Six lattices were not enough

A growing tip on a cylindrical stem sits in a slot. Two organs make its walls: the one that sits a smaller counted number of places back, and the one that sits the larger counted number back. Those two are its chain-neighbours, and between them is the gap the rule puts the next organ into.

Removing either wall alone is a cheap removal everywhere it has been tried. Removing both together is not the sum of removing each, and the difference — the interaction — was the measurement that thread was built to make.

What six lattices said #

They said the interaction is large and its sign is unaccountable. It ran from −25.8° to +132.9°: three lattices well above zero, three well below, and nothing in the row — not the rise, not the counted pair, not which branch of the ladder the lattice sat on — that put the two groups on different sides of anything.

That is a real result about a design and a weak one about the world. Six numbers either side of zero is what a quantity with structure looks like when it has been sampled six times, and it is also what a quantity with no structure looks like. Six rows cannot tell those apart, and the sentence that was written down — with no ordering by rise, counted pair or branch — is a list of three columns that were looked at, not a statement that no column would work.

It is the same shape as the shortest hop that turned out to be a coin flip: a rule scored on a handful of rows, coming out at chance, and the honest reading being that the rows were too few rather than that the rule was wrong.

The candidate that was named #

The last time a quantity here looked free, the column doing the work turned out to be position inside the rung — how far down its own rung a lattice had been grown, measured in the logarithm of the rise rather than in the rise itself. It is the variable this site keeps finding underneath things, and it was the obvious one to test.

It is also a variable the six lattices could barely address. Four of them sit on one golden rung, at 34%, 61%, 84% and — for the coarsest — 89% of the way down a different one. Two sit on Lucas rungs. That is not a sweep of a rung; it is four points on one and a scatter elsewhere.

The design #

Three lattices on every rung of the ladder, at 15%, 50% and 85% of the way down it in the logarithm of the rise. Eight rungs across the two branches gives twenty-four new lattices, and keeping the original six gives thirty.

Three positions is the smallest number that can show a trend rather than a difference. Two points make a line whatever they do; three can fail to, and a rung whose three lattices do not lie on a line is a rung saying something the design would otherwise have had to assume.

The positions are measured in the logarithm of the rise because the ladder is geometric. A rung near the coarse end spans two hundredths in the rise and one near the fine end spans two thousandths, so an absolute step is three different instruments on one ladder and a proportional one is a single instrument.

What thirty rows are powered to say #

A design should state what it could find before it is run, so that a null result is a measurement rather than a shrug.

Thirty rows scored against a continuous column give a rank correlation, and at thirty points a correlation has to reach about 0.36 before it would be produced by chance one time in twenty. That is a moderate association — the sort a scatter with a visible slope in it has — so the design can find an ordering by position and cannot find a weak one.

Six rows could reach nothing: at six points a rank correlation needs 0.89, which is one swap away from perfect. So the earlier design was not merely under-sampled in the column it needed; it could not have established any ordering at all, in that column or any other, however the six had come out. That is a stronger criticism of it than “three above zero and three below” and it is the one worth carrying, because it applies to every column the six were sorted by rather than only to the three that were tried.

Where the six actually sat #

The original six also sit in one part of a rung, and it is the part this collection has learned to be suspicious of.

Their positions are 34, 61, 84 and 89 per cent on the golden rungs, with two more on Lucas ones. Nothing below a third of the way down, and three of the four golden rows past four fifths.

That matters because every handover on this ladder sits in the coarse half of its own rung, between six and forty per cent. So the six slot lattices are almost all grown past their rungs’ handovers, on the side where the larger contact step is the shorter one — and the coarse third of every rung, where the ordering is the other way round, is sampled by nothing at all.

That is the same one-sidedness the ablation census turned out to have, arrived at independently and for the same reason: a rise chosen to give an unambiguous counted pair is a rise chosen away from a rung’s edges, and the edges are where the coarse half is.

The new design’s 15 per cent row is the repair. It is the first lattice in this thread grown on the coarse side of a handover on purpose, and if the interaction’s sign has anything to do with the ordering of the two contact steps, that row is where it shows.

Why not the ends #

Nought and one are not sampled, and the reason is that a rung’s boundary is where the ladder sweep has already declared the counted pair to be in transition. A lattice grown there has a pair in doubt, and the whole design names two offsets from that pair. Doubt about which organs the walls are would be doubt in every cell of the two-by-two rather than in one of them.

Fifteen and eighty-five per cent are far enough in that the pair is settled at every one of the thirty, which is checked rather than assumed: the walls are counted from the points of the very stem the cuts are made in.

Keeping the six #

The six original lattices are rows of the new table rather than a thing it replaces. Every number the earlier design reported is recomputed here by the same call and answered from the same store, so the two tables cannot disagree about a row they share.

A design that re-sampled the ladder from scratch would have made every comparison an argument about two designs. This way, if the older six now read differently, the reading has changed and not the arithmetic — and they do not.

What it cost #

Four seconds a lattice, and ninety runs: three cuts and their controls at each of thirty rises. The expensive thing in this thread has never been the cutting; it is the stems the cuts are made in, and those are shared between the cells of a two-by-two by construction.

That is worth saying because the reason the design had six lattices was never cost. It had six because six looked like enough for a table with a sign in it.

The first reading: the range widened #

Thirty lattices run from −104.3° to +135.2°. The new extreme is at the negative end and it is nearly four times the most negative of the six.

A widened range is the least interesting thing a bigger sample can do and it is worth pausing on anyway, because it says the six were not a spread of a distribution. They were part of one, and not the part that reaches furthest.

And the table came apart into two groups #

The second reading is the one that matters, and it is not a spread at all. On six of the thirty lattices, removing both walls costs exactly what removing the larger one alone costs — 35.9° and 35.9°, 12.0° and 12.0° — agreeing to the last digit of the grid the azimuths are placed on.

On those rows the smaller wall is free. Taking it away as well changes nothing.

Which makes their interaction arithmetic #

The interaction is both, less the smaller and the larger added. When both equals larger, that is minus the smaller — exactly, by cancellation, with no measurement left in it.

So the −104.3° that widened the range is the smaller wall’s own cost written with a minus sign in front. It is a cell of the table restated, not a seventh observation, and a table that leaves those rows in has its three most negative entries produced by algebra.

Setting them aside is not discarding them #

They are a finding of their own, and they get their own account. Three of the six are the whole of one short rung and the other three are the fine ends of two rungs whose coarse ends are not free at all. Whatever puts a lattice into that regime is worth knowing.

What they cannot do is carry a claim about an interaction, because on them the quantity is not measuring one. Leaving them in would be the same error as reading a summary statistic past the point where it still has a quantity — the number keeps being produced long after the thing it was a number for has gone.

Twenty-four measurements #

That leaves twenty-four lattices where both walls are really there. Thirteen are strongly positive, at +85.1° to +135.2°; eleven are negative or nearly zero, at −25.1° to −0.5°. There is nothing between −0.5° and +85.1°: a gap of eighty-five degrees with no row in it.

A gap is worth more than a threshold. It means the two groups would still be two groups wherever anybody drew the line, and nobody had to choose where.

The preview #

Every rung’s lattices fall on the same side as each other. All of them, across both branches and a factor of thirteen in the rise. Which rung a lattice is on decides the sign, and where in the rung it sits does not.

That is the answer to the question the design was built to ask, and it is a negative: the candidate was refused, and the column that does the work is the one the candidate was going to be tested against.

What position does do #

It moves the interaction by three to fourteen degrees across a whole rung, and it moves it downwards on six of the seven rungs that carry more than one measured lattice. Against a spread of a hundred and sixty degrees across the ladder that is a drift, not a sorting.

And it decides something else entirely: both of the rungs that go free go free at their fine end. So position sets whether the slot has a second wall in it, and the rung sets what removing both of them costs when it has.

What six could not have shown #

Two of the original six are in the free regime and were counted as negative interactions. One more sits on a rung whose measured lattices are all near zero anyway. So of the six, three carried the positive signal and three carried something else, and the something else was two different things.

That is exactly the shape of a sample that produces “no ordering by anything”: it mixes regimes. The ordering was there and the design could not see it, which is a different failure from the ordering not existing.

What the bigger design still cannot do #

It cannot say why a rung’s larger counted number should decide the sign of an interaction between two removals. It has a rule that sorts twenty-two of the twenty-four rows and an exception on the twenty-third and twenty-fourth, and no account of the mechanism behind either.

It also cannot rule out that three positions on a rung are three positions. A quantity that moves at the fifth decimal place of the rise would look flat here exactly as position looked flat in the six — and this site has already found one that does, on a band whose answer changes at a rise of its own.

The honest form of the claim is therefore about a resolution as well as about a table: at three positions a rung, the interaction is set by the rung and drifts by under fifteen degrees inside it. A finer sweep of one rung would test that, and it has not been run.

The instrument is the same instrument #

Every cell of every two-by-two is a displacement of one organ against a control that shares its history to the last digit, on a stem whose heights are prescribed by the rise. Nothing here grows a stem at an angle and asks whether it looks right, and nothing compares two runs that do not share a past.

That is what makes thirty lattices comparable with six rather than a second experiment. The design got wider; it did not change.

Two lattices where the pair wrecks what neither wall does #

There is a third thing in the table, and it is the strongest form the interaction takes. On two of the thirty, either single removal heals — the stem recovers its divergence and its counted pair — and removing both wrecks it.

The earlier design had one such lattice. Thirty has two, at the fine end of the golden 3/5 rung and the fine end of the Lucas 3/4, and the second of them is also one of the free rows, which is a combination nothing predicted.

What a wider table is for #

Not precision. The interaction at any one lattice was already measured to the grid the azimuths sit on, and thirty lattices do not make one of them more exact.

What it buys is the ability to be wrong in a stateable way. With six rows, “no ordering by anything” is a sentence about the sample. With thirty and a gap of eighty-five degrees in the middle of them, a rule that sorts the table can be written down, scored, and found to miss on one rung — and the rung it misses on can be named.

Why the walls are counted rather than named #

The two offsets are read off the very stem the cuts are made in, not from a table of what a lattice at that rise ought to have. Those two would agree nearly always, and the reason for insisting is that the whole design consists of naming two offsets: an offset read from the wrong stem names an organ that is nobody’s neighbour, while every other number in the row stays perfectly plausible.

That failure has already happened once in this thread, when a stem grown at the wrong starting angle returned a plausible answer to a question nobody had asked. It cost a table that had to be thrown away, and the check that would have caught it is the one now run on every row.

The one line #

Six lattices reported an interaction with no ordering in it. Thirty report two regimes and a rule: six rows where the second wall is free and the number is arithmetic, and twenty-four where every rung falls on one side of zero and the side is set by the rung’s larger counted number.

The candidate the design was built to test — position inside the rung — moves the interaction by a few degrees and decides nothing about its sign. It does decide whether there is a second wall to remove.