The rung was not the instrument

The earlier work closed with four leavings and one of them was a ceiling:

The reading needs three multiples of the smaller parastichy number inside the lag window and needs the window short enough that the larger number is not itself a candidate spacing. At 5/8 and 8/13 that is satisfiable; at 13/21 there is no window that satisfies both, so the instrument as written has a ceiling one rung above where it works. That is arithmetic rather than statistics and no amount of stem fixes it.

Every clause of the first sentence is right. Everything after it is wrong, and the arithmetic that was supposed to prove the ceiling disproves it in two lines.

The band is never empty #

Write the two requirements as inequalities. Three teeth of the main comb inside the window: L ≥ 3m. The larger number not itself a candidate: L < 3n. So the window has to satisfy 3m ≤ L < 3n, and since m < n at every rung there is always room. At 8/13 the band is [24, 39) and thirty sits in it, which is why the readout worked. At 13/21 the band is [39, 63) and thirty does not, which is a statement about the default rather than about the rung.

Widen the window to 45 and the requirement is met. That is not a repair to the instrument; it is the instrument being used at the setting its own arithmetic prescribes.

The mistake is a small one and it is the kind that gets written into a plan file at the end of a round of work: the two constraints were both correctly identified, neither was written down as an inequality, and the conclusion that they are incompatible was reached by reasoning about one example. It survived a round of work because a leaving is not gated.

What was actually stopping it #

Widening the window is necessary and it is not sufficient. Two things about the stem had to change with it, and both are more interesting than the arithmetic.

The seed has to be a piece of lattice. Every fixed-rise run on this site starts from eight nodes of ideal lattice at a stated divergence, which at a rise of 0.005 spans four hundredths of a circumference — a stretch of pattern the rule can continue. At a rise of 0.0019 the same eight nodes span fifteen thousandths, which is a cluster: eight points nearly on top of each other with no lattice structure in them at all. The rule has nothing to continue, the run wanders, and the recorded divergence scatter comes out at 115° — no pattern anywhere in it.

Forty seed nodes at that rise span the same height eight do at 0.005, and the run settles to a scatter of half a degree.

And the azimuth grid has to be finer than the pattern. With the seed fixed, the readout still failed on every stem at every disturbance from 0.08 to 0.18. The rule samples 384 azimuths when it takes its minimum — a step of 0.94° — and at this rung the pattern needs finer resolution than that. What happens is worth describing precisely, because it is not a rounding error: the minimum lands on the same grid point run after run, every divergence comes out at exactly the same value, and what is left to autocorrelate is a deterministic repeat. The readout refuses it as a cycle rather than a sample, which is a refusal the earlier work built for exactly this failure and did not connect to this rung.

So the claim goes #

The instrument has no ceiling at 13/21. It reads the rung on every stem at every disturbance tried, with the pair confirmed by a counter that is shown coordinates and no angles.

And the reason the earlier work could not read it was neither arithmetic nor statistics. It was two settings — a seed length and a sample count — that had been constants since the founding essays and had never been asked whether they were binding.

That is the third time this site has recorded a defect of that shape. reach and maxWindow were one number in the rising model until a sweep of a non-binding parameter reported the same answer at every setting and was read as robustness. A recency cut-off in the interaction-range work turned out to manufacture a lattice the model did not otherwise have. And now a sample grid sets which rungs an instrument can reach.

The common form: a parameter of the program, mistaken for a property of the model. Each was found by changing the parameter, and each had passed every check this collection runs, because a check asks whether a computation is correct and none of these computations was incorrect.

The failure is bimodal, which is how it hid #

One detail of the measurement is worth pulling out, because it explains why the problem looked like a ceiling rather than like a setting.

A run at the fine rung does not degrade gradually as the seed is shortened or the grid coarsened. It either settles onto a lattice — scatter half a degree, pair read cleanly — or it never finds one at all, and comes back with a scatter of over a hundred degrees. There is nothing in between. At twenty-four seed nodes, four runs of five settle and one does not, and the mean scatter across the five is 23°, which is a number describing no run that happened.

So a sweep that averages over seeds sees a quantity that moves smoothly from 0.5° to 115° as the setting changes, and reads it as a pattern degrading. What is actually happening is a fraction of runs falling off a cliff. The two look identical in any summary and are completely different failures: one says the instrument is reaching its limit, the other says some runs are broken and the rest are fine.

That is worth a general note, because this site averages over seeds everywhere. A mean over an ensemble whose members are bimodal is a number describing none of them. Where a run can fail outright, the thing to report is the fraction that succeeded and the distribution of what the successes gave — which is what the figures in this essay do, one mark per stem, and it is why the cliff is visible in them.

The band widens as the rung climbs #

Non-empty is the weakest true thing to say about [3m, 3n), and the stronger statement reverses the original claim rather than merely refusing it.

Write the band’s width. It is 3n − 3m = 3(n − m), and on the ladder the difference of a consecutive pair is the pair below it, so the width at 5/8 is 9, at 8/13 it is 15, and at 13/21 it is 24. The band does not narrow towards a ceiling. It grows, and it grows at the same rate as the numbers in it.

The relative version is the one worth carrying, because it is rung-independent. The band runs from 3m to 3n, and n/m tends to φ, so the band is always [3m, 3φm) to within a percent above the coarsest rungs — a fixed 61.8% of its own lower edge, at every rung for ever. There is no rung at which a window placed a fifth of the way into the band misses it, which means the instrument’s window requirement is not a difficulty that accumulates with depth. It is one multiplication.

That has a consequence for the rule this essay ends with. Reading at 3m + 6 works at 5/8 (21 against a ceiling of 24), at 8/13 (30 against 39) and at 13/21 (45 against 63), and the margin grows every time — but an additive margin against a multiplicative band has a floor rather than a ceiling. The condition 3m + 6 < 3n is 6 < 3(n − m), which fails when n − m is 2 or less: at 2/3 the band is [6, 9) and 3m + 6 is 12, outside it. So the rule as stated is safe for m of four and above and wrong at the two coarsest rungs, where the correct window is nearer 3m + 2.

This is a small correction and it is exactly the kind the essay is about. The original ceiling was reached by taking one rung’s arithmetic for the general case; the replacement rule was reached by taking three rungs’ arithmetic for the general case, and it also has a domain. Writing the inequality down is what makes the domain visible, and it was the step missing from both.

What the bimodality costs a sweep #

The cliff described below is not only an explanation of how the ceiling hid. It changes what a sweep of any of these settings has to report, and the arithmetic is short enough to give.

At twenty-four seed nodes, four runs of five settle at about half a degree of scatter and one fails at about a hundred and fifteen. The mean of those is 23°, which is the number the earlier sweep saw — and it is a number no stem produced, sitting forty times above the successes and five times below the failure. Read as a degradation it says the pattern is half-formed. Read correctly it says four stems were perfect and one was not a stem.

The reporting fix is to give the success fraction and the successes’ spread separately, and that fix has a price in sample size which is worth stating so it is not adopted casually. A mean over five runs is a usable estimate of a smooth quantity. A fraction over five runs is not: four of five and five of five are two draws from binomials whose 95% intervals both run past half, so five stems cannot distinguish an instrument that fails a fifth of the time from one that never fails. Separating those to any useful precision takes tens of stems, not five.

So the honest form of this essay’s result is that the fine rung reads on five stems of five at the corrected settings — which establishes that the rung is reachable and does not establish a failure rate. The first is what the refuted ceiling was a claim about, so it is the claim that needed making. The second is open, and it is the quantity a survey would actually need, since a survey specification has to say how often its instrument returns nothing.

The general note follows and this collection should apply it more widely than it does. Where a run can fail outright, an ensemble has two statistics and reporting one of them is reporting neither. Every fixed-rise sweep here averages over seeds, and the sweeps whose runs cannot fail — a hop length, a settled divergence — are fine. The ones at risk are the sweeps of things that are read rather than measured, and a readout that refuses is exactly a run that can fail.

What a leaving is worth #

Two of this essay’s four leavings from the previous one turned out to be wrong in the same way: a claim written into a plan file at the end of a round of work, on the basis of reasoning rather than of measurement, and carried forward as though it had been established. The other was the control that the work here withdrew.

That is not an argument against writing leavings down. A leaving is how a round of work hands its unfinished business to the next one, and unfinished business that is not written down is unfinished business that is forgotten. It is an argument for marking which kind each one is:

A leaving that reports a measurement — the dip’s coefficient runs 1,400 to 3,300 and three denominators do not support a q-dependence — is a fact with a number attached and it held up.

A leaving that reports a conclusion — there is no window that satisfies both, and no amount of stem fixes it — is an argument, and an argument that has not been run against the machinery is a hypothesis wearing a result’s clothes.

Both kinds belong in a plan. Only one of them should be quoted in the next work’s essays without being checked first, and the work here checked two of them and found both wrong.

What it cost, and why the default did not change #

Tripling the sample count triples the cost of every noisy run, and this site grows a great many of them. So the finer grid is used where it buys a rung and the default stays at 384, which is a decision that needs the evidence that it is safe.

The evidence is in the next essay: at the two rises the site has already argued from, the readings are identical at both grids, five runs of five. Nothing already written depends on the sample count. What does change at the finer grid is the deliberately marginal rise where the two instruments were reported as disagreeing — and a stem with no answer answering differently on a different grid is the object behaving as it was said to.

Why nobody had looked #

The rung was never attempted, and the reason is worth recording because it is not laziness.

Reading at 13/21 costs more than reading at 8/13 in three ways at once. The stem has to be longer, because a comb at spacing 13 needs its teeth out to lag 39 and the sampling band falls as one over the square root of the sequence length, so the same clearance needs more angles. The seed has to be longer, which means more of the run is transient rather than measurement. And the sample grid has to be three times finer, which triples the cost of every node.

Multiplied together that is roughly an order of magnitude in computation per stem, against a rung that no essay needed. So the earlier work asked whether the instrument could reach it, reasoned about the lag window, concluded it could not, and moved on — which is a perfectly ordinary thing to do with a question that would cost a day to answer and does not block anything.

What made it worth answering now is that the same three settings turned out to be the whole content of the answer. A ceiling that is really three constants is a different object from a ceiling that is arithmetic: the first says the instrument is fine and the defaults are stale, and the second says a whole class of plants is unreadable. A survey designer told the second would have excluded every specimen at a fine rung, which is most of the interesting ones — the fine rungs are where sunflower heads and pine cones live.

What the reading needs, stated as a rule #

The instrument now has three settings that depend on the rung rather than on the site’s history, and each has an arithmetic reason:

The lag window must satisfy 3m ≤ L < 3n. In practice: read at a short window first, take the spacing it returns as an estimate of m, and re-read at 3m + 6.

The seed must span a fixed height rather than a fixed number of nodes — about 0.076 of a circumference, which is 8 nodes at a rise of 0.01 and 40 at 0.0019. A seed shorter than that is a cluster and the run never finds a lattice.

And the azimuth grid must resolve the pattern, which at the fine end means about 1,150 samples rather than 384. The requirement scales with how finely the rule has to discriminate between competing minima, which tightens as the rise falls.

The first is a consequence of the readout’s own construction. The second and third are consequences of the model. All three were constants, and none of them should have been.