The rung decides the sign

Twenty-four lattices, one two-by-two each, and the interaction between removing the two walls of the slot: thirteen at +85.1° to +135.2° and eleven at −25.1° to −0.5°. Between −0.5° and +85.1° there is nothing.

Eighty-five degrees of empty table is worth more than any threshold anybody could have chosen. It means the two groups would still be two groups wherever the line went, and nobody had to put it anywhere.

The six rows where the second wall is free are not in this count. On those the interaction is the smaller wall’s own cost with a minus sign, produced by cancellation rather than by measurement, and leaving them in fills the negative end of the table with arithmetic.

Every rung agrees with itself #

The lattices are not scattered across the gap in some order of their own. They come in rungs, three or four to a rung, and every rung puts all of its lattices on one side. There is no rung with one lattice above the gap and another below it.

That is eight rungs, both branches of the ladder, and a factor of thirteen in the rise between the coarsest and the finest. Whatever decides the sign is a property a whole rung has, and it does not weaken across the rung.

Which is not what the design expected #

The design was built to test position inside the rung, which is the variable this site keeps finding underneath things. It is refused: knowing where in its rung a lattice sits tells nothing about the sign, and the same position on two different rungs gives +135.2° and −16.6°.

What position does is move the interaction three to fourteen degrees down the length of a rung. Against a spread of a hundred and sixty degrees that is a drift.

Four candidates, scored #

Five accounts were written down before the table was read: position inside the rung, the rise itself, which branch the lattice is on, the smaller counted number and the larger counted number. Each is a rule that puts a row above or below zero, and each is scored on all twenty-four.

Position gets fourteen. The rise gets sixteen. The branch gets thirteen, which is a coin. The smaller counted number gets nineteen. The larger gets twenty-two.

The rule #

Rungs whose larger counted number is 8, 11 or 13 give positive interactions. Rungs whose larger number is 3, 5 or 7 give negative or null ones. That is the golden 5/8, the golden 8/13 and the Lucas 7/11 on one side, and the golden 2/3, the golden 3/5, the Lucas 1/3 and the Lucas 4/7 on the other.

It is not a rule about the branch: it puts a Lucas rung with the golden ones and two golden rungs with the Lucas one. It is not a rule about the rise either: the Lucas 3/4 rung and the golden 3/5 rung overlap in rise and sit on opposite sides.

The exception #

The Lucas 3/4 rung has a larger counted number of 4, so the rule says negative, and its two measured lattices are +90.5° and +85.1°. That is the whole of the rule’s error: two rows out of twenty-four, on one rung, in the same direction.

An exception on one rung is a different thing from a rule that misses at random. It has a name and it was already labelled, for reasons that have nothing to do with this measurement.

Two claims, and they are not equally secure #

The essay makes two statements and they deserve very different confidences, which is easy to miss because they are made in adjacent sections.

The sign is a property of the rung. Twenty-four lattices in eight rungs, three or four to a rung, and every rung internally unanimous. If each lattice’s sign were a coin, a rung of three agreeing happens one time in four, and eight rungs all agreeing happens one time in sixty-five thousand. That is as strong as anything in this thread, it needs no rule and no candidate list, and it is what the eighty-five degrees of empty table is a picture of.

The larger counted number is the property. This is much weaker than twenty-two of twenty-four suggests, because the twenty-four are not twenty-four independent outcomes — the first claim has just established that they are eight. So the rule is scored on seven of eight rungs, not twenty-two of twenty-four.

Seven of eight, by a threshold on an ordered variable, with four rungs on each side: under a null where the four positives fall anywhere among the eight, a threshold getting seven or more right happens about one time in four. And the rule was chosen from five candidates.

So the honest reading is that the sign belongs to the rung — decisively — and that which property of a rung it belongs to is undetermined. The larger counted number is the best of five guesses and is not distinguishable from a lucky one.

Which the ladder cannot fix #

That is not a call for more lattices. Adding a fourth position to every rung strengthens the first claim and does nothing for the second, because the second is already scored on rungs rather than on rows.

What the second claim needs is more rungs, and the arithmetic says how many. Twelve rungs split evenly, all consistent with a threshold, would come out at about one chance in a thousand; allowing one exception it is one in twenty-five. Eight rungs cannot get there however cleanly they behave.

This ladder carries eight rungs between the coarsest rise a stem still patterns at and the fine end where it stops settling. So the larger-number rule is not establishable on this subject’s own ladder, and the only thing this collection can do with it is refuse it — by finding a rung that breaks the threshold, which is one row rather than twelve.

The Lucas 3/4 rung is already such a row, and the next section is about why it is being treated as an exception rather than as that refusal.

Why five candidates and not one #

Writing the candidates down before reading the table is what makes twenty-two of twenty-four a score rather than a description. A rule found by looking at the rows and then quoted against them has no denominator: there are always rules that sort twenty-four rows, and the question is how many were available.

Five were written down, and four of them are the columns anybody would reach for first. That the winner is the fifth, and wins by six rows over the runner-up, is the whole of the evidence — and it is the discipline this site applies to a survivor rule, where the account that sounded right scored at chance.

What the sizes look like #

On the positive rungs the pair removal moves the next organ 117° to 164°, while each wall alone moves it 1.4° to 49°. On the negative rungs the pair moves it 31° to 78°, and the larger of the two single removals moves it 24° to 34°.

So the difference is not that the negative rungs have a small interaction. It is that on the positive rungs the pair does something categorically different: it throws the organ most of a turn, where neither wall alone moves it a tenth of one.

Which is a threshold in the fate, not in the arithmetic #

Forty-five degrees is where this thread separates a cheap removal from an expensive one, and it is a gap in the data rather than a line anybody drew: the two populations measure 9° to 31° and 63° to 168°.

On the positive rungs the pair removal is past that line on every lattice. On the negative rungs it is under it on seven of the eleven. So the rule that sorts the interaction also sorts whether the pair is a cheap removal, which is a second reading arriving at the same partition.

And a third reading agrees #

What the cut stem keeps is a third instrument. On the positive rungs the pair removal leaves the smaller counted number’s own family standing — the five-hop on a 5/8 lattice, the eight-hop on an 8/13, the seven-hop on a 7/11 — and destroys everything else.

On the negative rungs it leaves either everything or a scatter, and on three of them it leaves the whole ladder of lags intact, which is what a stem that healed looks like.

Three quantities — a displacement, a fate and a set of surviving hops — measured independently and cutting the table the same way. That is the shape this collection trusts most: two routes arriving at one number, with a third behind them.

A larger number is a smaller step #

The obvious thing about 8, 11 and 13 against 3, 5 and 7 is that they are bigger, and the obvious meaning of a bigger counted number is a finer arrangement: more organs round the stem before the pattern repeats, so a smaller angle between neighbours in a chain and a shorter step across the surface.

That is measurable rather than rhetorical. On the positive rungs the smaller wall’s removal costs 14.5° to 25.8°; on the negative rungs it costs 29.5° to 65.9°. So the walls are cheaper to remove where the pattern is finer, and the pair is more expensive.

Which suggests a mechanism, and does not establish one #

A fine arrangement has a narrow slot with two close walls. Removing one leaves the other, and the organ shifts a little. Removing both leaves an opening several organs wide and the organ falls right across it.

A coarse arrangement has a wide slot already. Removing a wall changes it by a fraction of what it already is, and removing both changes it by a bit more. That is a story that fits, and this collection does not accept a story that fits as a result — it is a description of the numbers, not a mechanism behind them.

What would test it #

The story predicts a quantity nobody has measured: the angular width of the slot after each removal. If the account is right, the width after removing both should be several divergence steps on the positive rungs and about one on the negative ones, and the interaction should follow the width rather than the counted number.

That is a measurement on stems already grown, and it would separate the story from the rule — because “the larger counted number is 8 or more” and “the slot opens by more than two steps” agree on this table and need not agree everywhere.

The rule is a fit and it should be said so #

Twenty-four rows, five candidate rules written down in advance, and the best of them getting twenty-two. That is a good score and it is a score over a small table, with a rule whose threshold — larger counted number of 8 or more — sits in a gap between 7 and 8 with no rung in it.

There is no rung whose larger number is 8, 9 or 10 other than the golden 5/8. So the threshold is fitted between two adjacent values and any number from 7.5 to 8 gives the same table. What would test it is a rung with a larger number of 9 or 10, and neither ladder has one.

Where a third sequence would help #

There are additive sequences other than the golden and the Lucas, and stems do settle on them — the settling table reaches several of them. A sequence containing 9 or 10 as a larger counted number would give exactly the rung this rule needs to be tested against.

Nothing in this thread has grown a stem on one deliberately. It is a design with twelve runs in it and it would either break the rule or double the evidence for it.

What the negative rungs are doing #

Nothing much, and that is worth stating plainly. The eleven negative rows run −25.1° to −0.5°, and four of them are within ten degrees of zero, which is where this thread stops calling an interaction one.

So the honest partition is not positive against negative. It is large and positive against small, with the small ones straddling zero in a way that is consistent with the two walls acting nearly independently on the coarse rungs.

Which changes what the earlier result said #

The first version of this measurement reported an interaction that was “reliably not the sum of its parts and not reliably larger than it either”. The second half of that is now wrong.

On the rungs where there is an interaction at all, it is always larger: thirteen lattices, every one of them positive by eighty-five degrees or more. The negative half of the earlier sentence was two arithmetic rows and a set of near-zeros.

Two rungs the design nearly missed #

The golden 2/3 rung and the Lucas 1/3 rung are the coarsest on either branch, and neither had a lattice in the six the design began with. Both are negative here, and the Lucas 1/3 is negative for the wrong reason — all three of its rows are ones where the second wall is free and the number is arithmetic.

So of the four rungs the rule calls negative, one contributes no measurements at all. The rule is really sorting three negative rungs against four positive ones, which is a smaller table than twenty-four rows sounds, and the essay says so rather than quoting the row count.

The instrument did not change #

Every cell is a displacement against a control sharing its history to the last digit, on a stem whose heights are prescribed by the rise, with the walls counted from the points of the stem the cuts are made in. Thirty lattices is the same experiment run more often.

That matters because the conclusion reverses a sentence. A reversal produced by a changed instrument is a comparison of instruments; this one is a comparison of sample sizes.

What is not explained #

Why a counted number should decide anything about an interaction between two removals. The rule sorts twenty-two of twenty-four rows and offers no reason, and the story about slot widths is a story until the widths are measured.

This collection has a standing position on that: a rule that sorts a table is worth recording and is not an explanation, and saying which is which is most of the discipline.

What a reader should not take from this #

That a finer arrangement is more fragile. The interaction is about a pair of removals and says nothing about how often a stem gets one: which offsets wreck a stem at all is a separate census with a separate answer, and the fine rungs are not more prone to wrecking than the coarse ones.

Nor that the larger counted number is doing anything causal. It is a label on a rung, and the rung carries a rise, a divergence, a slot width and a step length, all of which move together down the ladder. Every one of those would sort this table as well as the counted number does, and the counted number is only the cheapest of them to write down.

The one line #

On the twenty-four lattices where the slot has two walls, the interaction between removing them is decided by the rung and not by where on the rung the lattice sits: positive and large where the larger counted number is 8, 11 or 13, small or negative where it is 3, 5 or 7.

Twenty-two rows of twenty-four, with the two misses on one rung and in one direction.