A plateau the instrument should have had

The counting window decides nothing on the settling table, across a factor of forty and 384 settled runs. That is a result about a table rather than about a window, and the difference is the whole of this essay.

The window has two real bounds. Below the first, the counter returns the previous rung of the same ladder. Above the second, it reports a coarser rung at every width there is. Neither bound produces noise, a refusal, or a wide error bar. Each produces a clean, plausible, wrong pair.

Where the floor comes from #

It is not an empirical curiosity and nothing about it was fitted. cylCount scores an offset only where it has six hops of that offset inside the band, and it refuses a band of fewer than twenty nodes outright.

So a window of w organs can return a family of at most w − 6, and the narrowest window that will read a given pair is max(20, larger + 6). That is an arithmetic prediction made from the counter’s own construction rather than from any measurement, and every one of the 384 runs in the table has its measured plateau at exactly that value.

What happens one organ below it #

The counter does not fail. It returns a different pair, and the different pair is a real one.

Thirty-seven of the 384 settled runs are read this way at twenty organs, and they divide into six pairs. 11/19 comes back as 8/11 and 12/19 as 7/12; 10/17 comes back as 7/10, 9/16 as 7/9, and both 8/15 and 7/15 as 7/8. Every substitute is a coarser rung of the run’s own ladder.

Why the wrong answer is the plausible one #

That is the part worth dwelling on, because it is not an accident of this counter.

The contact families of a lattice are closed under addition, and the two smallest members of whatever window is visible generate the rest. The families of a stem carrying 11/19 therefore include 8 and 11 genuinely; they are not noise and they are not an error. A band too short to hold nineteen hops still holds eight and eleven, so the counter reports the pair it can see, and the pair it can see is the previous rung of the ladder the stem is on.

An instrument whose failure mode is to return a slightly coarser reading of the same object cannot be caught by looking at its output. The output is what a coarser stem would have given.

Where the switch is #

Exactly where the arithmetic puts it, measured one organ at a time on the table’s own stems.

11/19 reads 8/11 from twenty organs to twenty-four and 11/19 from 25. 12/19 reads 7/12 until twenty-four and 12/19 from 25. 10/17 reads 7/10 until twenty-two and 10/17 from 23. Nineteen plus six is twenty-five and seventeen plus six is twenty-three, with nothing adjusted to make it so.

Nothing reports the substitution #

There is no flag, no widened interval and no refusal. A run read at twenty organs returns a pair, the pair sits on an additive sequence, and the sequence is one the table already holds.

That is what makes the failure silent in the strong sense: not that it is hard to notice, but that there is nothing to notice. Every downstream check this collection applies to a count — that it is two integers, that they are coprime, that they are consecutive terms of an additive sequence, that the sequence has a limit angle near the destination — passes on the substituted pair as readily as on the true one.

The census reads it as a smaller table #

The one place the substitution shows is in aggregate, and only against a reading that is already known to be right.

Read at twenty organs, the table at forty starting angles reports 26 arrangements and 14 additive sequences where it holds 31 and 15. Five destinations return something the standing window never gives, five arrangements go with them and one sequence goes with those, and every number in that column is a count of real, legible pairs. A reader with only that column has a smaller, tidier, internally consistent table and no way to tell.

How much of the table is anywhere near the floor #

Most of it is nowhere near. The larger counted number across the 384 runs is 5 on 92 of them and 7 on 90, and the whole distribution runs 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 and 19 with the mass at the coarse end.

Only 37 runs need more than twenty organs, and the eleven that need twenty-five are the finest rise at every falloff exponent. So the floor is a bound that four fifths of the table never approaches, and the fifth that does is concentrated in one corner of it — which is the shape that makes a bound easy to miss and easy to underestimate once found.

The ceiling nobody had looked at #

The second bound is not a window at all, which is why varying the window could never find it.

cylCount considers offsets up to 60 and no further, and says nothing when that binds. A window of two hundred organs allows a family of 194, so the ceiling is 3.2 times tighter than the setting that had been under suspicion. Two settings decide what the counter can return, and the one that had never been named is the one that binds.

What a ceiling failure looks like #

Also like an answer. For every golden rise finer than 0.0002 the counter returns 34/55, at every window from a hundred organs to two thousand — a pair that is stable, consistent, and one rung of the ladder too coarse.

Raise the ceiling and the same stems return 55/89 at a rise of 0.00008, 89/144 at 0.00003 and 144/233 at 0.00001. Those are three different answers where the counter had been giving one, and the one it had been giving is a genuine family of every one of those lattices rather than a misreading.

Three pairs it cannot read at any width #

The ceiling is a real bound rather than an unreached one, and the demonstration is on ideal lattices built at a stated divergence and rise where the pair is known before the counting starts.

55/89, 47/76 and 76/123 are read at no window at all until the ceiling is raised. Every one of them is then read at the width the floor rule predicts, which is the same arithmetic holding on the far side of a bound that had been hiding it. How wide a window a known pair needs is measured across both ladders, and that measurement is what puts the two bounds on the same axis.

The five destinations that move #

They are not a scattered handful. Read at twenty organs, exactly five of the nineteen destinations return something the standing window does not: 47.9°, 106.0°, 132.3°, 151.0° and 157.3°.

Those are precisely the destinations whose larger counted number is 15 or more, with no exceptions in either direction. The floor is not a probabilistic thing that catches the occasional awkward run; it is a sharp condition on a quantity that can be read off the answer, and it partitions the table’s destinations into two groups with nothing in between.

Why this table is inert #

Because its largest counted number is 19, and 19 is a long way inside both bounds.

At two hundred organs the window allows 194, which is ten times the largest number the table returns, and the standing window sits eight to ten times above the plateau every run in the table has. On the other side, 19 is 32 per cent of the way to the ceiling. There is no bound anywhere near this table, which is why seventeen windows across a factor of forty found nothing.

A setting on a plateau, and a setting that is not #

The word plateau is doing specific work here and it is worth separating from harmless.

The reading window in the ablation work was varied and it changed the verdict on three rows. That setting sits on a slope: every value of it gives a slightly different answer, so there is no width at which the answer stops depending on the width, and the right response is to report the dependence.

This one sits on a plateau between twenty-five organs and eleven hundred, which is a qualitatively different thing to be able to say. A setting on a plateau can be defended by pointing at the flat part. A setting on a slope can only be reported.

Right by luck rather than by design #

Two hundred organs is a good setting for this table. It is a good setting for no stated reason: it was not chosen against the floor, it was not chosen against the ceiling, and the quantity that decides whether it is safe — the largest number the counter is going to return — is not known until after the count.

That is the honest description of every default of this shape. It is not wrong, it is not justified, and the two are indistinguishable from the output.

What the setting should have been #

Not a fixed number. The narrowest sufficient window is max(20, larger + 6), computed from the counter’s own answer, and reading every run at its own plateau instead of at two hundred organs moves no published number at any sampling.

So the instrument should have had a plateau built into it: count once at a generous width, read the larger of the two numbers it returns, and check that width against the floor that number sets. A count that fails the check is a count to re-take, and re-taking costs three milliseconds. What that buys is not accuracy on this table — there is none to buy — but a count that knows when it is out of range.

Nothing has to be regrown #

The check is cheap in the way that matters, which is that it needs no new stems.

Counting one window once is about three milliseconds, and the whole seventeen-window sweep over the table reads back from what was already computed in six tenths of a second. The expense in this collection has always been growing the stems; the counting is a rounding error on top of it, and re-reading an answer at a second width to see whether the first width was wide enough is cheaper than the sentence explaining why nobody did.

That is the argument for making the floor check part of the counter rather than part of a protocol somebody remembers.

And what the ceiling should have been #

Harder, and the essay should say so rather than propose a number.

A ceiling has a cost the floor does not: the counter scores every offset up to it on every band, so doubling it doubles the work. Sixty is a defensible choice for a collection whose counted numbers run to 19. What is not defensible is that it is silent — an offset ceiling that binds is a fact about the answer, and reporting it costs one comparison.

Five different ceilings are in use across this collection’s libraries: 90, 60, 40, 34 and 30. None of them is stated as a measurement anywhere, and a stem counted under one and recounted under another is two functions doing different work under one name.

The default is inherited, and the bound is not #

Eight further libraries in this collection take a count through the same wrapper at the same two hundred organs, on stems grown for other questions entirely.

Whether each of them sits on the plateau is a separate measurement, because the floor is set by the counted number rather than by the window: a library working at finer rises counts larger families and its floor is higher, while one working on short stems may not have two hundred organs of settled patch to read. The null measured on the settling table transfers to none of them. What transfers is the check, which is one comparison between a setting and an answer, and which none of the eight currently makes.

The founding bug had this shape #

The collection has been caught by exactly this before, and the record is worth putting beside the two bounds.

The site’s first spiral counter returned 21 and 34 for a head whose nearest families are 34 and 55. Both numbers were real families of that head, twenty-one spirals were drawn and there were twenty-one of them, and every check that read the picture agreed with it. What caught it was a recovery step that could refuse — asked which divergence angles make 21 and 34 the shortest offsets, it found none.

The floor here produces the same class of error and there is no recovery downstream to refuse it, because a coarser rung of the same ladder is a consistent pair with a real angle behind it.

An instrument that cannot refuse #

That is the general statement and it is the one worth carrying out of this thread. An instrument that can refuse has a second output, and the second output is where errors show up first.

A window has no second output. It returns a pair at every width, and below the floor it returns a pair that is better-behaved than the true one — smaller numbers, a shorter sequence, a tidier census. Every property a reader would use to sanity-check the answer points the wrong way.

What this does not claim #

No run in the settling table is past the ceiling and no published count in this collection was taken below the floor. Nothing here withdraws a number.

The floor is demonstrated on the table’s own stems and the ceiling is not: the ceiling failure is shown on lattices built at rises finer than the table goes, which is a demonstration on constructed objects rather than a finding about the settling work. What the two share is only their silence, and that is a property of the counter rather than of either sweep.

What would refute the floor rule #

A run whose measured plateau is not max(20, larger + 6). There are 384 of them and every one is at that value, and the rule was written down from the counter’s construction before the plateaus were read rather than fitted to them afterwards.

The stronger refutation would be a pair read below its own floor as something other than a coarser rung — a refusal, or a pair from a different sequence. Six substitutions across thirty-seven runs are all coarser rungs of the run’s own ladder, which is what makes the failure predictable as well as silent.

The two bounds are not symmetric #

Worth separating, because they are easy to state as one thing.

The floor moves with the answer: it is a property of the pair being counted, it can be computed once the count exists, and a run can be re-read at a wider window for nothing. The ceiling does not move with anything. It is a fixed refusal to look, it cannot be detected from the answer, and the only way past it is to raise it and count again.

So the floor is a checkable condition and the ceiling is a scope limit. A count should report the first and a library should state the second.

What a survey would have to record #

Three numbers, none of which this collection’s counts have carried: the width the count was taken at, the largest offset the counter was willing to consider, and the larger of the two numbers it returned. The first two are settings and the third is the answer, and the check is one comparison between them.

That is the same discipline as reporting what a summary throws away — a quantity the instrument already has, and that nothing downstream can reconstruct once it is dropped.

What a reader should carry #

That a count has a range, that this counter’s range runs from the larger counted number plus six up to sixty, and that both ends of it fail by returning a neighbouring rung rather than by complaining.

The settling table sits in the middle of that range by luck. Being in the middle of a range is the only reason a sweep of the window returned a null, and it is not a property anyone chose.

The one line #

The counting window has a floor at the larger counted number plus six and a ceiling at offset sixty; a window one organ too narrow returns the previous rung of the same ladder and a family past the ceiling is reported as a coarser rung at every width there is — and neither failure says anything at all.