A width read off a staircase

Fourteen fractions were measured for the previous essay, twelve inside three families and two outside them. Ten returned a dip half-width that agrees between two head sizes to within three and a half per cent, once the law’s own n² scaling is applied. One agreed to twenty-one per cent, for a reason already found here.

And two disagreed by factors of 1.6 and 2.4.

They are 9/34 and 24/55, and this essay is about why — because the answer turns out to be a property of the statistic rather than of those two fractions, and it sets a resolution on every width in the thread.

The first guess, which was wrong #

The obvious explanation, and the one written into this collection’s own working notes before the diagnosis was done, is the neighbourhood. This thread’s standing difficulty has been that a dip is measured against a background and a background is a stretch of neighbouring angles; an earlier essay found that a single background sample 0.2° from 21/55 lands inside 13/34’s dip, reads low, and inverts the verdict.

So a fraction whose nearest rational sits unusually close would have its shoulder lifted by that neighbour’s dip, and the half-way crossing would land somewhere that has nothing to do with its own width.

That is not what is happening. 24/55’s nearest rational is 17/39, a sixth of a degree away; 9/34’s is 14/53, at a fifth of a degree. The offsets a width is read over run to five hundredths of a degree. The neighbours are three times outside the band.

What is actually happening #

Print the profile — the second moment against the offset from the rational — and the shape is not a curve.

At 24/55 on a head of 1,279 organs, reading outwards from the exact rational, μ₂ runs

0.094 0.094 0.094 0.094 0.094 0.094 0.094 0.095 0.095 0.112 0.130 0.131 0.131 0.132 0.132 0.133 0.134 0.200 0.261

Nine identical values, a jump, two more jumps, six identical values, another jump. It is a staircase: flat treads separated by risers — the same shape the whole curve has, seen at a hundredth of the scale.

That is not a numerical accident. The second moment is a mean of squared departures from a common side count, over a finite head, and a cell’s side count is an integer. As the divergence angle is moved continuously, nothing changes until some cell in the tessellation gains or loses a neighbour, and then the statistic jumps. Between those events it is exactly constant.

It is worth seeing the same shape at a fraction that measures cleanly, so that the staircase is not read as a property of the two failures.

At 21/55 on a head of 1,279 the profile runs 0.094 nine times, then 0.095 four times, then jumps to 0.134, 0.172, 0.173, 0.174, 0.175, and on up. Same structure: long treads, sudden risers. At 13/34 on 791 organs it runs 0.071 twelve times, then 0.072 three times, then 0.119, 0.119, 0.120, 0.120, 0.121, 0.122, then 0.192.

Every profile in this thread is a staircase. What differs between the fractions that measure and the fractions that do not is only where the half-way level happens to fall.

Why that makes a half-width unreadable, sometimes #

The half-width is defined as a level crossing: find where the profile has climbed half way from the dip’s floor to its background, and report the offset.

On a staircase, a level crossing has two cases.

The level falls inside a riser. The crossing is where the riser is, the interpolation between the two sampled offsets either side is a good estimate of it, and the answer is reproducible.

The level falls on a tread. There is no crossing there at all. The interpolation returns the offset of the next riser’s foot, which is a property of where the tessellation happens to change rather than of the dip’s shape — and when the head size changes, the tessellation changes at different angles, so the treads move and the reported width moves with them.

At 24/55 the half-way level is 0.125. The profile goes 0.112 at an offset of 0.0025° and 0.130 at 0.003°, so the level falls inside a riser and is reported at 0.00285°. On the larger head the level is 0.090 and the profile goes 0.084 at 0.0037° and 0.097 at 0.0045°, giving 0.00405°. Both are honest readings of where a riser is; they are not readings of the same quantity.

For comparison, 21/55 on the same two heads gives 0.00554° and 0.00234°, which after the n² scaling agree to one per cent.

There is a tempting misreading to head off. It is not that the two refused fractions have narrower dips than the others — 24/55’s floor and background are ordinary, and its width, whatever it is, is somewhere in the range its family occupies. It is that the half-way level lands badly for them. A fraction with a perfectly typical dip can be unmeasurable by this method and a fraction with an unusual one can measure cleanly; which happens is decided by the arithmetic of where the tessellation flips, and that has nothing to do with the dip.

There is a second repair available and it is taken up elsewhere: define the width by an integral over the dip rather than by a level, which is insensitive to where the treads fall because it adds them all up. That is the better instrument and it brings a free parameter of its own — a limit — which is why it is a separate essay rather than a paragraph here.

Which means the refusal carries no information about the fraction. It is not a finding about 9/34 or 24/55. It is a finding about the method, and the two fractions are the two that happened to expose it.

The repair is a bracket rather than a number #

A level crossing on a staircase is not a badly measured number; it is a quantised one. The width can only come out at a riser, because the risers are the only places the profile crosses anything, so the reported value is drawn from a discrete set fixed by where the tessellation happens to flip.

That changes what the right output is. A quantity that can only take values at particular angles should be reported as the interval between the riser below the level and the riser above it — a bracket rather than a point — and a fraction whose bracket is wide compared with the width itself should be refused with its bracket attached rather than dropped.

The difference is not cosmetic. A refusal says nothing; a bracket says the width of 24/55 lies between two stated offsets, which is a bound, and a bound is a measurement. Both of the two fractions this essay is about would then contribute to the family comparison instead of leaving a gap in it, at the cost of contributing a range rather than a value.

It also has a consequence the previous essay was already relying on without saying so. Four fractions of one denominator have their risers in four different places, because each has its own tessellation flipping at its own angles — so a family mean is not four noisy readings of one number but four differently-quantised readings, and averaging them beats the tread spacing in the way averaging usually does not beat a systematic error. That is the reason a family of four is worth measuring rather than one fraction four times, and it is an argument the family design deserved and had not been given.

The resolution this puts on the whole thread #

The useful form of the finding is not “two fractions failed”. It is that the method has a resolution, it can be stated, and every width in the thread should be read against it.

The tread spacing near the half-way level is what sets it. On these heads the risers near the crossing are a thousandth to a few thousandths of a degree apart, which is the same order as the widths being measured — 5.5 × 10⁻³ at 21/55, 1.1 × 10⁻² at 13/34. So the thread has been operating within a factor of a few of its own resolution throughout, and it did not know — which is the shape of every defect this collection keeps finding: a number that was correct, plausible and reported without the scale it should have been read against.

Two consequences follow, and one of them is reassuring.

The measured widths are good to about a tread. Which is to say a few tenths of themselves for the narrow ones and a few per cent for the wide ones — and that is consistent with the agreement actually observed between head sizes, which is what makes the account believable rather than merely available.

And the law is safe, because the law is a scaling over a factor of eleven in denominator and a factor of two and a half in head size. A resolution of a few per cent to a few tenths does not touch a factor-of-twelve trend, which is why the earlier result stands and only its precision is at issue.

Why the check was two head sizes and not two anything else #

The disagreement that produced this essay came from measuring each fraction at two head sizes and comparing through the law’s own scaling. That choice is worth defending, because it is the only one of the available checks that would have worked.

Comparing two fractions would not: the whole question of the previous essay is whether two fractions should agree, so a disagreement is ambiguous between a defect and a result.

Comparing two background methods would not: that was already done here, and the background was fixed, and a staircase in the profile survives any choice of background level.

Comparing two sampling grids on the offsets would not, for the reason given below: the risers are already resolved.

Two head sizes work because the law says exactly what should happen — n²w is constant — and because changing the head changes where the tessellation flips without changing the dip. It is a comparison with a prediction in it, which is what makes a disagreement diagnostic rather than merely a spread.

One number is worth recording for whoever does the repair, because it is the thing that makes the area version attractive. The dip’s depth — the distance from its floor to its background — is a difference of two numbers each of which is stable to a per cent or so, and it is reproducible at every fraction measured here, including the two whose widths are not. 24/55’s floor is 0.094 at 1,279 organs and 0.060 at 1,980, and its background 0.218 and 0.179; the ratio of the two falls in the same proportion as every other fraction’s.

So the depth is measurable where the width is not, and an area is a depth times a width. A quantity built from the profile as a whole rather than from one level crossing on it would inherit the depth’s stability, and the two refusals in this essay would become measurements.

What would fix it #

Three repairs, none of them run here, and it is worth being explicit about why not.

Read the width from an area rather than from a crossing. The integral of the profile’s departure from its background is a smooth function of the head size even when the profile is a staircase, because integration does not care where the risers are. It is the right repair and it brings a worse problem with it, and because it changes the definition of the quantity it cannot be done without re-measuring every width the level crossing produced — which would mean reporting a different law from the one this set out to test.

Average over head sizes. Cheap and partial: it reduces the tread’s influence without removing it, and it costs a factor of three in computation on the thread’s most expensive machinery.

Or sample the profile densely enough to find the risers. This does not help. The problem is not that the risers are missed — the sampling already resolves them — but that the level being sought does not always fall on one.

The first is the one a later essay should do, and doing it would settle whether the residual ordering by crowding in the previous essay is real or is a second face of this same effect.

What a staircase says about the dip itself #

Set the measurement aside for a paragraph, because the staircase is not only an obstacle. It is a statement about what a rational divergence does to a head, and it is a sharper one than the dip.

At an exactly rational angle the organs fall into q radial rows, and the tessellation is as regular as that arrangement allows. Move the angle by a millionth of a degree and nothing happens at all — not a small change, no change: the same cells have the same neighbours and the statistic is identical to the last digit. The arrangement’s combinatorics are unchanged over a finite interval of angles.

That interval is the first tread, and its width is the dip’s true floor: the range of angles over which a head of n organs is combinatorially the rational head. Everything the dip’s width is trying to capture is a coarse-grained version of that.

It also says what a real plant would have to do to sit in a dip. Not to have a rational divergence — no measurement could establish that and no plant could hold it — but to have one within the first tread, which for a head of a thousand organs at denominator 55 is a few thousandths of a degree. The dip is not a region of near-rationality; it is a region of exactly the same tessellation, and its edges are the angles at which the first cell flips.

That reading was available from the earlier data and nobody looked. It is the more interesting half of what this essay found while diagnosing a failure.

Where this fits in the collection #

This is the fourth defect on this site whose symptom was a plausible number.

A spiral counter returned the two smallest index offsets instead of the two shortest surface hops, and was believed until a recovery step refused. Ticks vanished from an axis with a descending domain, and two figures carried no gridlines for a long time while every check passed. A single background sample landed inside a neighbour’s dip and inverted a verdict. And a width has been read off a staircase at a resolution comparable to itself.

All four share a shape: the output was well-formed and about something else. No assertion failed, because nothing had been asserted about the thing that was wrong.

The habit that catches them is not more assertions of the same kind. It is measuring the same quantity a second way and requiring the two to agree — a recovery against a count, a second window against a first, two head sizes against each other. The refusal in this essay exists because somebody made the width measure itself twice, and the whole of the finding is the disagreement.

One more thing is worth recording, because it is the sort of detail that gets lost between one piece of work and the next, and then costs a day. The two refusals were not found by a gate. There is no check on this site that asks whether a width is measurable; what exists is a check that asks whether the reported widths agree, and it exists because the previous essay needed a stability criterion for a different reason — to decide which fractions could enter a family comparison.

So the refusal is a by-product of a threshold introduced for bookkeeping. That is worth noticing rather than celebrating: a defect whose discovery depended on somebody needing an unrelated criterion is a defect that could as easily have gone unnoticed for longer still, and the collection’s other three of the same shape were all found the same way.