What the pair costs

An instrument is worth what it costs to use. This site has priced several — how many specimens a frequency claim needs, how many junctions distinguish two branching laws, how many internodes the single-number readout wants — and the prices have been more useful than most of the positive results, because a specification is something a person can act on.

So: what does the pair cost?

Internodes #

Two hundred and fifty divergence angles from one stem, at the rise where the readout works best, with no reading error at all. Five stems out of five return the pair the position counter finds at 250 and above; four out of five at 150 and at 200.

The single number cost sixty. The pair costs about four times that, and the reason is visible in the two scores rather than being a matter of statistical taste: at this rise the main comb averages 0.64 and the second averages 0.50, and both have to clear a threshold that falls only as one over the square root of the length. The weaker comb sets the price.

Two hundred and fifty internodes is a real requirement and it is worth saying what kind of plant supplies it. Not a rosette, which has no internodes to speak of. Not a cone or a capitulum, where the organs are on a disc and there is no sequence to walk up. A long unbranched shoot with visible leaf scars, and enough of it that the parastichy pair does not change over the stretch measured — which is the rate condition, and it has its own essay.

The protractor #

The binding constraint on the single-number readout turned out to be precision rather than length, and it is worse here.

A reading error enters the recorded divergences in a particular way, and the way matters. An observer marks where each organ is and subtracts to get the gaps, so an error of ε on one organ’s azimuth enters two consecutive divergences with opposite signs. That makes the error a moving average of variance 2ε², which puts a large negative correlation at lag one — where nothing is being read — and adds variance at every lag.

The pattern’s own contribution to the correlation at a comb’s lags is untouched by the error. What changes is what it is divided by. So every correlation dilutes by the ratio σ² / (σ² + 2ε²) for a pattern scatter σ, which is the law the earlier work measured to within three per cent while there was signal.

Now set that against the threshold. The threshold a comb has to clear falls as one over the square root of the number of divergences. So recovering a diluted comb needs the threshold to fall by the dilution factor, which needs the length to grow by the square of it: N(ε) = N(0) · (1 + 2ε²/σ²)².

There is nothing free in that. The dilution law came from the earlier work, the threshold’s dependence on length is the sampling band’s, and neither was fitted to anything measured here.

The prediction against the measurement #

The pattern scatter on these stems is 0.70°. Feeding that in, the law says the pair should need 250 internodes at no reading error, 395 at a quarter of a degree, 1,027 at half a degree and 2,741 at three quarters.

Measured, it needs 250, 250, 400 and 1,100.

The shape is right and the scale is not. The measured curve rises steeply, monotonically, and by roughly the amount predicted between the last two points — a factor of 2.75 measured against 2.67 predicted. What is wrong is that every prediction is too large, by 1.6, 2.6 and 2.5, in the same direction at every point.

A consistent over-prediction is a different kind of error from a scattered one. It is the signature of one input being systematically mis-measured rather than of the law being wrong, and the input in question is not hard to identify: σ is taken as the root mean square of the whole divergence series, and only the part of that variance which is correlated carries the comb. A larger effective σ moves every prediction down by the same factor. Solving for the σ that would fit gives about 1.4°, or twice the measured RMS, which would mean roughly a quarter of the variance is incoherent — plausible and unverified.

That reading can be checked against the same four points and it does not survive, which is worth doing before it is carried forward.

The law’s shape and its level want different values of σ. Take the ratio between the two largest errors: predicted, it is 2.67 at σ = 0.70 and measured it is 2.75 — agreement to three per cent. At σ = 1.4 the same ratio is 1.57, which is off by a factor of 1.75. So the ratio between adjacent points fits the measured scatter and refuses the fitted one.

Now take the levels. At σ = 1.4 the prediction at half a degree of error is 394 internodes against a measured 400, which is nearly exact — and at three quarters it is 619 against 1,100, which is out by 1.8. So the fitted σ fixes one point and breaks the next.

A single mis-measured σ therefore cannot be the whole of it, because moving σ trades the shape agreement for the level agreement and there is no value that buys both. The account named above is the natural one and the four points refuse it.

What that leaves is a discrepancy with no candidate rather than a discrepancy with a plausible one, which is a worse position and the honest one. The two other inputs — the 250 at no error, and the assumption that the threshold falls as one over the square root of the length — have not been varied, and one of them is where the account will have to come from.

It is recorded that way rather than repaired by fitting σ. A fitted σ turns a prediction with nothing free in it into a curve with one parameter, and a curve with one parameter that passes through four points is not evidence of anything. The useful content survives either way, and it is worth converting into the number a survey would actually use. Written out, the cost multiplier is (1 + 2ε²/σ²)² — so what matters is the reading error relative to the plant’s own scatter, and the multipliers are easy to tabulate. Read to a quarter of the scatter and the stem needs to be 1.3 times longer. Read to half of it and 2.3 times. Read to the scatter itself and nine times.

On these stems the scatter is 0.70°, so a quarter of it is 0.18° — and the specification that follows is a single sentence: measure each organ’s azimuth to better than a fifth of a degree and the two-hundred-and-fifty-internode figure stands; measure it to half a degree and no stem is long enough.

The useful content survives either way: a coarser protractor is paid for at something like the fourth power of the error, so doubling the reading error costs about sixteen times the stem.

Why a quarter of a degree is the number to quote #

At a quarter of a degree per organ the pair costs nothing extra: 250 internodes, the same as with no error at all. At half a degree it costs 400. At three quarters it costs 1,100 — and at one degree the readout does not clear at any length tried.

So the specification has a cliff in it rather than a slope, and the cliff sits just past half a degree, which is around the pattern’s own scatter. That is not a coincidence: the dilution is a ratio of the reading error to the pattern’s scatter, so an instrument that reads a pattern to worse than the pattern’s own departures is measuring its own noise.

A field protractor against a stem will not do a quarter of a degree. The earlier work said so about the single number and the sentence stands unchanged: this is photogrammetry, or a stem cleared and imaged, and it is the line in the specification that has changed what a survey would cost more than any other.

What is bought for the price #

It is worth setting the cost against what the cheaper instruments give, because the pair is not the only thing a sequence carries.

• Sixty internodes and a rough protractor give the lag-one correlation, which separates noise arriving before the placement from noise arriving after it.
• Sixty internodes and a quarter of a degree give the smaller parastichy number, from the main comb alone.
• Two hundred and fifty and a quarter of a degree give the pair, and therefore the branch.
• A photograph gives the pair too, when the spirals are traceable, and gives it in an afternoon.

The last of those is the comparison that keeps the result honest. Where the spirals can be counted, counting them is enormously cheaper and this instrument is a curiosity. Its value is where they cannot: material where the organs are gone and only the scars remain, a stem too long or too irregular to photograph as a whole, or a specimen where the question is not the count but whether the sequence has a memory in it — which no photograph can answer, and which is what the mechanism essay is about.

The specification, in one paragraph #

For the record, and in the form the survey file keeps them:

To read a parastichy pair from angles alone. One shoot, unbranched, with at least 250 consecutive internodes over which the pair does not change — which requires a rate slower than about 250 nodes per rung, measured in its own essay. Azimuths to a quarter of a degree per organ; half a degree costs sixty per cent more stem, and one degree does not work. The plant must be disturbed enough for its divergences to vary — a scatter of half a degree to one degree in the model — and not so disturbed that the lattice is gone. Report the spacing of the main comb, the residue of the second, the pair that follows, and the three clearance tests, so that a refusal is distinguishable from an absence of data.

Why the threshold is sharp rather than gradual #

The length curves have an odd shape and it is worth explaining, because a reader who expects a gentle rise in reliability will read the figure wrongly.

At no reading error the readout is right on four stems of five at a hundred and fifty internodes, five of five at two hundred and fifty, and five of five at every longer length tried. It does not creep up; it steps.

That is what a threshold on a mean does. The comb score is roughly independent of length — a longer sequence estimates the same correlation more precisely rather than a larger one — while the threshold it has to clear falls as one over the square root of the length. So the question at each length is whether a fixed number has crossed a falling line, and the answer flips over a narrow range of lengths rather than improving steadily.

Which means the number to quote is a threshold rather than a rate, and the honest form of the specification is “two hundred and fifty internodes” rather than “more is better”. Past the threshold, extra stem buys nothing at all: the pair at eleven hundred internodes is the same pair as at two hundred and fifty, with a slightly larger margin.

What extra stem does buy is tolerance of a worse protractor, which is the trade the four curves in the figure are. That is not the same thing as buying reliability, and conflating the two would produce the advice “measure a longer stem when the answer is unclear”, which is right only if the uncertainty comes from the protractor.

What the three tests cost separately #

The readout has three clearance tests and they do not cost the same.

The main comb’s test is nearly free. On every stem where the pattern is a lattice at all, the main comb clears comfortably — 0.64 against a threshold of 0.19 at the working rise — and it is the test that fails only when there is no pattern.

The second comb’s test is what sets the length. It is the weaker signal and the threshold is the same, so it is the binding constraint at every length and every protractor error in the sweep.

The margin test costs the least in length and the most in refusals. It fails at one rise out of the five tried, and at that rise it refuses three runs of five — all three of which would have been wrong. So its price is paid in coverage rather than in stem: it makes one rung of the ladder unreadable rather than wrongly read.

That decomposition matters for anybody deciding whether to relax a test. Relaxing the second comb’s threshold buys shorter stems and immediately buys wrong answers, because the second comb’s score at short lengths is not small — it is uncertain, and an uncertain number that clears a lowered bar is a coin flip with a decimal point. Relaxing the margin test buys one more rung and buys the three wrong answers back with it.

What the price says about the instrument #

Four times the length for the second number is a steep price and it is worth asking whether it is intrinsic.

Part of it is: the second comb is genuinely weaker, because the correlation reaches the second family through a longer chain or a weaker coupling, and no amount of cleverness makes a smaller signal larger. Part of it is the three clearance tests, which are deliberately conservative — the margin test alone turns three wrong answers into three refusals at one rise, and a version without it would report at shorter lengths and be wrong more often.

That trade is the one this site keeps making and it is worth naming as a choice rather than a fact. A readout that reported at 150 internodes with four right answers in five and one wrong would be cheaper and would look better in a table. What makes 250 the number quoted is that at 250 the wrong answers are gone, and a number that is sometimes wrong with no way of telling which is not a measurement.

The price beside the others #

Set against the other prices this site has computed, the pair is expensive and not absurdly so.

A frequency claim about how often plants are Fibonacci needs tens of specimens. A junction measurement that distinguishes Murray’s law from Da Vinci’s needs one even fork or two thousand twigs. The noise-discrimination test needs sixty internodes. The pair needs two hundred and fifty internodes and photogrammetry, and the noise test’s own ceiling means its sixty have to come off a stem quiet enough to carry the reading at all.

What all four have in common is that the cost is dominated by one term and the term is different in each case: specimens, junction shape, sequence length, angular precision. A survey that budgeted for all four uniformly would overspend on three and underspend on one, which is the ordinary failure of a specification written without arithmetic behind it.

A later note: the same length, one more plant #

Everything this page prices is per stem, and the work after it added a requirement that is per sample. A single stem cannot distinguish a lattice from a disturbance that repeats every m organs, because such a disturbance names its partner from its own shape and names a different one on each stem.

The lengths on this page are unchanged. What is added is six plants rather than one — three to exclude the forgery, six to make the ratio of the two combs a measurement — and the arithmetic for that is in What a forgery has to know.