Two readings from one stem

There is a worry that has been written into three consecutive note left with the work on this site, in almost the same words each time, and it has never been measured.

The earlier statistic reads the correlation between one divergence and the next. It is decisive below about 0.9° of recorded scatter and gone by 1.2°: it wants a quiet plant. The spectrum readout reads the periodicity of the whole thing. It has nothing to autocorrelate at 0.05° of scatter and is clean at 1.1°: it wants a disturbed one. Worse, the first was measured on a stem whose rise was falling and the second needs a rise held fixed, because a band over which the parastichy pair is constant is forty to eighty nodes at the site’s usual rate and forty nodes cannot show a period of eight.

Two statistics, apparently two plants, and a mixture problem that cannot be closed by having two numbers if the two numbers cannot be taken off one specimen.

This essay takes them off one specimen.

Both, on the same sequences #

The experiment is the obvious one and its only virtue is that it was actually run. Hold the rise fixed, sweep the disturbance from 0.02 to 1.0 in azimuth degrees per node, five seeded stems at each amplitude, and compute both statistics from the same autocorrelation of the same list of angles.

There is no trade. Across the whole range where the pair is readable, the lag-one correlation sits between −0.50 and −0.65 with no trend in it, which is four to six times the sampling band. One stem supplies both, comfortably, at every amplitude where either works.

Where the conflict came from #

The sign.

The earlier work reported a lag-one correlation of +0.54 with no noise at all and called it the sequence’s own memory — a lattice correcting itself, a node placed to one side leaving a gap the next falls into. The earlier work found that explanation is about a lattice, so it ought to hold of one that is not going anywhere, and tested it: at a fixed rise the noiseless rule converges exactly, every divergence identical, and under a disturbance the lag-one correlation is −0.68.

The +0.54 belongs to the rise declining. It is the pattern chasing an equilibrium that is moving under it, and it appears as a threshold in the rate — negative to about fifty-two nodes per rung, positive from fifty-eight.

So the two statistics were never asking for different plants. They were being read on stems that differed in the rise, and the quantity that changed sign was mistaken for a quantity that changed availability. The worry in three separate notes was a worry about a sign convention.

Which leaves a condition on a real shoot #

A real shoot’s rise falls — that is what climbing the ladder is — so a plant is a rising stem, and the sign of its lag-one correlation is set by how fast it climbs.

The threshold measured here is about fifty-five nodes per rung: slower than that and the correlation is positive, faster and it is negative. So a botanist reading a sequence off a shoot cannot interpret the sign without knowing the rate, and the two plants that would otherwise look like opposite results are one rule at two rates.

That sounds like a new requirement and it is not, because the rate is measurable from the same sequence. A shoot climbing the ladder changes its comb spacing as it goes — the smaller counted number steps from three to five to eight — so the number of internodes between two spacings is the rate, read off the angles with no coordinate and no second instrument.

So the specification gains a line and no measurement: report the rate, in internodes per rung, beside the lag-one correlation. With it the sign is interpretable; without it a positive and a negative reading are indistinguishable from two different plants, which is exactly the confusion three separate notes recorded.

It also says what a shoot near the threshold is worth, and the answer is: little. A plant at fifty-five internodes per rung sits where the correlation crosses zero, so its lag-one value is small whatever its noise is doing, and the statistic that separates two kinds of disturbance has nothing to separate them with. That is an eligibility criterion of the same kind as the scatter ceiling, on a quantity nobody had thought to record.

The window’s two ends, and they fail differently #

The pair is readable from a disturbance of about 0.1 to about 0.4, which is a recorded divergence scatter of roughly 0.4° to 1.0°. Both ends are hard edges and they fail in ways that are not equivalent.

The loud end is safe. At 0.6 and above the lattice is gone: the divergences wander over the whole circle, the recorded scatter is above a hundred degrees, and every run refuses. A botanist looking at such a stem would not call it patterned either, and the instrument’s coherence test is the same judgement made arithmetically.

The quiet end is dangerous, and this is the finding of the essay that was not looked for. At a disturbance of 0.02 the rule locks onto the grid of azimuths it is sampled on. Every divergence comes out at 137.8125° — which is exactly 147 of 384 grid steps — on every seed, to the last digit. What is left to autocorrelate is a short deterministic repeat rather than a sample of a process, its main comb is 0.98, and the pair it names is 8 and 11 where the counter says 8 and 13.

It reports, and it is wrong. That is a worse failure than a refusal and it took a sweep to find, because at the amplitude the earlier essays use it does not happen.

The cap, and what it does not fix #

A correlation above 0.95 means the sequence is very nearly deterministic, which no real measurement of a real plant will be. So the readout refuses above it.

That removes the failures at the quietest amplitude — all five runs — and three of five at the next. One wrong answer survives at 0.05 and one at 0.15, and both are left in the record rather than tuned out, because moving the cap far enough to catch them also refuses runs that are right.

The rest of the requirement is a condition on the specimen and not something the angles can enforce: the plant has to be disturbed. That is an odd thing to have to ask for and it is worth saying why it is not a defect. The comb is a measurement of how a disturbance propagates. A pattern with no disturbance in it has nothing propagating, so there is nothing to measure — in the same way that a perfectly still fluid reports nothing about its viscosity.

The grid artefact, and what it says about the model #

The quiet-end failure is a property of the simulation rather than of plants, and it is worth being clear about which.

The rule samples 384 azimuths and takes the minimum, so its output is quantised to 0.9375°. At large disturbances the noise moves the chosen sample around and the quantisation is invisible. At small ones it does not, and the run settles onto a single grid point and stays there. The “sequence” is then a repeat with a period set by the seeded stretch.

A real apex has no grid. So the specific failure — locking to 137.8125° on every seed — will not happen to a plant. What will happen to a plant is the general version of it: a measurement quantised more coarsely than the pattern’s own variation produces a sequence with structure that is the quantisation’s rather than the plant’s. A protractor read to the nearest degree, on a plant whose divergences vary by half a degree, is exactly this failure with a different grid in it.

So the cap earns its place twice. It guards the model against its own sample grid, and it guards a real measurement against a coarse protractor — and in the second case it is the only guard there is, because the dilution law that prices reading error assumes the error is fine and random, which a rounding is not.

What is now available from one stem #

Setting out the readings, in the order of what they cost:

• the recorded scatter, from a ruler, which says whether the other two are worth attempting;
• the lag-one correlation, from sixty internodes, which on a fixed rise is the rule’s self-correction at about −0.6 and on a rising stem is the pattern chasing the rise at about +0.55 — and whose sign therefore reports which of those the specimen is;
• the parastichy pair, from two hundred and fifty internodes at a quarter of a degree.

All three from one list of angles up one stem. That is the thing three separate notes doubted, and it is settled — for a stem held at a fixed rise, which no plant is.

The remaining question is whether it survives a stem that climbs, and it does, on a condition that turns out to be about the shoot rather than about the statistics: the window read has to fit inside one rung of the ladder. That is the next essay, and it is where the mixture problem finally gets an answer with a shape.

The window is wide, and that is the surprise #

It is worth dwelling on how much room there is, because the worry that prompted the experiment implied there might be none.

The pair is readable from a disturbance of 0.1 to 0.4, which is a factor of four in amplitude. The recorded scatter across that range runs from about 0.4° to about 1.0°, which is most of the range in which the site’s noise work has found a lattice to exist at all. And the lag-one correlation is not merely present across it — it is between −0.50 and −0.65, which is four to six sampling bands, at every amplitude.

Compare that with the two statistics’ individual ranges as the essays that introduced them measured them: the lag-one statistic decisive below 0.9° and gone by 1.2°, the comb absent at 0.05° and clean at 1.1°. Those two ranges overlap over most of a decade, and the overlap is where both work. The note left with it that recorded the conflict were reading the two endpoints of one range against the two endpoints of the other and concluding that the ranges were disjoint, when what they had was two overlapping intervals described from opposite ends.

That is a small error of reading and it survived three restatements, which is worth recording as a caution about how a worry propagates. Each restatement was a paraphrase of the one before, none of them was a measurement, and the paraphrase got sharper each time — “two statistics with opposite requirements” in one round of work became “they want opposite specimens” in the next and “one a disturbed plant on a steady rung, one a quiet plant on a rising stem” in the third. The last of those is the most specific and the most wrong.

What the two statistics are about, restated #

Since the sign is what caused the trouble, it is worth writing down plainly what each number means on which kind of stem.

On a stem at a fixed rise, the pattern has an equilibrium and stays at it. A node displaced to one side leaves a gap the next node falls into, so consecutive divergences alternate: the lag-one correlation is negative, around −0.6, and it is the rule correcting itself. The comb is present, and its spacing is the parastichy number.

On a stem whose rise is falling, the equilibrium is moving. The pattern is always slightly behind where the current rise would put it, so consecutive divergences are displaced in the same direction: the lag-one correlation is positive, around +0.55, and it is the pattern chasing. The comb is absent, because the parastichy number is changing.

So the sign of the lag-one correlation reports which regime the specimen is in, and it does so before either instrument is attempted. That is a use for it nobody had, and it costs nothing — it is the same number, read for its sign rather than its magnitude.

The methodological note #

This essay’s result is that a worry recorded three times was not a real constraint, and the reason it survived three rounds of work is worth naming: it was never measured because it was never cheap to measure the two together. Each statistic lived in its own library, each library had its own runs, and the comparison was made by reading two numbers out of two docstrings.

The fix was to compute both from one call. bothStatistics takes a list of angles and returns the lag-one correlation, the comb scores and the pair, all off a single autocorrelation, precisely so the two cannot be quoted from different runs by accident. That is not a clever piece of engineering; it is a shape that makes a particular mistake impossible, and the mistake had been made three times.

What is left of the worry #

Nothing about the fixed rise, and one thing about the rising one.

At a fixed rise the two statistics come off one stem across a wide window of disturbance, and the conflict recorded three times was a misread sign. That much is settled.

What remains is that a real plant’s rise is not fixed, so the comb has to be read over a window and the lag-one statistic over the whole climb — and those are different lengths of the same stem with different requirements. The next essay measures what that costs, and the answer is a condition on the shoot’s rate rather than on either statistic.

Why the sweep is five stems and not fifty #

Every point in the disturbance sweep is five seeded stems, which is small, and it is worth saying why it is not larger.

The quantity being measured at each amplitude is not a mean with an error bar. It is a count — how many of five returned the counted pair — and what the sweep establishes is where that count is zero, where it is five, and roughly where it changes. Five is enough to distinguish those three states and it is not enough to locate the boundary precisely, which the essay does not claim to do: the window is quoted as “about 0.1 to about 0.4” rather than with edges.

Fifty stems per point would locate the edges and would cost fifty times as much, on a sweep whose points are already grown stems of nine hundred nodes each. The edges are not worth that, because a real plant’s disturbance is not a number this model can predict — what matters is that the window is wide, and five stems at seven amplitudes says so.

A later note on which rungs this reaches #

This page reads the pair at three rises and stops at 8/13, and the work after it recorded that as a ceiling. It is not one. With the lag window widened to 45, the seed lengthened so that it spans a piece of lattice rather than a cluster, and the azimuth grid tripled, the same readout returns 13/21 on five stems out of five at every disturbance from 0.08 to 0.18.

Nothing on this page changes: the two published rises give identical readings at both grids, five runs of five. What changes is that the instrument’s settings are now functions of the rung rather than constants — the essay that measured it states all three.

Two readings that are one reading #

A warning that belongs beside this essay’s argument. Two statistics off one sequence are two measurements only if neither is a function of the other, and one pair this collection proposed failed that test: a comb and a slow wander are the same autocorrelation function evaluated at a lag and at a block size.