The window was not carrying it

A seed measured in whorls grew stems from Lucas seeds of fifteen to eighty whorls, at one, two and three organs a whorl, and found the rate at which each gives the seed up proportional to the seed’s length — nearly. Divided by the seed, the edge read 2.05 at fifteen whorls, 2.09 at twenty, 2.12 at thirty, 2.16 at forty, 2.19 at sixty and 2.20 at eighty. A number that nearly scales and creeps is usually a transient on top of a scaling, and that essay named the transient it thought likeliest.

Every stem was read with a counting window of twenty-six folded nodes. On a fifteen-whorl seed that window is most of the seed, and on an eighty-whorl seed a small part of it. So the creep might be the instrument: a narrower window should lift the short seeds’ ratios towards 2.2 and leave the long ones alone. The measurement is the same eighteen stems read through several windows, and it has a problem before it starts.

Thirteen nodes is not a window #

The question named windows of thirteen, twenty-six and fifty-two nodes. The counter that reads these stems is the one that needs no order of arrival: it follows each family round and keeps a family only when it links most of the window. Thirteen organs are too few for that at any position along a stem.

On an ordinary stem seeded with forty whorls, windows of thirteen, sixteen and eighteen nodes return a count at no position at all. Twenty returns 72 positions, twenty-six 71, thirty-nine 67 and fifty-two 64; the wider windows lose a few only because they need more organs below their first reading. So the narrowest window that exists is twenty, and the measurement uses 20, 26, 39 and 52 — which still halves and doubles the window every earlier reading used.

Four windows, one edge #

For each of the eighteen stems the edge is found as before: a scan of rates across a bracket set by the seed, then a bisection on whole organs of rate between the last rate that keeps the seed and the first that loses it. The only thing changed is the window the final count is read through, and since a window changes how a grown stem is read and not how it grows, every stem is grown once and read four times.

Seventeen of the eighteen stems lose the seed at exactly the same organ of rate through all four windows. The narrowest window and the widest, a factor of 2.6 apart, agree to the organ on every ordinary and bijugate stem and on five of the six trijugate ones. The window is not what makes the ratio change with the seed.

The one that differs #

The exception is a trijugate stem seeded with thirty whorls. Through windows of 20, 26 and 39 nodes its edge is at 201 organs of rate; through 52 it is at 207, two whorls of rate later. The widest window reaches furthest back down the stem, and on this one stem it goes on reporting the seed’s pair for two whorls of rate after the narrower windows have stopped; why this stem and no other is not established here.

That is a real effect of the window and it is small: one edge in seventy-two readings, moved by about one per cent. It does not run in the direction the creep would need, since it moves a middle-length seed rather than lifting the short ones.

So the earlier explanation is withdrawn #

The essay that found the creep said its likeliest cause was the window — that a short seed’s first readings are taken while the seed is still most of what there is — and predicted that a narrower window would lift the short seeds. The prediction was specific enough to fail, and it has. A window of twenty nodes and a window of fifty-two give the fifteen-whorl stems the same edge as twenty-six did, to the organ, at every jugacy.

That leaves two places the creep could come from: the way the edge was located, or the stem.

What the grain was hiding #

The edge was located on a grain, and a grid is a setting like any other: the grain is not small. The earlier search stopped once its bracket was two whorls of rate wide — two organs on an ordinary stem, four on a bijugate one and six on a trijugate one. The search here stops at two organs at every jugacy. On an ordinary stem one organ of rate moves the edge by 0.962 whorls a rung, and divided by a fifteen-whorl seed that is six hundredths of the ratio, which is nearly half of the whole creep.

So the edge is found again on a real-valued rate, bisected to a small fraction of an organ inside the two-organ bracket. The ratio never falls in the process and rises by up to 0.058, on the ordinary thirty-whorl stem, from 2.117 to 2.175. The bijugate fifteen- and twenty-whorl stems rise by 0.044 and 0.048. The long seeds hardly move: an organ is a small share of eighty whorls.

The same stems on a fine rate #

On a fine rate the ordinary stems read 2.076, 2.133, 2.175, 2.166, 2.186 and 2.184 at seeds of fifteen, twenty, thirty, forty, sixty and eighty whorls. The bijugate stems read 2.065, 2.141, 2.175, 2.197, 2.215 and 2.216. The whole-organ readings had put the ordinary thirty-whorl stem level with the twenty, at 2.117 both, which made the creep look steady; read finely the thirty-whorl stem is already most of the way to where the long seeds are.

Not a creep, a rise that stops #

What is left once the grain is taken out is a different shape from the one first reported. From fifteen whorls to thirty, the ordinary stems’ ratio rises by 0.100 and the bijugate stems’ by 0.110. From sixty to eighty it moves by 0.0016 and 0.0005. The rise is concentrated in the shortest seeds and has stopped by sixty whorls; above that the edge is proportional to the seed.

That is the shape a transient has when it belongs to the stem rather than to the reading. A seed of fifteen whorls is lost at a smaller multiple of itself than a seed of sixty, and a seed of sixty at the same multiple as a seed of eighty.

The straight line, read again #

The earlier essay drew a straight line through all eighteen whole-grain edges, $\text{edge} = 2.2373 \times \text{seed} - 3.079$, and found it straight to within 2.67 whorls a rung. The same line through the fine edges is much tighter. For the ordinary stems it is $2.206 \times \text{seed} - 1.48$, straight to within 0.57 whorls a rung, and for the bijugate stems $2.249 \times \text{seed} - 2.30$, straight to within 0.45.

That reframes the creep as a single number. A line with a small negative intercept divides into a ratio that rises as the seed grows: slope plus intercept over seed. For the ordinary stems that is 2.206 less 1.48 over the seed, which is a tenth short at fifteen whorls and a fiftieth short at eighty, and nearly all of the approach falls below thirty whorls because a reciprocal does most of its moving early. The measured rise and the measured levelling are the same fact, and the grain had spread that fact out by about five times its own size.

Why short seeds are lost early #

Nothing here derives the level or the rise, but the direction of the rise has a plain reading. The rate edge is where a stem’s history stops being able to hold the Lucas lattice against a rise that keeps falling. A short seed is a short history, and a history shorter than a certain number of whorls has less to hold with per whorl as well as fewer whorls — its first readings still include the rows the seed was built with rather than rows the growth has made consistent with them.

The rate decides the branch is the essay about what a rate does to a history, and a bijugate stem grown by the same rule found a lag of a hundredth of a transition behind the static prediction at every jugacy: both are statements that growth carries its past forward over a definite distance, and a seed shorter than that distance is lost sooner.

The levels are not the same #

The second thing a fine rate shows is that the jugacies do not settle on one number. At an eighty-whorl seed, where every jugacy’s edge is a clean threshold, the ordinary stem’s ratio is 2.184, the bijugate stem’s 2.216 and the trijugate stem’s 2.215. The bijugate and trijugate agree to a thousandth, and the ordinary stem sits 0.03 lower, about one and a half per cent. The straight lines say the same thing in their slopes, 2.206 for the ordinary stems against 2.249 for the bijugate ones.

It is a surprising place for a difference to appear. A bijugate stem is an ordinary stem folded twice round, which is why its transitions are the ordinary ones at half the rise and why its seed was expected to be lost at the same rate in whorls, and to within two per cent it is.

That difference was invisible on the earlier search’s grain of two whorls of rate, on which all three fifteen-whorl stems read 30.80 whorls a rung and all three thirty-whorl stems 63.52 — identically, because at that grain three nearby edges round to the same step.

What that does to the earlier claim #

The earlier essay reported stems of every jugacy losing the seed at the same rate in whorls for the same seed length in whorls, “identically across the three” at fifteen and thirty whorls. On the grain that search used that is exactly what the numbers were, and the conclusion it supported — that the seed is measured in whorls rather than organs — stands untouched: at the same number of organs the jugacies still differ by a factor of more than three.

What does not stand is the implied equality at long seeds. Read finely, an ordinary stem keeps its seed to a rate about one and a half per cent faster than a bijugate or trijugate stem with the same number of whorls, and that is a small, consistent difference between an ordinary stem and a stem that grows more than one organ at a time.

A trijugate edge is not always a line #

The fine bisection assumes that a seed kept at one rate is kept at every faster rate, and it checks that at five points across its bracket. At three organs a whorl with a forty-whorl seed that assumption broke: kept at 270.75 organs of rate, lost at 272.5, kept again at 274.25. So every stem was scanned every half organ of rate, from two whorls below its whole-organ edge to two above.

Every ordinary and bijugate stem changes once: kept below a rate, lost above it. At three organs a whorl the stems seeded with fifteen, twenty, forty and sixty whorls change three times. Each has a stretch where the seed is lost and then kept again at a slightly faster rate, from one organ of rate wide at sixty whorls to four at twenty — never more than about a whorl and a third.

What a flickering edge is #

A trijugate stem places three organs in each whorl, and whether its final count reads the Lucas family depends on how the last few whorls fall against the window. Within a whorl’s worth of rate, the stem that ends a little further through its rise and the one that ends a little short can differ in which three organs the final window catches, and at the edge that is enough to change the count.

So a trijugate edge is a band about a whorl wide rather than a line, and a fine rate cannot locate it more precisely than the band. The trijugate ratios at fifteen, twenty, forty and sixty whorls are therefore left on the two-organ grain, which the band is about as wide as, and only the clean trijugate stems at thirty and eighty whorls are read finely: 2.161 and 2.215, rising between them as the others do.

What is now known about this edge #

It is proportional to the seed in whorls above about sixty whorls, at a ratio of about 2.18 on an ordinary stem and 2.22 on a bijugate or trijugate one. Below sixty whorls the ratio is lower, rising by about a tenth from fifteen whorls to thirty. The counting window does not move it; the grain of the rate it is searched on does, by up to six hundredths at short seeds on an ordinary stem; and at three organs a whorl it is a band about a whorl wide.

The window nobody varied moved a different window in a different measurement and found it changed nothing either. Two settings moved, two nulls, and one of the two inherited numbers — the grain — turned out to be what mattered.

What this does not say #

It does not derive either level, or the rise below thirty whorls, or why an ordinary stem settles lower. It does not say why a trijugate edge flickers and a bijugate one does not, beyond the reading that the final window catches different organs. The design is unchanged from the essays before it — the folded rise falls from 0.12 to 0.004, the window strides four nodes — and nothing here varies those.

A model seed is an exact lattice of stated length, and a plant’s is neither. A Lucas seed counts whorls is the reason the question is asked at all, and it is a question about a model’s history, not about any plant’s.

The claim, reduced #

The counting window carries almost none of the creep in a Lucas seed’s edge-to-seed ratio: seventeen of eighteen stems lose the seed at the same organ of rate through windows from 20 to 52 nodes, and a window of thirteen cannot be read. Part of the creep was the whole-organ grain the edge was found on. The rest is a rise of about a tenth between fifteen and thirty whorls that has stopped by sixty, at a level about 1.5 per cent lower on an ordinary stem than on a bijugate or trijugate one, whose edges at three organs a whorl can be bands a whorl wide.

What would withdraw it #

Two windows from 20 to 52 nodes giving different edges on more than one stem. A fine edge outside its whole-organ bracket. An ordinary or bijugate stem whose kept rates flicker across its edge. A fine ratio that still moves by a hundredth between sixty and eighty whorls on an ordinary or bijugate stem. Each is checked whenever the stems are read.

Still open: why an ordinary stem settles lower #

The one new difference is between a stem that places one organ at a time and stems that place two or three: an ordinary stem keeps a long Lucas seed to a rate about one and a half per cent faster, in whorls a rung, than a bijugate or trijugate stem with the same number of whorls. The bijugate and trijugate levels agree to a thousandth, so the difference is between one and more than one rather than a trend in the jugacy.

The measurement that would locate it is the same long seeds at four and five organs a whorl, to see whether the level stays with the bijugate and trijugate stems, and the ordinary stem grown with its organs placed in pairs that are forced to the same height — a stem that is ordinary in its angle and jugate in its timing — to see which of the two properties the level follows.