A hard edge has no step to read

A whorl feels the far side of its stem found that a bijugate stem grown by the free placement rule is, node for node, an ordinary stem whose repulsion counts every neighbour twice round — once where it is and once a whole folded turn away — and that this second count, the far image, is why a whorled stem keeps a long Lucas seed slightly longer than an ordinary one. Two essays then cut the repulsion off at a range. Cutting the far side in half keeps its step used a gaussian, and the far copy’s weight sets the step an exponential, and between them they found a ruler: the step lasts while the cut-off leaves the nearest neighbour’s far copy, about 3.2 spacings round, more than about three hundredths of its weight, and the exit from the seed changes from 2/4 to 2/3 where it leaves about a half.

That essay wrote the ruler as a rule for any profile and named the profile that would test it hardest. A hard cut-off counts every image within its range in full and every image beyond it not at all, so the far copy’s weight is one or nothing, and the rule predicts both changes at the same place: the step and the exit should go exactly where the far copy crosses the edge, at 3.2 spacings. A hard edge is not a falloff had found a reason to doubt it — a hard edge’s set of neighbours depends on where the candidate is, so that a cut at three spacings made no pattern at any width where a smooth cut made a clean one — but nobody had grown the stems on a seed.

The stems, under a hard edge #

The design is unchanged from the window was not carrying it, whose counting window and seed lengths every reading since has used: Lucas seeds of sixty and eighty whorls, a folded rise falling from 0.12 to 0.004, the placement rule’s inverse-cube repulsion read in a window six spacings wide, and the rate edge — the largest number of whorls a rung of rise at which the stem still reads the seed’s pair at its end — found by a scan of twelve rates, a bisection and a fine search. An ordinary stem on a Lucas seed keeps that ladder when its rise falls fast and gives it up when it falls slowly, and the edge is measured, as a seed measured in whorls settled, in whorls a rung against the seed’s length. Every image term is now counted in full if it lies within $\lambda$ local spacings and dropped if it does not, for $\lambda$ from six down to one. With no edge the grower reproduces the earlier stems node for node.

The first surprise is in the search itself. At six spacings and four and a half every stem changes from keeping the seed to losing it once across its edge, as every gaussian and exponential stem did. At four, one does not quite. From three and a half down, almost none does, and a rate edge — the one number every earlier reading was built on — stops existing.

The rate edge breaks up #

Scanned at twenty-four rates instead of twelve, the edge’s break-up is plain. At six spacings both stems keep the seed at every rate up to one and lose it at every rate above: one change along the scan. At four the stem counting once round still does; the stem counting twice round loses the seed at one rate on the keeping side of its edge and then keeps it again. At three and three quarters both stems have holes — three rates lost among the ones that keep — and the last kept rate has moved down. At three and a half the same, at fewer organs again.

At three spacings and below there is no edge in any useful sense. The stem counting once round changes between keeping and losing eight times across the twenty-four rates, the stem counting twice round ten. At two and a half spacings almost every rate loses the seed and four scattered ones keep it.

At two spacings the order inverts at one end. The stems whose rise falls fastest — over the fewest organs — lose the seed, which under every smooth cut-off was the safe side; the middle of the scan keeps it; and the slow end loses it, as before. A seed that is kept only in a window of rates is not a seed with an edge.

Where it breaks up, all at once #

Stacked, the nine ranges show the edge holding at six and four and a half spacings, beginning to break at four, broken at three and three quarters and three and a half, clean again — briefly — at three and a quarter, and gone from three down. The clean row at three and a quarter is worth noticing and not worth much: on the sixty-whorl seed both stems have a clean edge there, with a step of 0.017, and on the eighty-whorl seed the finer search finds the edge unclean after all. Between three and a half and three the stems pass through a narrow range where they behave, flanked by ranges where they do not.

Under the exponential every one of 44 stems had a clean edge at every range from six spacings down to half of one. The hard edge breaks the rate edge at four, a range at which the smooth shapes had not even begun to lose the step.

Counted at the seed, the edges read admit 19, 15, 12, 12, 11, 11, 8, 8 and 5 near images and 17, 12, 10, 8, 7, 6, 4, 1 and none of the second images, from six spacings down to two. The counts do not sort the clean ranges from the broken ones: the clean row at three and a quarter admits the same eleven near images as the broken row at three and a half and one second image fewer. What decides a stem is not how many images its first organ sees but which ones cross the edge as each later candidate moves round the stem, and that is a property of the whole growth, not of the seed.

The levels do not rise #

Under every smooth shape the level — a stem’s rate edge as a ratio to its seed, averaged over the two seeds — rose at every shortening of the range, because a rule that feels fewer neighbours has to change more slowly to keep a seed at all. The hard edge’s levels do not. The stem counting once round reads 2.1843 at six spacings, against 2.1848 uncut; then 2.1749 at four and a half, lower; then 2.2275 at four. The stem counting twice round reads 2.2133, 2.1766 and 2.2684. A cut that removes neighbours has made the stem keep its seed less well at four and a half spacings and better at four.

Seed by seed the readings are no steadier. The step on the sixty-whorl seed is 0.0282, −0.0039 and 0.0430 at six, four and a half and four spacings; on the eighty-whorl seed 0.0299, 0.0072 and 0.0389. The two seeds agree at six and at four to within a few thousandths and disagree in sign at four and a half, which is what a quantity looks like when it is near a jump: a few organs’ difference in where the seed ends decides which side of the jump the stem is on.

What the edge cuts through first #

The reason is the geometry the edge cuts through. Measured from the first organ above the seed, the seed’s nearer images stand at 1.01, 1.05, 1.21, 1.67 spacings and on, with a cluster at 3.04, 3.13 and 3.23 and the next at 3.70. Their second images — each neighbour’s copy a whole folded turn the other way round — begin much closer than the far copy the ruler was read against. A neighbour half a turn round has both its images half a turn away, 1.76 spacings at this rise, and the seed’s second images start at 2.03, then 2.56, 2.61, 2.89, 3.04 and 3.18. The nearest neighbour’s own far copy, the one the smooth cut-offs weighed, sits at 3.2 on the grown stems, behind at least five other second images.

So a hard edge brought in from six spacings meets, in order, the outer near images at 3.70, the cluster of near images and second images between 3.0 and 3.25, and the nearest second images at 2.5 to 2.9 — all before or at the far copy. Each of those is a neighbour that exists for some candidate positions and not for others, because the distance from a candidate to an image changes as the candidate moves round the stem. The essay on the hard edge’s pattern called that a neighbour set that depends on where the candidate is. Here it means that the repulsion a stem feels changes by a jump as the candidate crosses each image’s edge, and a jump in the repulsion is a jump in where the next organ goes.

Below half a turn, one stem #

At the short end the hard edge does have a clean answer, and it is exact rather than measured. No second image can lie nearer than half a turn round the stem, and half a turn is 1.76 local spacings at the seed’s end, more as the rise falls. So an edge shorter than that counts no second image anywhere on the stem, and the stem counting each neighbour twice round is the stem counting it once.

Grown, it is. Under a hard edge of one and a half spacings and of one and three quarters, the two stems are the same stem at every one of 511 nodes at one rate and of 681 at another — not similar, identical in every organ’s position. At two spacings they are identical at the faster rate and part at node 91 at the slower, where the rise has fallen to 0.076 and half a turn is 1.81 spacings: the edge has just reached the second images of neighbours half a turn round. So the step is exactly nothing below about 1.8 spacings, by construction, and the far copy at 3.2 spacings has nothing to do with it; between 1.8 and four spacings the step is whatever the stems’ broken edges make of it, which is not one number.

That is the cleanest possible failure of the ruler’s prediction. It placed the step’s end at 3.2 spacings, where the far copy crosses the edge. The step is certainly gone by 1.8, where the nearest second image crosses it, and it is not a quantity between the two.

The step, where there is one #

Where every stem does have an edge the step can be read, and it is not what the ruler says. At six spacings it is 0.029, close to the 0.031 of an uncut stem. At four and a half it is under two thousandths, and its two seeds disagree in sign: the sixty-whorl stem’s step is −0.0039, the eighty-whorl stem’s +0.0072. At four it is 0.041, larger than uncut — the gaussian too had a step a fifth larger than uncut at three spacings. At all three ranges the far copy, at 3.21 to 3.23 spacings, is inside the edge and counted in full.

A step that goes from 0.029 to nearly nothing to 0.041 while the far copy’s weight stays at one is not set by that weight. What changes between six, four and a half and four is how many second images are counted — seventeen of the window’s neighbours have theirs inside the edge at six spacings, twelve at four and a half, eight at four — and which near images are cut.

The exit changes early #

The exit is the pair a one-image stem grown one per cent past its edge first reads instead of the seed’s. Under the gaussian it is 2/4 down to four spacings and 2/3 from three, where the far copy’s weight falls from 0.52 to 0.31; under the exponential 2/4 at six and 2/3 from four, where the weight falls from 0.58 to 0.45. Under the hard edge it is 2/4 at six spacings and 2/3 at four and a half and at four, with the far copy counted in full throughout.

The rule predicted the exit’s change at 3.2 spacings. It has happened by four and a half, a spacing and a quarter farther out, with no change at all in the far copy. The two smooth shapes agreed on a half because, for each, the weight of the far copy and the weights of the near images move together as the range shortens; a hard edge moves them separately, and the exit follows the near images.

What the ruler was #

The weight ruler was a restatement for the two shapes that produced it, as its essay said: the three hundredths and the half were read off the same stems that confirmed them. A third profile was the test, and the hard edge has failed it twice over — on the exit, which changes a spacing and a quarter early, and on the step, which has no edge to be read from at the range the rule names. What survives is narrower. For smooth cut-offs, which shrink every image’s weight continuously and in order of distance, the far copy’s weight is a good clock for both changes. It is a clock because under those shapes it moves with everything else; under a shape that does not, it is just one image among many.

That is a statement about the instrument as much as about stems. How far a primordium reaches found nothing in a plant that fixes the repulsion’s exponent; nothing fixes its range or its profile either. A cut-off is a hypothesis about how a placement signal falls with distance, and a neighbourhood is a hypothesis made the same point about which neighbours count. The smooth shapes and the hard edge are different hypotheses, and the stems are not indifferent between them below four spacings.

Two seeds, two images, one window #

The stems are the earlier essays’ stems, on two seeds, counting one image and two; the third image is not grown. The window stays at six spacings at every range. The scan reads twenty-four rates, so a hole narrower than one rate step is invisible, and the finer edge search at three and a quarter spacings already shows that. Nothing here says what an inhibitor’s profile is on a plant; the hard edge is the limit no diffusing substance reaches.

What would rescue the ruler #

A hard-edged stem whose exit changes at 3.2 spacings and not before. A step that falls where the far copy leaves the edge, with the rate edge clean on both sides of it. A rate edge that stays clean down to three spacings. Each is checked on every stem above, and each fails.

Still open: an edge with a ramp #

The hard edge breaks the rate edge because every image crossing it changes the repulsion by a jump. A cut-off that falls from one to nothing over a short ramp — a few tenths of a spacing — is the bridge between the two kinds: smooth enough that no image appears or vanishes at once, sharp enough that the near images and the far copy are cut in a different order from the gaussian’s. The measurement is the same stems under a ramped edge at the ranges where the hard edge broke up, asking how wide the ramp has to be before every stem’s edge is clean again, and whether, once it is, the step and the exit go where the far copy’s weight says or where the near images’ do.