The order decides the count

The lengths are free established one half of what a vertex’s numbers are for: the lengths of its creases carry none of the information about whether it folds and all of the information about what the folded paper looks like. This is the other half, and it is stranger. The angles carry almost none of it either.

Reading the conditions one at a time #

The claim sounds implausible until the four conditions are read separately, at which point it stops being surprising and starts being obvious in hindsight.

Developability asks the sectors to sum to a full turn. No letter appears in it. Kawasaki asks the alternating groups to sum to half a turn each. No letter appears in that either. Both are settled before a single mountain or valley has been written down, and every vertex in this essay satisfies both by construction.

Maekawa asks for mountains minus valleys to be exactly two in absolute value. No angle appears in it. It is a condition on a word of letters and nothing else, and the reason the constant is two is a winding argument about the folded cross-section rather than a fact about any particular vertex.

That leaves the big-little-big lemma, and it reads the angles through the narrowest possible aperture: a sector strictly smaller than both its neighbours must be flanked by creases of different letters. It asks which sectors are strict local minima. It never asks how small they are, how much smaller than the neighbours, or what any other sector is doing.

So the whole apparatus reduces to one combinatorial object: the set of positions at which a sector is a strict local minimum. Two vertices with that set in common admit exactly the same letterings.

Not the same number of them. The same ones.

It is worth being clear about what has and has not been claimed. Nothing here says two such vertices look alike or fold to the same object — the lengths decide what the folded paper looks like and the angles decide it further. What is claimed is that the subject’s own tests cannot separate them.

Measured, and then computed a second way #

Over 759 degree-six vertices in fifteen groups, no group held two vertices admitting different letterings. That is the measurement, and on its own it is only a measurement — a coincidence of the sampler, conceivably, or a bug shared by the two things being compared.

What makes it evidence is the second routine. The counting routine never looks at a vertex. It takes a degree and a set of positions, walks the cycle of creases carrying a running count of mountains minus valleys, forbids the two creases beside each named position from agreeing, closes the cycle against the letter it started with, and returns a number. It shares no line of code with the enumeration and no data with the vertex, and it agrees with every group.

The last row of that figure is worth a sentence of its own. A vertex at which every sector is strictly smallest than both its neighbours is impossible on a cycle — a sector cannot be smaller than a neighbour that is smaller than it — and the count says so by returning zero rather than by refusing the question. That is the right behaviour for a routine that is supposed to know nothing about vertices: it answers a combinatorial question combinatorially, and the geometric impossibility shows up as an empty answer.

Where the count is allowed to change #

If the count depends only on which sectors are smallest, then deforming a vertex changes the count exactly when it changes which sectors are smallest — that is, at a tie.

This is where the result stops being a curiosity and starts explaining something the site has already published. The dial that decides nothing measured the admissible lettering count of a twist tessellation across its whole angle band and found it constant at sixteen, moving only at the single angle where two sectors tie. That essay reported the constancy as a fact about the big-little-big lemma reading which sector is smallest and never how small.

The order type is the general form of that fact. The twist’s count is constant because its arrangement of smallest sectors is constant, and the arrangement is what the conditions see. Turning the dial moves every angle in the pattern and moves nothing the conditions are looking at.

That reading also predicts where the exception in that essay came from. On a triangular twist the count moved once, from 128 to 64, at exactly the angle where two sectors tie — and a tie is precisely a wall between two arrangements. The count did not respond to the angle; it responded to the crossing, and the crossing happened to be at an angle.

What the arrangement can be #

There are not many arrangements, and that is worth making concrete before the rest of the argument leans on it.

An arrangement is a set of positions round a cycle of n sectors, no two of them adjacent — two neighbouring sectors cannot both be strictly smaller than the other. At degree four that leaves seven possibilities: none, each of the four single positions, and the two opposite pairs, with the four adjacent pairs excluded. At degree six the random sampler reached fifteen in 759 draws.

A forty-five-degree grid reaches one. Every degree-six vertex that grid admits has the same arrangement, which is why a box-pleated design’s vertices all admit the same number of letterings whatever their sectors look like, and why the catalogue of them is as short as it is. That is a strong restriction hiding inside a construction rule that says nothing about arrangements at all.

How many arrangements there are #

The closing section asks for the count of possible arrangements as a ceiling on what any local test can distinguish, and the count is a sequence with a name.

An arrangement is a set of positions round a cycle with no two adjacent, which is an independent set in a cycle — and the number of independent sets in a cycle of $n$ vertices is the $n$-th Lucas number. Seven at degree four, eighteen at six, forty-seven at eight, a hundred and twenty-three at ten.

Seven is the degree-four list above, item for item. Eighteen at degree six is why a sampler reaching fifteen in seven hundred and fifty-nine draws is a sampler that has very nearly exhausted the possibilities rather than one that has scratched them: three arrangements went unvisited, and they are presumably the ones that need three widely spaced minima at once.

Which is a ceiling, and a low one #

That count is the essay’s requested bound and it is worth reading against the thing it bounds.

A vertex of degree $n$ has $2^n$ letterings — sixteen at degree four, sixty-four at six, two hundred and fifty-six at eight. The number of vertices the four conditions can tell apart is the Lucas number, which grows as the golden ratio to the $n$: 1.618 to the power of the degree rather than 2.

So the conditions’ resolving power falls behind the thing they are resolving, exponentially. At degree four they distinguish seven classes among sixteen letterings. At degree twenty they would distinguish about fifteen thousand among a million.

That is the sharpest form of this essay’s claim. It is not merely that two particular vertices can share a lettering set; it is that the whole apparatus has a fixed and modest capacity to tell vertices apart, the capacity is a Lucas number, and it is outgrown by everything it is applied to.

It also bounds the design question the closing section raises. A grid whose vertices all share one arrangement is not sacrificing a small part of a large space — it is choosing one of seven at degree four, one of eighteen at degree six. The space the conditions can see was never large, and the grid takes one point of it.

And the paper #

The conditions are one thing. Whether the paper actually folds is another, and it has been known since the crimping rung that above degree four the two part company: letterings pass every condition in the subject and have no flat folded state.

So there are two counts — how many letterings pass and how many fold — and the natural expectation is that the first is an invariant of the arrangement and the second is not. The reduction gives a reason to expect it: crimping a sector away leaves a merged sector of prev + next − here, and where that lands among the remaining sectors depends on their actual sizes, so the second step of the reduction sees more than the first did.

The expectation is not borne out, and it has not been settled either. Over every group measured — 759 vertices at degree six, 130 at degree eight, and a deliberate hunt over 1,900 more for two vertices of one arrangement with different fold counts — the fold count has been constant inside every group. No counterexample has been found and no argument has been given, which is a state this site is obliged to report as it is rather than tidy.

What is settled is that the two counts are different functions of the same object. At degree eight there is an arrangement whose conditions admit thirty-two letterings and whose paper folds sixteen, and another whose conditions admit sixteen and whose paper folds all of them. The arrangement decides both numbers and does not make them equal.

Which theorem was checked, and how #

Three claims, each with its own check.

The invariance is checked as sets and not as counts. Two vertices of one group must admit the same list of letterings; comparing totals would pass a bug that swapped one lettering for another, and comparing lists does not.

The count from the arrangement is checked against three cases that can be worked out by hand. Four creases with no named minimum admits eight, because the letterings with mountains minus valleys equal to two are exactly the four with one odd letter and the four with the other. Naming one minimum halves it. Naming all four admits nothing. A routine that walks a cycle and returns plausible numbers is the easiest kind of thing to get subtly wrong, and those three are the cases where wrong is visible.

The degree-four corollary falls out rather than being measured separately. At degree four the conditions are the whole answer, so the fold count equals the pass count there, so the fold count is an invariant of the arrangement there for a reason rather than by observation. That is the earlier result arriving as a consequence, which is the shape a ladder is supposed to have.

Where the model stops #

The invariance is a statement about the four conditions this site computes, and those are not every condition the subject has. A generalised big-little-big lemma exists that constrains any sequence of sectors bounded by two larger ones rather than a single minimum, and a version of it that read more of the angles would break the claim as stated. What the essay establishes is that the four conditions this site checks see only the arrangement, which is a fact about a stated checker and is exactly the kind of fact that checker’s own essay was written to keep honest.

The second limit is the one already named: whether the fold count is an invariant of the arrangement is open here. The hunt for a counterexample runs at degrees six and eight and stops there, because the brute-force stacking search this site checks the reduction against refuses a degree above nine, so the population where a counterexample is most likely is the one that cannot be searched.

What the picture cannot show #

Nothing in these pictures shows a folded state. Every figure is of a flat pattern or of a count, and the folded object is a second view that has to be computed rather than read off. A shading cannot show what it is not. The hero figure marks the strict minima and gives the sector sizes round the outside, and a reader will inevitably read the sizes as though they were doing something. They are doing one thing only: deciding the shading. Every other property of the picture — how sharp the small sector looks, how the vertex sits on the page — is invisible to every condition in the subject.

Nothing here shows a lettering either. The counts are counts of words in two letters, and the pictures are of angles, and the whole point is that the two do not see each other.

The generalisation #

The useful way to state this is as a separation between two kinds of quantity. A vertex has continuous data — the angles — and discrete data — which of them are smallest, in what cyclic order. The conditions are functions of the discrete data alone. The paper, as far as anything measured here can tell, is too.

That makes the space of vertices a stratified object: not a continuum with a property varying over it, but a finite set of cells, with the answer constant on each cell and changing only on the walls between them. The walls are the ties. And a tie is where the reduction has to search, which is the same wall approached from the other side — the arrangement changes there, so the count can change there, and the procedure loses its forced move there.

Who found it, and when #

The ingredients are all old. Kawasaki’s condition dates from Husimi’s work in the 1970s, Maekawa’s parity from the same period, and the big-little-big lemma from Justin in 1986. That the lemma reads only a comparison is visible on the face of its statement, and any careful reader of it knows that the count of admissible letterings is combinatorial.

What does not appear to have been written down is the consequence: that the count is a function of an arrangement, that the arrangement is a small finite object, and that a transfer count over the cycle computes it without reference to any vertex. It is the kind of result that is easy to have known without ever having stated, and stating it turns a family of separate measurements — the twist’s constant sixteen, the tie’s doubling at degree four, the grid catalogue’s six vertices — into readings of one thing.

Where the ladder goes next #

The open half is the interesting half: whether the fold count is an invariant of the arrangement, and if so why, given that the reduction’s own arithmetic says it should not be. Settling it needs either a counterexample at degree ten — which needs a stacking search this repository does not have — or an argument that the merged sector’s position among the survivors is itself decided by the arrangement.

There is also a design question waiting in it. If a grid’s vertices all share one arrangement, then every vertex a box-pleated model contains admits the same number of letterings, and a designer choosing between them is choosing between objects the conditions cannot distinguish. What the grid settles measured what a grid costs a packing; this is what it costs a vertex, and the cost is that a whole dimension of choice collapses.

The other direction is to ask how many arrangements there are. At degree six the random sampler reached fifteen and the forty-five-degree grid reached one; the number of possible arrangements is a combinatorial count over cyclic sequences with no two adjacent minima, and it bounds how many genuinely different vertices the subject’s conditions can tell apart at each degree. That bound is a ceiling on what any local test could ever distinguish, and it is small.