A theorem with an unstated hypothesis

Every account of flat-foldability quotes the same short list: developability, Kawasaki’s condition, Maekawa’s condition, the big-little-big lemma, and the two-colouring of the panels.

Four of those five are stated completely wherever they appear. One is not, and which one it is follows a rule.

The four that are complete #

Developability, Kawasaki, Maekawa and the big-little-big lemma are conditions at a point.

Each takes the creases meeting at one interior vertex, reads the angles between them or the letters on them, and returns a verdict about that vertex. Nothing else about the sheet enters.

So there is no hypothesis to state. A vertex on a square, a vertex on a cylinder and a vertex on a Möbius band are the same object with the same conditions, and a theorem about vertices is a theorem about vertices.

That is why those four survive every sheet this collection can build unchanged, and it is why their statements have never needed correcting.

The one that is not #

The two-colouring is a condition about the whole sheet, and it is quoted in two forms, both of which are incomplete.

The panels of a flat-foldable pattern take two colours. That presupposes the panels have two colours to take — that the paper has two sides, consistently, everywhere — and a Möbius band does not.

A closed path on a flat-foldable sheet crosses an even number of creases. That is false on a Möbius band, where the condition is that the count is odd.

The correct statement is that the signs multiply to one round every closed path, with a crease contributing minus one and an orientation-reversing identification contributing minus one as well.

On any sheet with two sides that reduces to the even count, which is every sheet anybody folds.

Why the omission survived #

Not through carelessness. Through the hypothesis being true of everything.

Nobody folds a sheet with one side. Nobody teaches one. Every diagram, every book, every model, every application: all discs, or discs with holes, and all of them orientable.

So the general form of the condition would have been a strictly weaker statement about strictly more objects, none of which existed, and there was nothing to gain by writing it.

That is the ordinary economics of a hypothesis. It gets stated when a counter-example is in the room, and not before.

Two kinds of incompleteness #

It is worth separating two things that both count as a statement being incomplete, because only one of them is a problem.

A statement with a hypothesis that is always true is complete in practice and incomplete in principle. Quoting it without the hypothesis produces a true statement about every object anybody has. That is the two-colouring’s position for the whole history of the subject, and nothing went wrong.

A statement whose hypothesis becomes false is a different matter. Once an object exists that violates it, the unqualified statement is a false statement, and anybody who learned it as unqualified will apply it and be wrong.

The transition between those two happens the moment somebody builds the object, and it happens silently: no textbook changes, no result is retracted, and the statement in circulation becomes false without anybody doing anything.

That is why the correction is worth writing down rather than merely being known. The unqualified form is in a great many places and will stay there, and the only defence is that the qualified form is also somewhere.

What happens when the hypothesis is missed #

Concretely, since the essay’s claim is that the omission has consequences.

The collection’s own consistency test for layer relations was imported from a literature where sheets are discs. Run on a glued sheet, it does not fail: it exhausts, proving in three, thirty-five and three thousand four hundred and fifty-five nodes that no lettering exists — and the letterings it proves impossible pass every other check the collection has, on ordinary patches, up to fifteen hundred creases.

That is the shape of the damage. Not a crash, not an error message: a well-formed proof of something false, produced at a cost that rises faster than the cost of being right.

And it was found by a disagreement between two computations rather than by anybody suspecting the test. A missing hypothesis does not announce itself; it produces confident answers that happen to be wrong, and the only way to catch it is to have a second computation that shares no code.

The four path conditions, and their status #

A tidy summary, since the essay’s structure is a split.

The two-colouring. Hypothesis missing in the standard statement, found here, corrected. The general form is a product of signs.

The closure of the folded motions. Hypothesis missing in this collection’s implementation, found here, corrected. The general form compares against the identification map.

Acyclicity of the layer relations. Hypothesis missing, imported from another literature, found here, corrected. The general form reads lattice steps.

The existence of a bottom layer. Hypothesis missing, still missing: the stacking enumeration has not been repaired, and the collision test has not either.

Three of four repaired, one outstanding, and all four were invisible for the same reason. That consistency is what makes the rule — path conditions carry hypotheses, point conditions do not — worth stating as a rule rather than as four observations.

The rule the omission follows #

With the split laid out, the rule is short and it predicts.

A condition at a point carries no hypothesis about the sheet, because a point of one sheet is a point of any other. Developability, Kawasaki, Maekawa, big-little-big.

A condition about paths carries a hypothesis, because a path can do things on one sheet that it cannot on another. The two-colouring, the closure of the folded motions, the acyclicity of the layer relations, the existence of a bottom layer.

Four of each, and the collection has now found the hypothesis missing in three of the four path conditions — the closure condition compared against the identity, the acyclicity test imported without its sheet, and the enumeration that starts from a bottom layer.

Why point conditions are safe #

The rule’s first half deserves an argument rather than an observation, since a point is a point sounds like an assertion.

Every sheet in this collection is a surface, and every point of a surface has a neighbourhood that looks like a disc. That is what being a surface means. So a small enough region round any interior point of any sheet is indistinguishable from a small region of a flat sheet.

A condition at a point reads only that neighbourhood: the creases meeting there, the angles between them, the letters on them. Nothing it reads can distinguish one sheet from another, so its verdict cannot depend on the sheet.

That is a complete argument and it covers all four of the vertex conditions at once. It also predicts that any future condition of the same kind will be safe, which is a useful thing for a rule to do.

The one qualification is the letters. Maekawa reads mountains and valleys, and on a non-orientable sheet those are not globally defined. But Maekawa reads their difference, which is unchanged by swapping both, so the condition survives the loss of the thing it counts.

The arithmetic is worth doing rather than gesturing at, because it is what decides which of the two conditions the loss of sides actually reaches. Maekawa’s statement is that the mountains and the valleys at an interior vertex differ by two, in one direction or the other. A sheet with one side denies a global choice of which face is up, and it denies nothing local: the neighbourhood of the vertex is a disc, so a side can be chosen there, and every crease at that vertex is then labelled against it. Choose the other side and every label swaps at once — every mountain becomes a valley and every valley a mountain — so a vertex reading four and two under one choice reads two and four under the other. The difference is two either way. What the condition counts is not defined on such a sheet and what the condition asserts is, and those are different sentences about the same formula.

That is a near miss rather than an exception, and it is worth noticing that the condition survives by accident of how it is phrased rather than by design.

Where the qualifier goes #

A small practical matter, since the corrections have to be written somewhere.

The clause is on a sheet whose every closed path bounds — or more briefly, on a disc — and the honest place for it is in the statement rather than in a footnote.

A flat-foldable pattern’s panels take two colours becomes a flat-foldable pattern on an orientable sheet has a consistent two-colouring of its panels.

A closed path crosses an even number of creases becomes the signs multiply to one round every closed path.

Both corrections lengthen the statement and neither changes its content on any sheet anybody folds. Which is exactly why the shorter forms are in circulation and will stay there, and why writing the longer ones down somewhere is the whole of what can be done.

What attribution has to do with it #

The essay is filed under attribution and the connection is worth drawing.

Attribution is about who found what and under what conditions, and a result’s conditions are part of the result. Quoting Maekawa’s condition without a hypothesis is correct; quoting the two-colouring without one attributes to the original a claim it does not make.

The people who established these results proved them about the objects they were studying, and the objects were discs. The over-general statement is not theirs; it accumulated in the retelling, as statements do when the qualifier says nothing.

So the correction is not to anybody’s theorem. It is to the version of it that circulates.

What the corrected statements are #

For the record, since the essay is a list of what needs a clause.

Developability, Kawasaki, Maekawa, the big-little-big lemma. No change.

The two-colouring. The signs multiply to one round every closed path, with a crease contributing minus one and an orientation-reversing identification contributing minus one. On a two-sided sheet this is the even count.

The closure condition. The reflections composed round a closed path equal the identification map that path induces. On a disc that map is the identity.

Acyclicity of the layer relations. A closed chain of relations is a contradiction when its lattice steps sum to nothing. On a disc there are no steps.

A bottom layer exists. On a sheet with a boundary. Not otherwise.

Five statements, four clauses, and every clause says nothing on a disc.

The subject’s other unstated hypotheses #

The two-colouring is not the only place, and a short survey puts it in proportion.

The paper does not stretch. Stated everywhere, correctly, as an idealisation. Nobody omits it.

The paper has no thickness. Likewise.

The pattern is on a disc. Not stated anywhere, and the subject of this whole phase.

The sheet is connected. Not stated, never violated, and nobody has built the counter-example.

The sheet is finite. Not stated, and a torus is the closest anything has come.

Two of those five are stated and three are not, and the three unstated ones are precisely the ones that are true of every object in the subject. The stated ones are stated because real paper violates them and everybody notices.

So the rule for whether a hypothesis gets written is not how important it is. It is whether anything violates it, and importance has nothing to do with it.

An older example, for scale #

The subject already has one well-known case of a hypothesis being lost in retelling, and it is worth setting beside this one.

A sheet of paper cannot be folded more than seven times circulates as a fact and is not one. The true statement is about a particular thickness and a particular size folded in half in one direction, and dropping those turns a measurement into a false universal.

That case is different in an instructive way. Its lost qualifiers are quantitative — how thick, how big, folded which way — and losing them makes the statement obviously suspect to anybody who thinks about it.

The two-colouring’s lost qualifier is structural — which kind of sheet — and losing it makes the statement look more general rather than less careful. Nothing about the panels take two colours invites the question on what.

So the two failures have opposite signatures. A lost quantitative qualifier produces a claim that sounds too strong; a lost structural one produces a claim that sounds exactly right.

The second is much harder to notice and it is the one this collection keeps finding.

One sentence #

Four of this subject’s five standard conditions are about a point, and a point of one sheet is a point of any other, so they carry no hypothesis and none is missing.

The fifth is about a path, paths behave differently on different sheets, and its usual statement is missing the sentence that says which.

Why this is not pedantry #

A statement that is true of every object anybody has is a true statement, and correcting it can look like tidying for its own sake. Two reasons it is not.

The first is that the objects are arriving. Every folded structure that gets manufactured is a closed sheet, and the number of people applying flat-folding results to tubes is not small. For them the unqualified two-colouring is a false statement about the object in front of them.

The second is that the omission cost this collection two working pieces of machinery. Both returned confident wrong answers, both were found by accident, and one of them was proving in thousands of nodes that letterings exist which do exist.

So the correction has already paid for itself in defects found, before any consideration of whether a statement ought to be exact for its own sake.

That is the argument for writing the clause down. Not that the short form is wrong for anybody who folds paper, but that the long form is what a computation has to implement, and a computation implementing the short form is a computation with a bug in it that nothing will report.

The record, for this phase #

Since the essay is about attribution, the phase’s own claims deserve the same treatment.

Not claimed: any correction to Maekawa’s, Kawasaki’s or anybody’s theorem. Every one of them is correct as proved, about the objects it was proved about.

Claimed: that the two-colouring’s standard statement is incomplete, with a counter-example; that three pieces of this collection’s own machinery carried the same omission and are repaired; and that a fourth still carries it.

Attributed elsewhere: the vertex conditions, the two-colouring, the acyclicity test, and the √3 bound for smooth Möbius bands. None of those originates here.

That is the honest ledger, and keeping one is most of what the attribution anchor in this collection exists for.

What a reader should do #

Practically, on meeting any statement in this subject.

Ask whether it is about a point or about a path. Point conditions are safe as quoted. Path conditions have a hypothesis and it is probably not written.

If it is about a path, ask which sheet. Almost always a disc, almost always unstated, and almost always correct because the object in hand is a disc too.

If the object is not a disc, look for the general form. For the two-colouring it is a product of signs; for the closure it is a comparison against the identification; for the layer relations it is a sum of lattice steps.

That is three questions and they take a moment, and they are worth asking whenever a folded object is closed rather than flat — which is every folded object that gets manufactured.

The habit that finds them #

There is a method here, discovered by accident and worth stating.

Build an object that violates something everything else satisfies. Then run every existing computation on it and see which ones give an answer that is wrong rather than absent.

That is what produced all four of these. The gluing construction was built to ask what a boundary costs a search, and running the collection’s machinery on the result turned up two tests that returned confident wrong answers and two statements that needed clauses.

None of it required suspecting anything. It required an object, and the object required a reason to build it that had nothing to do with any of this.