A test imported without its hypothesis

There is a way of getting things wrong that is not carelessness and is hard to defend against. A rule is stated in a source, with its hypotheses; it is carried into new work; and the hypotheses, being about the setting rather than about the rule, are the part that does not travel.

This collection has now done that once with a cure and once with a test. The cure was named first — randomised restarts, imported from the search literature along with the heavy tail they exist to fix, when the tail was a property of the search’s own arrangement. The test is this essay, and it is the more consequential of the two.

The rule #

Every crease of a folded sheet says which of its two panels lies above the other. Those statements are a relation on the panels; a cycle among them is a contradiction, so a lettering whose relations contain one describes no folded state.

That is the collection’s cheapest and most-used negative. It reads a crease list once, it works on patterns of two hundred panels where an enumeration of stackings gives up at a dozen, and it is the only negative answer available on a real pattern.

It is also true, in the setting where it is stated, and its truth has two halves. A cycle means no order — that half is obvious and holds everywhere. No cycle means an order exists — that half is the useful one, and it holds because a finite acyclic relation can always be extended to a list.

Finite. The hypothesis is in the second half, and it is about the sheet rather than about the rule.

Where it came from and what it carried #

The layer-ordering literature is about pieces of paper. A folded state is a map from a bounded sheet to the plane; the panels are the finitely many regions the creases cut it into; the non-crossing conditions are relations among those; and a valid folded state is an ordering of them.

Nothing in that setting needs the word finite, because nothing in it is otherwise. A sheet of paper has an edge, and every theorem, every algorithm and every complexity result is about a bounded region. The hypothesis is not omitted from the sources; it is built into what a folded state is, and it never has to be stated because there is nothing outside it.

So a reader taking the rule away takes a true statement with an invisible condition attached, and applies it wherever the words fit.

Where it stopped fitting #

A tessellation is not a bounded sheet. It is a rule for filling the plane, and every essay here about a twist tessellation is about an infinite object, described by drawing a square of it.

For as long as this collection has existed that description was the only one available. The squares are bounded, so the rule applied to them correctly, and every result taken off one is as good as it ever was. The gap opened the moment the pattern itself could be built, and the first thing asked of the new object was the old question.

The answer was that the square twist tessellation has no consistent lettering at all, proved by exhaustion in thirty-five steps.

It has one. The lettering the proof excludes passes every vertex condition and forces no loop on ordinary patches of one, four and nine periods, on four tilings.

It is worth being precise about how large the wrong answer is, because a false negative sounds like a near miss and this is not one.

The rule does not merely fail to find the lettering. It closes its whole tree and reports that none exists — the strongest conclusion the machinery can produce — and it does so at every size it can finish: three steps at one period, thirty-five at four, three thousand four hundred and fifty-five at nine. Three proofs, all valid, all of something untrue.

And the cost of producing them grows faster than the cost of the right answer, so the failure is expensive as well as confident. Being wrong here costs between four and fifty-six times what being right costs, depending on the tiling.

Why it was invisible #

Three reasons, and the third is the one worth carrying.

Nothing outside the domain existed. Every pattern the collection had ever searched was a bounded sheet, so the rule was correct on every input it had ever seen. A hypothesis that is never violated is a hypothesis that never announces itself.

The failure looks like a result. The wrong answer is not a crash or a warning; it is a closed search tree, which is the strongest kind of result the machinery produces. An exhausted search reports a proof, and the proof is valid reasoning from a premise nobody had written down.

And the words did not change. A cycle in the layer relations means something on a disc and something else on a torus, and the sentence is identical. The relations are still relations; the cycle is still a cycle; the panels are still panels. What changed is that “the panels” now names infinitely many things, and no phrase in the rule notices.

What a borrowing owes #

This collection has a habit of recording where an idea came from, and the habit needs a second half.

Found before it was designed, the same vertex found four times, the tail named somewhere else — the thread is full of cases where a thing here has a prior name elsewhere, and recording that is a matter of honesty about priority.

What this adds is that a borrowing also carries a domain, and the domain is the part most likely to be lost, precisely because it is the part the source never has to say. A theorem’s hypotheses are stated. A field’s background assumptions are not, because within the field they are always true.

So the second half of the habit is: when a rule is brought in, write down what it is a rule about. Not what it says — that comes with it — but what class of object it was established for, which is usually not written anywhere and is usually obvious to everybody in the source and to nobody outside it.

What would have caught it #

Three things might have, and it is worth ranking them by how much they cost, because the cheapest one is the one to keep.

Building the object. What actually caught it, and the dearest: a new construction, a new module, a season of work. It is the only one that was certain to work, and it is not a routine to be applied.

Reading the growth rate. The exhaustion cost grows roughly a hundredfold per size on a pattern whose panel count is growing by fours, while the same drawing cut out of the plane costs sixty per cent of a step per panel at every size. A search doing something structurally different from reading, on an object that is not obviously harder, is a question worth asking — and the question is what exactly is this proving, which has an answer.

Writing the domain down at the time. The cheapest by an enormous margin and the only one that scales. When the rule was first used, one sentence — this is a statement about a sheet with finitely many panels — would have sat in the notes for years doing nothing, and would have fired the moment anything else was handed to it.

The third is the practice this essay recommends and the first is what it took.

The two borrowings, compared #

The cure and the test failed differently and the comparison is instructive.

The restart curve was correct arithmetic on a distribution that the collection’s own search created. The literature’s heavy tails are real, restarts are the right answer to them, and the tail here came from one line of randomisation. So the import was of a remedy for a condition that had been manufactured locally — the mathematics was fine and the diagnosis was wrong.

The cycle test is the reverse. The diagnosis is right — a cycle really is a contradiction — and the sufficiency is what has a hypothesis, and the hypothesis was left behind. So the import was of a correct theorem applied outside its class.

Two failure modes, both from taking something true elsewhere and using it here, and neither detectable by checking the imported thing itself. The first needed a measurement of the local instrument; the second needed an object outside the class to exist.

What is owed to the source #

Nothing, in the sense of an error to report: the layer-ordering results are correct as stated and this is not a correction to them.

Something, in the sense of an attribution. The rule is not this collection’s invention and the domain is not this collection’s discovery. Neither is the mathematics of orderings that the replacement rests on, which is a borrowing of the same kind and is recorded as one. That a folded state of a bounded sheet is an ordering of finitely many panels is the definition the field works with, and the reason it does not carry the word finite is that the field is about paper.

The contribution here is narrower and worth stating exactly: an instance outside the class, with a witness. A periodic crease pattern on which the acyclicity test returns a false negative, and a lettering it rejects that folds every finite piece of paper it is written onto. That is the sort of thing a collection like this can supply — not a theorem, an example.

What the collection now does differently #

Three things, and they are small.

An exhausted search is reported as what it is: this rule found nothing, in this many steps, rather than the pattern has none. The two are the same statement when the rule is right, and only the first is ever established by running a search.

The cycle test stays in place where it is sound, which is on every disc, and is consulted first because it is cheap and sufficient. Only what it rejects costs the fuller decision, so nothing that was fast is slower.

And the periodic case has its own answer written down, with its own domain attached: on a sheet with no edge a closed walk in the relations is a contradiction only when its lattice steps add to nothing, and a walk that travels is a stack with no bottom.

Why this is a history essay #

A reader might reasonably ask why a mistake about a search belongs in a thread about where ideas came from, and the answer is that the mistake is entirely about provenance.

The rule is not wrong. The implementation is not wrong. Nothing was mis-copied and no number was mis-read. What went missing is a fact about the source: which objects the statement was established for, which is a piece of context rather than a piece of content, and which is exactly the sort of thing this thread exists to keep track of.

The other essays in the thread record who found something first. This one records that a thing carried more than its statement, and that the part it carried invisibly turned out to matter more than the part written down.

That is a different kind of debt to a source and it is worth naming as one. A collection that borrows a result owes the result an attribution; a collection that borrows a method owes it a domain.

The general shape #

A rule with an unstated domain is a trap that springs only when somebody builds something new, and building something new is what this collection does.

Every addition here brings constructions, and each construction is a chance to hand an old instrument an object it was not designed for. The tessellation with no edge is the clearest case so far because the object is genuinely outside the class rather than merely unusual within it — a torus is not a large disc.

The defensive habit that follows is not to distrust imported rules. It is to ask, of any instrument being pointed at a new construction, what class of object it was built for, and whether the new thing is in it. That question has an answer far more often than it is asked, and here it had been unasked from the beginning.

Where else this collection may have done it #

The honest end of an essay like this is a list of places to look, and there are three.

Anything that walks outward from a seed. The folded state is built by starting at one panel and reflecting into its neighbours until the panels run out. On a sheet with no edge they do not run out, and the construction here works only because a period is drawn as a bounded rectangle first. Every quantity computed that way inherits the assumption.

Anything that counts. A crease count, a panel count, a vertex count: each is a count of objects, and a cut turns one object into two. The counts are correct on the sheet they are taken from and are not the pattern’s, by an amount that falls as the reciprocal of the patch’s size and never to zero.

Anything with a smallest or a largest. The bottom of a stack, the shortest crease, the outermost panel — extremes are exactly the quantities an infinite object need not have, and the bottom layer turns out to be entirely a fact about where the paper stopped.

None of those is a defect found; they are the shape of question this essay recommends asking, and asking it is cheap now that there is an object outside the class to ask it about.

Which theorem was checked, and how #

The claim that the imported rule fails is not established by argument. It is established by a witness: a lettering the rule rejects, written onto ordinary clipped patches and handed to the four vertex conditions and a folded sheet rebuilt from coordinates, passing everything at every size on four tilings.

The claim that the rule is right on a disc is not checked here at all, and does not need to be. It is a standard result and every one of this collection’s negatives that was established on a bounded sheet still stands on it.

And the replacement rule is checked against something it must refuse: two panels with one relation each way, neither leaving its own copy of the period, is a walk that genuinely closes, and the decision has to call it a contradiction. Without that, a rule that says fine to everything would look exactly like a rule that is right.

What the picture cannot show #

A missing hypothesis is not a thing. There is no figure of it, and the closest any drawing here gets is a table of two columns disagreeing, which shows the consequence rather than the cause.

Nor can a picture show why nobody noticed, which is the interesting half. The rule produced correct answers on every object it had ever been given, for years, and a record of years of correct answers is exactly what a figure of an unnoticed assumption would have to look like.

And no drawing here settles whether the same slip is present elsewhere in the collection, which is the question a reader ought to be left with. Every instrument in use was written for a bounded sheet, because until now that was the only thing there was; how many of them have an unstated hypothesis of the same kind is not known, and finding out means asking each of them the question rather than looking at a picture.