How many organs a pair needs

The counter that reads a stem is handed a window: a number of organs at the top of the run, with everything below it ignored. That number has been two hundred everywhere it appears in this collection, and moving it by a factor of forty changes not one answer in the settling table.

A null is where a question gets interesting rather than where it stops. A setting that decides nothing on one table can decide everything on the next, and the useful form of the answer is not two hundred was safe but here is what a window has to be. There is such a rule. It is not a constant, it is not fitted, and it can be read off the counter’s own machinery before a single stem is grown.

The counter’s two requirements #

The instrument works by scoring index offsets. For each candidate offset it walks the hops of that offset inside the band it was given and asks how straight and how short they are, and it will not score an offset at all unless it can see six of those hops inside the band. Six is not a tuning parameter that happened to work; it is the number of samples the score is an average over, and an average over fewer would be an average over noise.

It has a second refusal, further out. A band holding fewer than twenty nodes is not counted at all, on the grounds that a patch that small does not have a lattice in it in any sense worth reporting.

So the floor is arithmetic #

Those two requirements are the whole of the rule. A window of w organs holds w − k hops of the offset k, because each hop joins an organ to the one k places above it and the top k organs have nothing above them to reach. Needing six of those hops means k cannot exceed w − 6. Turn it round: to read a pair whose larger member is m, the window must be at least m + 6, and at least 20 whatever m is.

The smallest window that reads a given pair is max(20, larger + 6). Nothing in that sentence was measured. It is what the counter is, stated as a number.

Why a known pair is the only honest test #

A rule derived from machinery and checked against the machinery’s own output is a circle unless the check is made where the answer is known first. So the check is made on ideal cylinders: lattices built at a stated divergence and a stated rise, whose pair follows from the construction, then handed to the counter at every window from twenty organs to two hundred as though nothing about them were known.

That is the same discipline the blind counters are built to — an instrument that is told the answer is not an instrument. Here it matters more than usual, because the quantity being measured is the point at which the instrument starts being right.

Thirteen of thirteen #

Sixteen rungs were built, eight on each branch, from a pair of 2 and 3 at a rise of 0.05 down to a pair of 76 and 123 at a rise of 0.00005. Thirteen of the sixteen the counter can read at all at its own settings, and for thirteen of those thirteen the smallest window that returns the true pair is exactly the larger count plus six, floored at twenty.

Not thirteen of thirteen within a tolerance. Exactly: the pair is wrong at one organ narrower and right at the predicted width, on both branches, at every rung. Nothing was fitted and there was nothing to fit — the rule has no free parameter in it at all.

The flat part is a refusal, not a property of the pair #

Seven of the thirteen sit at twenty organs: the golden rungs up to 8 and 13, and the Lucas ones up to 7 and 11. Their arithmetic floor is lower than twenty — a pair of 2 and 3 needs nine organs by the hop rule — and twenty is where they are read because twenty is where the counter stops refusing.

That distinction is worth keeping straight when reading the figure. The flat stretch on the left is not a fact about small pairs. It is the band refusal, and it would move if the refusal moved. The sloping part is the fact about pairs.

What thirteen and twenty-one costs #

A pair of 13 and 21 needs 27 organs and a pair of 2 and 3 needs twenty. That is the whole practical content of the rule and it is the sentence the collection has never been able to write: how much stem a given count requires before it can be taken at all.

It scales the way the ladder does, which is to say it barely scales at all against what the pair itself is doing. Climbing four rungs of the golden branch multiplies the counted numbers by about seven and the window they need by three, from twenty organs to sixty-one. The window is linear in the count while the count is geometric in the rung, so the expensive part of reading a fine lattice was never the reading.

The three the counter cannot read at all #

Three of the sixteen return the wrong pair at every window from twenty organs to two hundred: the golden 55 and 89, the Lucas 47 and 76, and the Lucas 76 and 123. That is not the window failing. The counter looks at index offsets up to 60 and no further, and a family of 76 is not on the list of things it is looking for.

So the instrument has two bounds and they fail in opposite directions. Below the floor it reports a coarser pair because it cannot see enough of the stem; above the ceiling it reports a coarser pair because it is not looking far enough along the offsets. Neither announces itself.

And they land on the rule once the ceiling moves #

Raise the ceiling and all three are read, at 95, 82 and 129 organs — which is the larger count plus six in every case, to the organ.

That is the strongest single line of evidence the rule has, and it is worth being clear about why. The three unreadable rungs were not used to derive anything; they sat outside the instrument entirely. When the obstruction is removed they fall on a rule computed from sixteen numbers none of which is theirs. A rule that survives being extended past the range it was stated over is doing more than describing that range.

Both branches, one rule #

Sixteen rungs, two branches, a rise range spanning three orders of magnitude, and one expression covers all of it with no branch term and no rise term.

The absence of a rise term is the part that could have gone otherwise. A finer rise packs more nodes into the same number of organs, so it would have been reasonable to expect the required window to depend on it. It does not. The counter counts organs and the pair it is looking for is a pair of index offsets, and neither of those knows anything about the geometry the offsets sit in.

The table’s own stems say the same #

The ideal lattices are the clean test and the settling table is the real one: 384 settled runs across nineteen destinations, grown at eight rises and four falloff exponents from forty starting angles, each counted at seventeen windows spanning a factor of forty.

Each run has a plateau — the narrowest window from which its answer never changes again. 384 of 384 plateaus are exactly max(20, larger + 6) for the run’s own pair. There are two distinct plateau values in the whole table, twenty organs and twenty-five, because the table’s counted numbers are small.

What the table actually asks for #

The distribution is lopsided and that is why the standing setting was never tested. The largest counted number anywhere in 384 runs is 19, and it occurs eleven times. Ninety-two runs count a 5 and ninety count a 7; a single run counts a 10.

So the widest floor anything in the table needs is twenty-five organs, against a standing window of two hundred. The setting sits eight to ten times above the largest demand made of it, which is why a sweep of the window returns a null and why that null is a fact about this table rather than about counting.

Thirty-seven runs and a legible wrong answer #

Read every run at twenty organs — the narrowest window in the sweep, and one this collection has never used — and 37 of 384 return a pair their window cannot hold.

Not one of them refuses. Every one of the 37 returns a pair, and every pair it returns is a real pair on the run’s own ladder: the rung below the true one. A run counting 11 and 19 returns 8 and 11. A run counting 12 and 19 returns 7 and 12. A run counting 10 and 17 returns 7 and 10, and every run counting 9 and 16 returns 7 and 9.

The failure has no signature #

This is the property that makes the floor worth stating rather than merely satisfying. A too-narrow window does not produce noise, a wide error bar, a refusal or an implausible number. It produces a clean, legible, wrong pair that sits on the same additive sequence as the right one, one rung down, and it produces it at every window below the floor and none above.

Anyone reading the output has a pair, a sequence and a destination, all mutually consistent and all wrong. The signature that a count was taken over too little stem is that there is no signature. Compare what happens when a count is taken at the wrong radius on a disc, where the answer changes across the head and the change is visible in the head.

Located to one organ #

A finer scan settles it. Running the same three stems at one-organ steps rather than at the seventeen listed widths puts the switch exactly where the arithmetic puts it: the 11 and 19 run reads the rung below from twenty organs to twenty-four and its own pair from 25; the 12 and 19 run switches at 25; the 10 and 17 run switches at 23.

Twenty-five is 19 + 6 and twenty-three is 17 + 6. A rule that predicts the switch to a single organ on three independent stems is not a summary of the data; it is the mechanism, and the data is a check on it.

What the standing setting is worth #

Two hundred organs is a safe window for this table and it is not a window anyone chose. It is safe because the table’s largest count is 19 and 19 + 6 is 25.

The honest reading is that the setting has been inert rather than correct, and the two are different claims about the same number. A collection whose stems went one ladder rung finer would need 31 organs and still be safe; a collection counting a family past 194 would be reading at a width that cannot hold it, with no indication of it anywhere. The setting the instrument should have had is the rule, computed from the answer the instrument itself returns.

The window is not what decides the pair #

Worth saying plainly, because a rule about how much stem a count needs invites the inference that the count depends on how much stem it was given. It does not. Above its own floor, every run in the table returns one pair and returns it at every width from twenty-five organs to eleven hundred.

The floor is a boundary between no answer worth having and the answer, not a gradient between two answers. That is what makes it a floor and not a resolution: past it, nothing further happens.

What the rule is not #

It is not a claim that a window of max(20, larger + 6) organs is a good window. It is the narrowest one that returns the right answer on an ideal lattice and on a settled stem, which is a different property from being robust to anything a real stem might do.

Nothing here was measured on a noisy patch, a stem still settling, a patch with a defect in it, or a lattice whose pair is changing across the window. Each of those is a reason to read wider than the floor, and none of them is priced by this measurement.

It is a statement about this counter #

The six hops and the twenty nodes belong to one instrument. A counter that scored an offset over four hops would have a floor of larger + 4, and the counter that is handed angles and no positions is not covered by any of this, because it is not given a band of organs in the first place.

So the transferable part is the form of the argument rather than the constant in it: an instrument’s minimum sample is computable from what the instrument requires per sample, and computing it is cheaper than sweeping for it.

The ceiling is the other bound and it is not this rule #

The floor is exact and the ceiling is a separate object with a separate story. Sixty offsets is a setting this collection inherited, and it is 3.2 times tighter than the largest family a two-hundred-organ window would allow, which is 194.

What it does when it binds is the same silent thing the floor does. Every golden rise finer than 0.0002 returns 34 and 55 at every window from a hundred organs to two thousand, and raising the ceiling turns those into 55 and 89, then 89 and 144, then 144 and 233 as the rise falls. The window was never the constraint there and no amount of widening it would have been.

Where the setting reaches #

Eight of this collection’s measurement libraries take a count at the standing window without ever naming it, because the count is taken through one shared call and that call carries the default. So the floor is not a property of one table; it is a property of every statement any of them has made.

All eight are safe by the same arithmetic and for the same reason — none of them counts past 19 — and safe-by-accident is what this measurement converts into safe-by-check.

What would refute it #

A single settled run whose plateau is not max(20, larger + 6) for the pair it returns. Not a run that is noisy near its floor, and not a run that refuses: a run that gives a stable answer from a window narrower than the rule allows, or that keeps changing above it.

Three hundred and eighty-four runs and sixteen ideal lattices have been asked and none has done either. The check is cheap enough to attach to any future sweep — counting one window takes about three milliseconds, and the whole seventeen-window sweep over 384 runs is three per cent of the cost of growing the stems it reads.

The refusals that make the answer worth having #

The reader that assembles a pair from these windows refuses four things and answers a fifth: a pair claimed from a single window, a window narrower than the floor for the family it claims, two windows that disagree, and two families that wind the same way. What is left — a pair two wide-enough windows agree on, whose families cross — is answered, with the sequence it sits on.

The fourth refusal is the one with a published consequence, and it is a separate argument. What matters here is that the refusals discriminate: a machine that refused everything would satisfy the same assertions and prove nothing at all.

What a rule buys that a constant does not #

A constant has to be defended once and then trusted. Two hundred organs was defended by nobody and trusted by every count on the settling table, and it happened to be right for a reason nobody had written down.

A rule is checkable at the point of use. It costs the counter one comparison against an answer it already has, it names the width at which the answer would have been different, and it fails loudly on the one case a constant fails silently on. The arithmetic that prices what a reported pair is worth has the same shape: a number that is a function of the pair rather than a habit attached to the instrument.

The one line #

A count of a lattice needs its larger family plus six organs, floored at twenty, and needs nothing else. Every one of 384 settled runs and thirteen of thirteen readable ideal rungs sits exactly there, on both branches, with nothing fitted — and a window one organ short of it returns the rung below, cleanly, forever, with nothing anywhere to say so.