The slide a counter holds constant

A rung is usually described as a plateau. The word is doing damage, and this essay replaces it with a measurement.

The damage is not in any single sentence. It is in what a plateau licenses without saying: that two stems on the same rung are the same kind of object, that a result measured on one transfers to the other, that “the 5/8 lattice” names a thing rather than a family. None of those is ever asserted outright, and all three are assumed by any census that takes one rise from each rung and treats the rows as comparable.

Across the 3/5 rung, swept at half a ten-thousandth — twenty-three rises, ten times finer than the ladder this collection normally draws — the settled divergence moves from 139.06° to 136.73°. It moves monotonically. There is no interval anywhere in the rung across which it is flat.

Across the 5/8 rung, swept at a thousandth, it moves from 136.55° to 137.87° — the same behaviour on a different rung, in the opposite direction, with the pair held throughout. Twelve rises there, twenty-three on the finer rung, and thirty-five settled stems between them: enough that the shape of each slide is read off a curve rather than inferred from its endpoints.

So a rung is a plateau in exactly one thing: the pair a counter returns. Beneath it the geometry slides continuously, and every quantity built from the geometry slides with it.

The two sweeps together are the point rather than either alone. One rung swept at a thousandth could be a fluke of that rung; two rungs, at two resolutions, an order apart in step size, both sliding smoothly and neither showing a flat stretch, is a property of rungs. And the finer of the two was run for an entirely different reason — to look for a locked band — so its evidence about the slide is incidental rather than sought, which is the better kind.

Why the direction differs #

The two rungs slide opposite ways and that is not an error.

The settled divergence approaches the golden angle from alternating sides as the ladder is climbed, which is what the continued-fraction structure of the limit requires. A rung whose pair is a Fibonacci pair sits on one side; the next sits on the other. So a slide down within one rung and up within the next is the signature of converging on a limit by alternation rather than from one direction.

That is a satisfying check rather than a new result. It means the slides are not numerical drift or a settling artefact; they are the ladder doing what the arithmetic says it should, resolved finely enough to see within a single step.

It also gives the slide a direction that can be predicted rather than merely observed, which is the difference between a curiosity and a handle. If the alternation is the convergent structure, then the sign of the slide on any rung is known in advance from the pair, and a sweep that came back with the wrong sign would be evidence of something wrong with the stem rather than with the expectation. Neither sweep here does.

The size of it #

A degree and a third across the 5/8 rung; two and a third across the 3/5 rung. Those are large numbers in this subject.

For comparison, the scatter of a settled stem’s own divergences — the wobble that decides whether it counts as settled — is a fifth of a degree at worst across the fine sweep. The slide within a rung is more than ten times the uncertainty of any single point on it.

And for a different comparison: a real plant’s divergences scatter by about half a degree between organs. The slide across one rung is larger than the measurement error of the field measurement the whole subject rests on — which means the difference between two stems at opposite ends of one rung is, in principle, visible on a specimen. It is not a difference that lives below the noise; it is a difference nobody has looked for, because the pair says the two specimens are the same and nothing prompts a second look.

The slide is the missing coordinate #

A slide that is monotone is not only a warning about plateaux; it is a ruler, and this is the constructive half of the finding.

A counted pair names a rung and is blind to everything inside it, which is the one hole a count has and the reason several results in this collection cannot be settled from a photograph. The slide fills it. Within a rung the settled divergence is monotone in the rise, so the divergence says where in the rung a stem sits, and it says it with resolution: two and a third degrees of travel against a settled stem’s own scatter of a fifth of a degree locates a stem to about a twelfth of a rung.

Both quantities are field-measurable, which is the part that matters. A pair is what a person counts on a photograph; a mean divergence is what a protractor gives from the same photograph over a few dozen organs. Nothing in the pairing needs a model, a rise measurement, or an assumption about how the specimen grew.

Why it takes both, and neither alone #

The two are exactly complementary, and the sweeps here show why in their own numbers.

The divergence alone is ambiguous. The 3/5 rung runs from 139.06° down to 136.73° and the 5/8 rung runs from 136.55° up to 137.87°, so the two overlap between 136.73° and 137.87° — which is most of both. A stem measured at 137.2° could be near the bottom of one rung or the middle of the other, and the divergence cannot say which. That overlap is not an accident of these two rungs: it is what converging on a limit by alternation means, since each rung’s range must straddle the limit’s neighbourhood to approach it from its own side.

The pair alone is blind, which is the finding this essay began from.

Together they are a coordinate. The pair selects the rung, which resolves the overlap; the divergence then locates the stem inside it. Neither quantity carries the other’s information and neither is redundant, which is a cleaner division of labour than this collection usually gets.

So the practical recommendation is one line and it costs a survey nothing extra: report the mean divergence beside the counted pair, with the number of organs it was averaged over. That is the same shape as the counting radius a survey specification already asks for — a second number that converts a robust but coarse reading into a located one — and it is the quantity that would let two studies of the same species be compared at all.

What it does not give is the rise itself, which is what several results here actually depend on. The divergence locates a stem within its rung and the map from that position to a rise runs through the rule; a person with a photograph has the position and not the map. That is a smaller gap than the one it replaces, and it is worth saying that it remains.

What slides with it #

Everything built from the geometry, which is most of what this collection measures.

The two contact steps. Their lengths are computed from the divergence and the rise, so both move, and their ordering reverses once inside the 5/8 rung.

The front’s depth. The run of organs at which a removal is felt grows as the rise falls, so the number of offsets that can wreck a stem grows across a rung.

And the answer to at least one experiment. At one offset the surviving family changes between the coarse end of the rung and the fine one.

That last item is the one that turns this from a vocabulary complaint into a result. If the slide moved only quantities nobody had drawn a conclusion from, it would be a footnote about description. It moves an outcome — a binary, reproducible outcome that an essay had reported as a property of a lattice — and that is what makes the interval worth insisting on.

What does not slide #

Two things hold across a rung, and they are the reason the count is worth taking at all.

Which lags exist. The two contact families are the two shortest steps, and which lags those are does not change inside a rung — only their lengths and their ordering do. So a wrecked stem’s menu of possible survivors is genuinely constant across the interval, which is the strongest result the ablation thread has and is untouched by any of this.

And whether the arrangement is a lattice at all. Every rise on both rungs settles, counts consistently, and passes the same test. The interval is an interval of lattices, not a range that shades off into something else at its edges — which is what makes sweeping across one a controlled comparison rather than a walk between two regimes.

Those two constants are what make the variable ones legible. A sweep in which everything moved would not be a sweep of anything; here the menu is fixed and the choice is not, and that separation is the whole reason the rung is the right unit to sweep along.

Three controls #

Both sweeps stay on one rung. The pair is measured at every rise and the generators refuse to draw a sweep whose rises do not all return the same pair, so a sweep that wandered over a boundary would stop the build.

Every rise settles. The slide is a slide between settled values, not a trajectory through unsettled ones. The worst wander on the fine sweep is 0.221°, comfortably inside the threshold — and, more to the point, an order smaller than the slide itself. If the two were comparable the slide could be an artefact of where in each stem the average was taken; a factor of ten between them rules that out without needing a separate control.

And the slide is not the grid. The azimuth grid the rule is evaluated over is finer than the slide by orders, so the divergences being reported are not quantised versions of a smooth curve. This is worth checking rather than assuming because a quantised slide would look identical at low resolution and would mean something entirely different — a staircase in the divergence would be a physical claim about preferred angles, which is the shape the locked band on the coarse rung actually has. Neither of these two rungs shows anything of the kind.

How wide a rung is, and why that matters more than it sounds #

The 5/8 rung runs from eighteen thousandths of rise to seven — a span of eleven thousandths. The 3/5 rung runs from about thirty down to nineteen, a span of eleven again. Higher up the ladder the spans are wider in absolute terms and narrower relative to the rise; further down they compress quickly.

That has a practical consequence for every sweep this collection plans. A step fine enough to put a dozen samples inside the 5/8 rung puts six inside the next one down and three inside the one after. Two rungs finer than the sweeps here, a rung sweep stops being a sweep — before the settling ceiling is reached, which is the other limit and bites later.

So the interval this essay insists on is not a fixed feature of the subject. It is widest exactly where this collection has done most of its work, and it shrinks towards the pairs real plants most often show — which is a reason to expect the sampling problem to matter less at the fine end and the settling problem to matter more.

That comparison points at something two other essays here both concluded was out of reach.

Where a stem sits inside its rung is the quantity almost every reading turns out to depend on, and it was described as unobservable without growing stems either side of a specimen until the pair changes — which can be done to a run and not to a plant. But the slide is itself a readout. The divergence moves monotonically across the rung, by a degree and a third here and two and a third on the rung above, and the divergence is the one quantity a specimen reports directly.

The arithmetic is friendlier than the half-degree scatter makes it sound, because a divergence is measured many times on one head. Scatter of about half a degree between successive organs, averaged over a hundred organs, gives a mean good to roughly a twentieth of a degree if the scatter is independent — which would cut a rung a degree and a third wide into something like twenty distinguishable positions. At a tenth of that resolution the reading would still separate the two ends of a rung comfortably.

The caveats are real and they are all of one kind. The estimate assumes the scatter averages down, and a systematic error — a tilted head, a consistent bias in how an angle is read off a photograph — does not average at all and would swamp it. Whether the scatter on a real head is independent between organs is a measurable thing nobody here has measured.

So the position within a rung is not recoverable from a single organ and may well be recoverable from a whole head, which is a considerably better situation than the one those essays describe. What it needs first is a rung swept finely enough to calibrate the conversion, which is what this one is.

That is the ordinary way a calibration comes about, and it is worth noticing when it happens: somebody measures one thing carefully for a reason of their own, and the measurement turns out to be the scale a different question was missing.

What to call a rung instead #

Not a plateau. The accurate description is longer and it is worth using: a rung is the interval of rises over which a spiral count returns one pair. It is defined by the instrument, not by the arrangement, and it is a range of geometries rather than a set of equivalent ones.

That phrasing costs nothing and it makes two mistakes harder to make. It makes “sample one rise per rung” look like the choice it is rather than the neutral act it seems, which is the sampling result of this round. And it stops “the 5/8 lattice” being spoken of as one object when it is a family of them.

Where this leaves it #

With a small, checkable, slightly deflating fact: the object this collection has been treating as a unit is an interval, and several of its results turned on where inside the interval they were measured.

None of the affected results is wrong in the sense of reporting a number that is not there. The shortest-hop refutation reached the right conclusion, the offset rule scores what it scores, and the census rows are all real measurements of real stems. What changes is what they are measurements of: not of a lattice, but of one geometry drawn from a range of them, chosen by a rise nobody recorded because nobody thought it varied.

The deflation is worth having because the alternative is a vocabulary that quietly asserts something false. A plateau has no inside. A rung does, it is wide enough to hold a reversal and an outcome and a degree of divergence, and most of the measurements in this thread have been spent there.

It is also, in the end, a compliment to the count. The reason a rung is an interval is that the count is robust — the same answer over a range of geometries — and robustness is what makes it worth taking on a plant with a hand lens. The measurement is sufficient for what it was invented for and insufficient for several of the questions this collection has since asked of it, and both halves of that are the same fact about what a topological reading is.