The ratio was the floor of a curve

When this collection finished retracting the comb, one number was left. Both a placement rule and a lattice with transported errors produce two combs in the divergence sequence, and both return the counted pair; what separated them was how the correlation divides between the two combs. The rule gave 0.65 and a distance-weighted transport gave 1.30.

That was measured at a rise of 0.005, and the note left with it said so:

The ratio 0.65 belongs to a rise of 0.005, where the 13-hop is shorter than the 8-hop by eight per cent. At a coarser rise the asymmetry has a different size and possibly a different sign, so a survey comparing a plant against “the rule’s ratio” needs a curve rather than a number, and nobody has swept it.

This essay sweeps it. There is a curve, it is the same curve on both rungs measured, and the reason offered above for expecting one is wrong.

Holding the disturbance fixed, which is not obvious #

A sweep over the rise is a sweep over two things unless the disturbance is held at something that means the same at every rise.

Every earlier comparison used a three-entry lookup — a quarter of a degree at 0.005, three tenths at 0.013 — and a constant at every rise in between. A constant amplitude in degrees is a larger disturbance at a fine rise, because the organs are closer together: at 0.005 the neighbours are 25° apart and at 0.032 they are 65° apart, so the same quarter of a degree is two and a half times as large a fraction of the spacing.

So the disturbance here is a fixed share of the local spacing — a hundredth of √h, in degrees — which comes to 0.25° at 0.005 and reproduces the site’s own figure there to two decimal places.

The curve #

Fifteen rises across two rungs, five stems each, with the pair taken from the positions rather than from the angle readout — because near a transition the readout starts returning the next rung’s pair while the arrangement is still on this one, and a sweep that took its pair from the readout would compare two different quantities and call the difference a trend.

Both rungs give a U. The floor is 0.71 on the 5/8 rung and 0.79 on the 8/13 rung, and on both it sits about two thirds of the way up the rung. Away from the floor the ratio climbs, and at the fine end of the coarser rung it reaches 2.8 — four times its own floor.

The rise of 0.005 it was measured at sits at 0.66 of the way up the 8/13 rung. It measured the bottom of a curve nobody had swept, which is the third time this collection has found an extremum reported as a constant: the second moment against the divergence angle turned out to be a staircase whose 0.253 was a plateau, and the tolerance of a lattice to noise turned out to be a scale rather than a number.

The numbers, since the shape is the argument:

On the 5/8 rung, reading from its coarse end down towards the transition, the ratio runs 1.10, 1.02, 0.78, 0.72, 0.71, 1.26, 2.13, 2.80 at rises of 0.0175, 0.0165, 0.0155, 0.0145, 0.0125, 0.0105, 0.0090 and 0.0080.

On the 8/13 rung it runs 1.26, 1.06, 0.79, 0.82, 0.90, 0.98, 1.05 at rises of 0.0066, 0.0058, 0.0050, 0.0046, 0.0042, 0.0038 and 0.0034.

The two are not identical. The floor is a little lower on the coarser rung and the climb towards the fine end is far steeper there — a factor of four against a factor of 1.3. What they share is the shape and the place the floor sits, and whether the depth of the climb is a property of the rung or of how close to the boundary the sweep was able to get is not decided by two rungs.

The second of those is worth taking seriously rather than listing. The 5/8 sweep reaches a rise of 0.0080 against a rung boundary at about 0.0069, so its finest position is a tenth of a rung from the edge; the 8/13 sweep stops at 0.0034 against a boundary near 0.0030, which is a similar fraction. So the two sweeps get about equally close and the climbs still differ by a factor of three, which argues against the depth being an artefact of reach — without settling it, since two rungs at one distance is one comparison.

The floor is not the handover #

The floor sits about two thirds of the way up a rung on both of them, and the obvious candidate for what puts it there is the one quantity this collection has already located inside a rung.

It is not that. Every handover on this ladder sits in the coarse half of its rung, between six and forty per cent, and the floors here are at about sixty-six. On both rungs the two are separated by a quarter to a half of a rung’s width — not adjacent, not within the sweep’s resolution, and not on the same side of the middle.

So whatever the ratio’s floor is, it is not the rise at which the two contact steps change places. That is a real negative and it is cheap: the handover’s position is a column computable from the rise, so the comparison costs nothing and it excludes the candidate anybody would have reached for first.

What it leaves is a second position-in-rung feature with no account, sitting at a different place from the first. Two features at two thirds and at a quarter of a rung is more structure inside a rung than this collection had, and neither is explained.

What the curve does to the discriminator #

The practical consequence is not that a survey needs a curve instead of a number. It is that the curve crosses the quantity it is being compared against.

The transport’s ratio is about 1.30 and does not move with the rise, because a kinematic lattice’s angle sequence contains no rise. The rule’s ratio runs from 0.71 to 2.80. So the two are far apart in the middle of a rung, close near its coarse end, and the wrong way round at its fine end — at the two finest positions sampled on the 5/8 rung the rule gives 2.13 and 2.80, both above the transport.

Counted over the fifteen positions swept, the discriminator works at thirteen and inverts at two. That is a usable instrument with a stated exclusion: the fine quarter of a rung, where a plant’s rule-ratio climbs past the transport’s and a reading above 1.3 means the opposite of what it means elsewhere.

And the exclusion is checkable before the sequence is even taken, because a plant’s position in its rung follows from its rise and the ladder’s own boundaries. So the curve does not make the discriminator harder to use; it adds one arithmetic step before it, and it is the reason the specification’s second outcome had to go rather than merely be widened.

The main comb is a hump, and the ratio is a quotient of two moving things #

A ratio hides its numerator and denominator, and both of these move.

The main comb — the correlation at multiples of the smaller parastichy number — is weakest at both ends of a rung and strongest in the middle: 0.13, 0.14, 0.20, 0.24, 0.34, 0.28, 0.20, 0.15 across the 5/8 rung. That is the pattern being most itself in the middle of its rung and least itself near a transition, which is the same thing the mixture thread found when it discovered that a window straddling a boundary reads worse than a window inside one.

The second comb does the opposite at the fine end: 0.14, 0.15, 0.15, 0.17, 0.24, 0.35, 0.42, 0.41. It rises as the main comb falls, which is what the two-family account predicts and is stronger evidence for that account than the ratio alone.

So the U is not one quantity varying; it is two quantities crossing. That matters for anyone who wants to use the ratio on a plant, because a stem whose main comb is 0.15 is a stem whose readout is close to refusing altogether, and a ratio computed just above the refusal threshold is a ratio of two small numbers.

Why it climbs, which is arithmetic about the comb #

The second comb is not a fixed set of lags. On a stem whose pair is m and n the main comb collects the multiples of m and the second collects the residue class containing n — so at the 5/8 rung it collects lags 3, 8, 13, 18, 23 and 28.

Thirteen is in that list, and thirteen is the larger member of the pair the stem is about to transition to.

As the rise falls towards the transition, the next rung’s family is already forming in the arrangement. The second comb picks it up, the main comb does not, and the ratio climbs. Near a boundary the ratio is not measuring the transmission of errors at all: it is measuring the transition.

What the blind readout says over the same sweep #

The ratio here is computed at the pair the positions give, which is why the sweep can be taken at all. It is worth recording separately what the angle readout — the instrument that is shown a list of numbers and told nothing — makes of the same fifteen sets of stems, because the two disagree in a way that is itself informative.

On the 8/13 rung the readout returns 8/13 on all five stems at all seven rises, including the two closest to a transition. On the 5/8 rung it returns 5/8 on all five stems at exactly two of the eight rises — the two nearest the floor of the U — and refuses at the other six.

The refusals are not random. Four of them are at the coarse end, where the main comb is 0.13 to 0.24 and barely clears its threshold, and two are at the fine end, where the second comb has grown until the search’s ordering breaks down and the readout can no longer decide which spacing is the main one. So the instrument reports exactly where the ratio is near its floor and refuses where the ratio is inflated — which is a piece of good fortune rather than a design, and it is the strongest argument in this essay for using the readout’s refusal as a guard.

A plant whose stem gives a clean two-comb reading is, on this evidence, a plant whose rise is near the middle of its rung. That is a testable side-claim and it is the sort of thing this collection has been wrong about before, so it is offered as a pattern in fifteen measurements rather than as a rule.

The geometric explanation, refuted on its own ground #

The plan’s reason for expecting a curve was the hop asymmetry. At a rise of 0.005 the 13-hop is 0.069 circumferences and the 8-hop 0.075, so the larger index offset is the geometrically nearer neighbour — and a d⁻³ interaction would weight them by the cube of that ratio, which is 0.80. Change the rise, change the asymmetry, change the ratio.

The asymmetry does change: across the two rungs it runs from 0.77 to 1.23, and it passes through one, so the two hops change places. But it does not change where the ratio changes.

Inside the 5/8 rung there are four rises whose hop asymmetry agrees to within a twentieth — 0.792, 0.793, 0.806 and 0.806 — and whose ratios are 0.71, 1.26, 2.13 and 2.80, a spread of a factor of 3.9. Inside the 8/13 rung there are five rises whose asymmetry spans 0.769 to 0.795 and whose ratios span a factor of 1.33.

Several rises with the same asymmetry and different ratios is a falsification rather than a weak correlation. Whatever moves the ratio inside a rung, it is not how far apart the two contact neighbours are.

What it costs the discriminator #

The rival’s curve across a rung is flat, and it is flat for a reason that is worth stating rather than measuring: a kinematic lattice’s angle sequence does not contain the rise. The arrangement is organ i at i times the divergence with an inherited error, the divergence cancels when consecutive angles are differenced, and the errors know only the two offsets they are inherited at. Two stems on the same rung with different rises give the same sequence statistics.

So the comparison is between a flat line at 1.29 and a U with a floor at 0.79 — and the U crosses the line.

At the coarse end of the 8/13 rung, a twentieth of a rung from the transition, the rule gives 1.26 against the rival’s 1.29. At the fine end it gives 1.05. Three of the fifteen rises measured put the rule within a fifth of the rival’s value, and at those rises the quantity distinguishes nothing at all.

There is a compensation, and it is a real one. At the coarser rung the rival does not merely give a different ratio: it gives no main comb at all. A disturbance transported at offsets 5 and 8 puts almost nothing on the comb through the origin at spacing 5 — a mean of 0.033 against a sampling band of 0.10 — while the rule puts between 0.13 and 0.39 there at every rise on that rung. At the 5/8 rung the discriminator is not the ratio but the presence of the main comb, and that is a much cruder and much more robust test.

What this does not do #

It does not restore the comb as evidence of a rule. That retraction stands: a comb is evidence that something was transmitted between contact neighbours, and both accounts transmit. This essay is about the size of the one quantity that was left.

It does not explain why the floor is two thirds of the way up rather than halfway. The coordinate here is logarithmic in the rise, because the ladder is geometric, and on that coordinate the floor sits at 0.61 and 0.66 on the two rungs. A symmetric contamination from the two neighbouring rungs would put it at a half. The asymmetry is in the same direction on both rungs and is worth about a sixth of a rung, and nothing here says where it comes from.

It does not establish that the curve is universal. Two rungs is two, and both are in the middle of the ladder. The 3/5 rung is coarse enough that the readout’s own thresholds start to bite — it was measured refusing three runs in five there — and the 13/21 rung needs stems this sweep does not build. Whether the floor is 0.79 at every rung or drifts with the pair is unmeasured.

And the floor is not 0.65. That is the next essay’s subject and it is not a detail: the number changes because the azimuth grid it was measured on was coarse enough to be a disturbance in its own right.

And it leaves the survey specification worse than it found it. Before this sweep, a botanist wanting to use the ratio needed a long stem, a good protractor and a plant that was not too quiet or too disturbed. Now they also need to know where on the ladder the plant sits, which means measuring the rise — the height gained per organ, in units of the stem’s circumference — on the same stretch of stem the angles came from. That is not a hard measurement, and it is one more thing that has to be right.