A refusal with a reason

That earlier work’s last sentence on this thread was that a refusal has four causes, the sequence distinguishes none of them, and that was the third failure to close the mixture problem — the first with a shape. This essay is the accounting that goes with the second window, and it closes three of the four.

It also finds something worse than the fourth, which is that agreement between the two windows is not the same as being right.

The four causes, and where each now stands #

Too quiet. The pattern is so undisturbed that there is nothing to autocorrelate — at a fixed rise the rule locks onto its own sample grid and the readout refuses the resulting repeat as a cycle rather than a sample. Now separated, by the scatter: three of five stems silent at a disturbance of 0.005, with a mean recorded scatter of 0.38°.

Too disturbed. The pattern has come apart and there is no lattice left. Now separated, by the same scatter reading at the other extreme: three of five silent at 0.9, with a mean scatter of 56°.

Too fast. The shoot climbs a rung in less than a window, so the pattern changes inside the read. Now separated, and this is the second window’s own contribution: agreement between the two windows happened 0 times in 25 at 130 nodes per rung and 28 times in 50 at 400 and above.

A window in the wrong place. The shoot is slow enough in general, but this particular window straddles a boundary that a window a little lower would sit below. Not separated. It presents as one-sided or silent, and neither the scatter nor the agreement rule distinguishes it from a shoot that is simply too fast.

So the accounting goes from four causes and no separations to four causes and three separations, which is progress of a kind that can be stated exactly and is smaller than the thread has been hoping for since the earlier work.

What the fourth would need #

A third window, and the arithmetic is straightforward enough to write down without building it.

Two windows separated by half a length establish that something changed between them; they do not establish where, because a change anywhere in the 125 internodes that one contains and the other does not produces the same outcome. Three windows at 0, 125 and 250 below the tip bracket the change: if the lowest reads and the upper two do not, the boundary is above the lowest window’s top, and the shoot’s rate follows from where it must be.

The cost is stem. Three windows of 250 overlapping by half need 500 internodes of readable shoot, and the specification already asks for 250. Doubling the requirement to separate one cause out of four is a trade a survey designer should be allowed to refuse, and stating it as a trade rather than as a recommendation is the honest form.

There is a second cost that is easy to miss. Five hundred internodes on one shoot is not merely more work; it is a different plant. A shoot with five hundred readable internodes below its tip has been growing for longer, has passed through more of the ladder, and is more likely to have changed rate along its length — so the third window is bought at the price of the assumption that the rate is constant across the read, which the two-window scheme does not need. The instrument that separates the fourth cause is measuring a stem over which the thing it certifies is less likely to hold.

That is not an argument against building it. It is an argument for measuring the rate-constancy assumption at the same time, which the three-window scheme can do — the two gaps between the three windows give two rate estimates, and their agreement is a check of exactly the assumption the longer read strains.

Agreement is not correctness #

This is the finding the essay would rather not have, and it is the reason the second window is presented as a certificate about the rate rather than as a certificate about the reading.

At the quiet end of the disturbance range — 0.01 and 0.02, where the pattern is barely perturbed — there are stems on which both windows report 8/10, agree with each other, and are contradicted by the position counter, which reads 8/13 from the same coordinates.

The mechanism is the overlap, and it was priced in the previous essay as a trade between independence and placement. Here is what the trade actually cost: two readings that share half their data can share a mistake, and the mistake they share is exactly the one a quiet stem produces — a second comb too weak to be decisive, with a nearby residue class winning by an accident that persists across 125 internodes.

So the certificate has a stated scope, and the scope is narrow:

Agreement certifies that the rung is longer than the window. That claim is about the rate, and a wrong pair reported by both windows is still evidence about the rate, because both windows had a single pair in them to misread.

Agreement does not certify the pair. For that the reading still has to be checked against something that does not share its data: a parastichy count on the same shoot, which is the site’s standing requirement and is now doing more work than it was asked to.

The wrong pair was refusable without a second instrument #

The pair both windows agreed on is 8 and 10, and that particular answer is worse than wrong in a way that is useful: it is impossible, and impossible in two independent senses, either of which the readout could have checked on its own output.

Eight and ten share a factor. A stem grown one organ at a time has contact families that are denominators of successive convergents of its divergence, and successive convergents are necessarily coprime — so a counted pair with a factor in it cannot arise on a spiral stem at all. It arises on a whorled one, where organs come several at a time, and these stems are not that.

And ten is not on this angle’s ladder. The convergent denominators of the golden angle are 3, 5, 8, 13, 21 and their continuation; ten never appears and cannot, because the expansion that generates them is fixed by the angle. That is the same check that would have refused an earlier misreading in this collection, applied to a different instrument.

So the quiet-end failure is not a demonstration that a readout can be confidently wrong in ways nothing catches. It is a demonstration that this readout emits answers it has the information to refuse, and does not. Adding either check costs a few lines and no computation, and it converts these stems from agreeing on a wrong pair to both refusing — which is the right verdict, and one the four-cause accounting already has a name for.

What the self-checks leave, and why the second window is still needed #

The obvious next question is whether the checks make the second window redundant, and the answer is no, in a way that says what each is for.

The self-checks refuse a pair that is off the ladder: not coprime, or containing a number the divergence’s expansion never produces. What they cannot refuse is a pair that is on the ladder and is the wrong rung — 5/8 reported on a stem that is 8/13. Both members are convergent denominators, the two are coprime, and every arithmetic test the output can apply to itself passes. Only a comparison with something outside the reading can catch it.

That is exactly the error a window straddling a rung boundary makes, and it is exactly what a disagreement between two windows reports. So the two mechanisms partition the failure space rather than overlapping:

Arithmetic self-checks catch off-ladder errors — the ones a weak second comb produces by naming a residue class that is not a family at all. They are free and they run on one window.

The second window catches adjacent-rung errors — the ones a moving pattern produces by being genuinely two things during one read. They cost a shifted window and they cannot be had from a single reading however it is checked.

And neither catches an adjacent-rung error on a stem whose rate is slow enough for both windows to sit on the same wrong rung, which is not a failure of either instrument but a statement that the stem really is at that rung over the read. That case is closed by the position counter and by nothing else, which is why the site’s standing requirement survives all of this unchanged.

Stating the partition is what makes the certificate’s scope defensible rather than merely cautious. The scope below is narrow because the second window is answering one question, and it answers that one completely.

The scatter reading is not free either #

The separation of the two silences rests on a recorded scatter, and a recorded scatter is not the plant’s scatter. It is the plant’s scatter with the protractor’s error folded into it, and the two add in quadrature.

That matters more at one end than the other. The disturbed silences come with 56° of recorded scatter, and a reading error of a quarter of a degree changes that by nothing measurable. The quiet silences come with 0.38° — which is larger than the quarter-degree reading error the specification asks for, but not by much. A botanist working at half a degree per organ would record about 0.63° on a plant whose own scatter is 0.38, and at three quarters of a degree would record 0.84 — at which point the reading is mostly protractor.

So the quiet diagnosis is exactly where the instrument in the hand becomes the limit, and it is worth putting plainly: a too-quiet plant and a well-measured ordinary plant are separated by a quantity smaller than most protractors resolve. The disturbed diagnosis is safe by a factor of a hundred and the quiet one is safe by a factor of about one and a half.

There is a way out and it is not free. The plant’s own scatter can be recovered by measuring the same organ twice — the repeat difference is pure reading error, and subtracting its variance from the recorded variance leaves the plant’s. That is a doubling of the field work for one number, and it is the sort of thing a specification should ask for only where it decides something. Here it decides whether a silent stem is evidence of an orderly plant or of a badly held protractor, so it is asked for on the silent stems and not on the others.

What a survey would actually report #

The point of an accounting is that it changes what goes in a table, so here is the table.

For each specimen: the two windows’ readings, the parastichy count, the recorded scatter, and one of six outcomes.

Read — both windows agree and the count agrees. The pair goes in the results.

Read, uncertified — both windows agree and the count disagrees. The pair does not go in the results, and the specimen is evidence about the instrument rather than about the plant. The quiet-end stems above are this case.

Transition — the windows disagree by one rung of the ladder. The specimen contributes a rate estimate rather than a pair, which is a different and perfectly good measurement.

Misread — the windows disagree by something that is not a rung. Discard, and count it: the rate of these is a property of the disturbance range the sample sits in.

Too fast or badly placed — one-sided. Discard, and count it; a survey whose one-sided rate is high is sampling shoots that are growing too quickly, which is a fact about the species and the season.

Silent — neither reads. Record the scatter and classify: below about a degree the plant is too orderly to have a readable sequence, above twenty it is too disorderly to have a pattern, and between them the stem is silent for the one reason still unseparated.

Six outcomes is more bookkeeping than a survey of this subject has ever carried, and every one of them was earned by a measurement that showed the previous scheme collapsing two different things into one.

What the specification says now #

Six requirements, in the order a botanist would meet them, with the work that produced each:

Two hundred and fifty internodes of readable shoot, from the work that found the pair. Five hundred if the third window is wanted.

A reading error under a quarter of a degree per organ, from the same round of work: half a degree takes the length to 400 and three quarters to 1,100.

A shoot slower than about 250 nodes to the rung, from the earlier work — and now checkable from the reading itself rather than assumed, which is this essay’s contribution.

The recorded scatter reported alongside every readout, from the earlier work, and now carrying a second job: it is what separates the two silences.

Two overlapping windows, and only agreement is reported, from the work here.

And several plants of the same species, with a parastichy count on each, from the forgery essays here — because one stem cannot distinguish a lattice from a disturbance that repeats, and the angle readout cannot certify itself.

One thing the accounting made visible #

Writing the six outcomes out produced a distinction the thread had not made, and it is worth pulling out because it changes what a refusal is.

Four of the six outcomes are statements about the specimen — this shoot is too fast, this one is too quiet, this one crossed a boundary. One is a statement about the instrument: the windows agreed and the count contradicted them, so the reading is wrong and the plant is fine. And one, the misread, is a statement about the disturbance range the sample sits in, because the wrong-partner failure clusters at a particular disturbance rather than at a particular shoot.

A survey that reports only the pairs it read throws all three away. A survey that reports the outcome counts is measuring three additional things for free: what fraction of its shoots grow too fast, what fraction of its readings its own instrument gets wrong, and where in the disturbance range its specimens sit. None of those has ever been reported for this subject, and none of them costs a single extra measurement — they are counts of what happened to the measurements already made.

That is the general shape of what a refusal is worth, and it is the answer to the question the earlier work left. A refusal is not a failed measurement. It is a measurement of something other than the thing being looked for, and the work is deciding which something.

What is left of the mixture problem #

Three rounds of work have carried it and this is where it stops being carried.

The original worry — that the two statistics of a divergence sequence want different plants — was settled one earlier in this collection and settled in the negative: at a fixed rise both come off one stem, and the apparent conflict was a sign error between a rising stem and a fixed one.

What replaced it was the four-cause refusal, and this essay leaves it as: three separated, one requiring a third window whose cost is stated, and a caution that agreement between overlapping windows is not a check on the reading.

That is not a closure. It is a problem with a shape and a price attached to every part of it, which is what a problem should look like before somebody decides whether to pay.