share of the repulsion the organ feels is 13.3 per cent uncut. It falls smoothly and fast as the range shortens: 11.0 per cent at six spacings, 8.8 at four, 6.7 at three, 5.2 at two and a half, 3.4 at two, 2.4 at one and three quarters, 1.5 at one and a half, and 0.2 at one. Halving the range from six spacings to three halves the far side’s share.

If the far image’s step in the edge followed the far side’s share, it would fall the same way: to five sixths of its uncut size at six spacings, half at three, a quarter at two. That is the reading the question expected, and it is the reading that fails.

The levels as the range shortens #

Each edge is found as the uncut ones were: a scan of twelve rates across a bracket set by the seed, a bisection on whole organs of rate, a half-organ scan from two organs below the edge to two above to check that the stem changes from kept to lost exactly once, and a bisection on a real-valued rate inside that. All sixty stems — ten ranges, three image counts, two seeds — change exactly once, and a level is the mean of the sixty- and eighty-whorl ratios, as the reading that found the levels defined it.

Uncut, the three levels are the published ones: 2.1848 for one image, 2.2155 for two, 2.2237 for three. Every level then rises as the range shortens — gently to three spacings, where the one-image level is 2.2025, and steeply below two, where it is 2.2474, reaching 2.3332 at one and a half and 2.5507 at one. A rule whose repulsion reaches only a spacing or two has to fall more slowly to keep a Lucas seed at all, and by a lot: at one spacing the stem’s decline has to be slowed to more than two and a half whorls a transition for every whorl of seed.

That rise is its own result and it is not the question. The question is the gap between the lines.

The far image’s step does not shrink #

The far image’s step — the two-image level minus the one-image level — is 0.0306 uncut. At six spacings it is 0.0308. At four it is 0.0332, at three 0.0364. Over the range in which the cut-off removes half the far side’s share of the repulsion, the step in the edge grows by a fifth.

It turns only below three spacings: 0.0301 at two and a half, 0.0243 at two, 0.0115 at one and three quarters. At one and a half spacings it is gone, and slightly reversed — the two-image stem’s level sits 0.0057 below the one-image stem’s — and at one and a quarter and one it stays within four thousandths of nothing either way. At those ranges a whorled stem and an ordinary one keep a long Lucas seed alike.

So the prediction’s second half holds and its first half does not. The step does vanish where the far copy falls outside the cut — at one and a half spacings the far copy of the nearest neighbour keeps a hundredth of its weight — but it does not shrink on the way. It holds, and then it collapses inside a factor of two in range.

Each seed on its own #

A level is a mean of two seeds, and a pattern made by averaging would not be worth much, so the two are worth reading apart. They agree. At three spacings the far image lifts the sixty-whorl seed’s ratio from 2.2020 to 2.2388 and the eighty-whorl seed’s from 2.2031 to 2.2391 — steps of 0.0368 and 0.0360, against 0.0296 and 0.0317 uncut. At two spacings the steps are 0.0277 and 0.0209. At one and a half both reverse: the sixty-whorl seed by 0.0023 and the eighty-whorl seed by 0.0092, the two-image stem keeping both seeds at 2.3274 or 2.3275 to the fourth figure.

So the growth and the collapse are each seen in both seeds, and the reversal at the short ranges is larger than the scatter between them. That last point matters because the reversal is small; it is a sign, and the sign is the same twice.

The third image goes first #

The third image is the copy one and a half turns round, and uncut it added 0.0082 to the level, a quarter of the second image’s step. It behaves as the question expected the second to: 0.0071 at six spacings, 0.0055 at four, 0.0035 at three, 0.0021 at two and a half, and under a thousandth from two spacings down. The farther copy is cut early and its effect falls steadily with the cut.

That difference between the images is worth stating carefully. A copy that is cut steadily loses its effect steadily; the far image, cut just as steadily, keeps its effect until it has almost nothing left. The two cannot both be proportional to the weight a copy keeps, and the third image is the one that is.

What the cut keeps, side by side #

Drawn as fractions of their uncut values, the far side and its step part company at once. At four spacings the cut keeps 66 per cent of the far side and 108 per cent of the step; at three, 50 per cent and 119; at two and a half, 39 and 98; at two, 25 and 79. Only at one and three quarters does the step, at 38 per cent, come down to near the far side’s 18, and below that the step is gone while the far side still carries a few per cent.

Set that against turning the far image’s weight up from nothing, on the uncut stem. There a quarter of the far image’s weight changed nothing, and the middle half of the dial carried nearly the whole step. A far image weighted a quarter carries about 3.7 per cent of the repulsion at the seed — about what the cut-off leaves at two spacings — and on the uncut stem that weight made no step at all, where the cut-off at two spacings keeps four fifths of it.

So the same share of the repulsion, held by the far side, does far more when the cut-off has also thinned the near side around it. A cut-off is not a far-image dial; it reweights the whole neighbourhood by distance, and the near copies a spacing or two away are cut too, a little. What decides the edge is evidently the far side’s weight against the near side’s shape, not its share of the total, and the share was the wrong number to predict from.

The step, as a rise #

The edge can be read as a rise: the ratio is $2\ln\varphi / \ln(0.12/h)$, where $h$ is the folded rise down to which a stem at its edge still holds the seed, so every level names a rise and every step names a difference of rises. Uncut, the one-image stem must hold its seed to 0.07725 and the two-image stem only to 0.07772: the far side buys 0.00047 of a rise, about five ten-thousandths, and the logarithm turns that into the one and a half per cent.

Read the same way under the cut-off, the far side buys 0.00047 at six spacings, 0.00051 at four, 0.00055 at three, 0.00045 at two and a half, 0.00036 at two, 0.00017 at one and three quarters, and nothing — a hair below nothing — from one and a half spacings down. The logarithm is not what makes the step grow at three spacings: the conversion barely changes over rises this close together, and the rise the far side buys grows by about a sixth, close to the fifth by which the step grows. Through a halving of its share of the repulsion, the far side moves the seed’s last held rise as far as it did uncut, and a little further.

Where the seed is lost #

The edge is a rise: a seed is kept exactly when it still holds the stem at a folded rise of about 0.077, and the ratio follows from that rise through a logarithm. Read that way, a one-image stem grown at its edge must hold an eighty-whorl seed down to 0.07757 uncut, 0.07786 at three spacings, 0.07858 at two and 0.08262 at one. The cut-off makes the seed’s last whorl give out sooner, which is the rising level said another way.

What happens after the seed is the part that changes with the range. An uncut ordinary stem grown just past its edge keeps reading the seed’s pair, 1 and 3, until a rise of 0.07226, and then reads 2 and 4 — counts with a common factor of two, on a stem that places one organ at a time, a pair the Lucas sequence of transitions never visits. Cut at six or four spacings it does the same. From three spacings down it leaves the seed by 2 and 3 instead, the Fibonacci pair, and earlier: at 0.07417 at three spacings, 0.07653 at two, 0.07780 at one and a half.

The two-image stem, uncut, already leaves by 2 and 3, at 0.07599; cut at two and at one and a half spacings it does too. So the uncut ordinary and whorled stems lose a long seed by two different routes, and a short enough range puts the ordinary stem on the whorled stem’s route. Whether that change of route is what closes the step is not settled here. The step is largest at three spacings, the first range at which both stems take the same route, and it closes only below two and a half, so the route alone does not fix its size.

What a rule with a range does to the whorled stems #

A grown bijugate stem passes the ordinary transitions at half the rises because a lattice wrapped twice round is the ordinary lattice. That much does not depend on any range. What the far image added was a small difference in how a grown stem holds a seed, a property of the placement rule rather than of the lattice — and under a cut-off it behaves as a property of the rule should, depending on the range.

At ranges of three spacings and more, a rule whose influence still reaches most of the way round the stem where the seed is decided, a whorled stem keeps a long Lucas seed to a ratio between 1.4 and 1.7 per cent above an ordinary stem’s. Below two spacings it does not, and a whorled stem and an ordinary stem with the same folded rate are indistinguishable by their edges. So the one and a half per cent is not a fact about whorls, and it is not a fact about an inverse cube with no range either; it is a fact about a rule that reaches at least two spacings.

That puts a number on what the far image needs, and it is about half a turn at the rise where the seed is decided, not a whole one. The nearest neighbour’s far copy is three and a quarter spacings off, and a cut that leaves it eight per cent of its weight still leaves four fifths of the step.

The model’s range, not a plant’s #

Nothing here says what a plant’s inhibitor reaches. The gaussian is one of three shapes, chosen because it falls fastest; two shapes read in the same unit have disagreed by half about where a lattice ends, and an exponential cut-off, which keeps far more of a copy three spacings away, could well move every range above. The spacing is the model’s, the square root of the rise, and on a real stem the spacing at the place a seed is decided is not known to a factor of two.

Nor does it explain the shape of the step. Why it grows by a fifth as the cut-off halves the far side, and why it then collapses inside a factor of two in range, is measured and not derived. The levels are means over two seeds each, and the grain of a single seed’s edge — up to a hundredth on neighbouring weights in the far-side essay — is larger than some of the differences read at the shortest ranges, where only the sign of the step is claimed.

Readings that would undo it #

A two-image stem under a cut-off that differs at any node from a bijugate stem grown under the same cut and folded. A step that falls in proportion to the far side’s share, to half its uncut size or less at three spacings. A step that survives at one and a half spacings, where the far copy keeps a hundredth of its weight. An ordinary stem that leaves its seed by 2 and 4 under a cut-off of three spacings or less. Each is checked against the measured stems whenever they are read.

A share of the repulsion is the wrong ruler #

The far side of a whorled stem’s neighbourhood does not need a repulsion with no range. A gaussian cut-off three spacings wide halves the far copies’ share of what an organ feels where the seed is decided, and the one and a half per cent the far image makes in a Lucas seed’s edge grows by a fifth. The step goes only when the cut-off is narrower than about two spacings and the far copy keeps a few per cent of its weight or less, and on the way the ordinary stem stops losing its seed by a whorled pair and starts losing it by the Fibonacci pair the whorled stem used. What the far side’s effect follows is not its share of the total.

Still open: whether the shape of the cut sets the range #

Every range here is a gaussian’s, the fastest-falling of the three cut-off shapes, which keeps less than a third of a copy three spacings off once its range is down to three. An exponential falloff — the profile of a substance that diffuses and decays, and the one with a mechanism behind it — keeps a tenth of the same copy at a range of 1.4 spacings and still a third at three. The next measurement grows the same one-, two- and three-image stems under an exponential cut-off and asks whether the step collapses at the range where the far copy’s weight falls to a few hundredths, whatever the shape, or at a range of its own; and whether the exit from the seed changes from 2 and 4 to 2 and 3 at the same weight of the far copy, which would make the route, rather than the share, the thing to watch.