The criterion cannot see friction

The integrity criterion is the best thing this site has. It decides, exactly and in linear time, whether a draft describes one cloth or several; it catches a failure that is invisible in the drawing; it has found real defects in real figures; and it works on constructions it was never designed for, including several with no matrix at all.

This essay is about what it cannot do, which is a specific and permanent thing rather than a list of unimplemented features.

The exact statement #

The criterion asks: can the strands be split into an upper set and a lower set such that at every crossing between them, the upper strand is on top? If so, the upper set lifts off and the fabric is not one fabric.

That is a question about the existence of a separation. Read as reachability it becomes a question about strong connectivity of a directed graph, which is where its exactness and its speed come from.

And it is the source of the limitation, in one sentence: a separation either exists or it does not. There is no continuum. There is no “nearly separates”, no margin, no measure of how close a fabric is to coming apart. The output is an integer and the integer is a count of components.

So a construction cannot be more or less integral in the criterion’s terms, and any question of the form “how firmly” is not a refinement of the criterion. It is a different question about a different kind of quantity, and no amount of work on the graph algorithm produces it.

Two constructions in the gap #

The two ladders of this field arrived at the same boundary from opposite directions, which is what makes it a boundary rather than an anecdote.

Pile. A tuft bound under one pick and a tuft bound under three are both connected. The criterion says one cloth for both, in identical terms, from the same implementation. One is specified for hotel corridors and one sheds within a year.

Leno. A gauze and an open plain weave at the same sett are both connected. The criterion says one cloth for both. One is a fabric and the other is a heap of threads that has not noticed yet.

In each case the criterion is correct. There is no separation in a V-fastened carpet and none in an open plain weave. The fabrics differ in a property the criterion does not measure, and the property is the same in both cases.

What has to come from outside #

The missing quantity is friction, and the model that supplies it is the capstan relation: a line wrapped through angle θ around a cylinder holds exp(μθ) times the tension at its free end.

Two things about that model deserve to be stated every time it is used, and this essay is the place to state them once properly.

It has a measured parameter and the parameter sits in an exponent. Yarn-on-yarn friction is not a material constant — it depends on fibre, finish, twist, moisture and crossing angle, and reported values for cotton run from about 0.2 to 0.4. At a wrap of three half-turns that range moves the answer by a factor of five. Any single number quoted from it is a number at a stated coefficient.

It is an idealisation applied a long way from where it was derived. A yarn wrapped round a yarn is not a rope on a bollard: the “cylinder” moves, flattens, and is under tension itself. The wrap angles are real geometry; the exponential relating them to force is a model with a name and a domain.

So the criterion’s verdict and the capstan’s number are different kinds of claim, and this field’s figures print them apart — “the criterion”, “the capstan, at μ = 0.3” — because putting an exact count and a modelled estimate in one typeface with one air of authority is the failure this site’s fourth invariant exists to prevent.

Once, the parameter cancelled #

There is one result in this field where the friction coefficient drops out entirely, and it is worth setting beside the others precisely because it is unlike them.

A plain weave’s wrap on its weft is twice the weave angle; a leno’s is π. Peirce’s geometry cannot produce a weave angle of ninety degrees at any spacing, so twice it never reaches π, so no plain weave reaches a leno’s grip — at any sett, in any yarn, at any friction. Both sides carry the same μ and it cancels.

What that shows is not that the capstan is trustworthy after all. It shows that a comparison between two constructions using the same model can be far more robust than either construction’s absolute number, and that it is worth looking for comparisons of that shape rather than settling for a ratio at a plausible coefficient.

Why the criterion is worth having anyway #

It would be easy to read all of this as a demotion, and it is not.

It catches what nothing else catches. 144 of the 22,874 four-by-four drafts describe more than one cloth, and not one of them looks wrong. No inspection, no rule of thumb, and no other measurement on this site detects them. A criterion with a narrow question and an exact answer to it is worth a great deal more than a broad one with a vague answer.

It certifies as well as filters. A double cloth asserted to be two must come out as two, which is what makes the first direction trustworthy.

It transfers. It was defined on a weave matrix and it never needed one. Braids, warp knits, pile fabrics, leno gauzes and double plush are all decided by the same implementation, because it consumes contacts rather than a matrix — and that portability is why this field could be written at all.

And it describes a process. Velvet’s manufacture is a step from one to two, on purpose. A criterion built to answer “is this cloth?” turned out to answer “what did the machine just do?”, which is not a question it was designed for.

An instrument with a sharp boundary and a known blind spot is a usable instrument. An instrument that gave a soft answer to every question would be worse in both directions.

The general shape of the mistake this prevents #

The failure this essay exists to prevent has a shape, and it is worth naming because it is not specific to cloth.

An exact criterion answers a well-defined question. The question is related to the one people care about, closely enough that the criterion is useful and closely enough that the two get identified. Then a case arrives where they come apart, and the criterion says what it always said — correctly — while the thing people care about has changed completely.

Stated that way it is recognisable. It is the shape of a test suite that passes while the product is unusable, of a structural calculation that is satisfied while the building is unpleasant to be in, of a thread count that is high while the sheet is thin and rough, and of a cover factor that rose past its own domain and came back down.

The defence is not a better criterion. It is knowing what question the criterion asks, in one sentence, and being able to say what it does not ask. For this one: it asks whether a separation exists, and it does not ask what force anything holds at.

What a good invariant is worth, and what it costs #

The pattern this field has run into three times is worth stating as a general observation about instruments, because it is the transferable part.

A narrow invariant with an exact answer is worth more than a broad one with a vague answer, and the reason is that its failures are legible. When the integrity criterion says one cloth and the fabric sheds, the disagreement is precise: here is what was asked, here is what was answered, and here is the question that was not asked. That precision is what made it possible to name the missing quantity, find a model for it, and keep the two apart.

Compare an instrument that returned a “fabric quality score” combining connectivity, anchorage, float length and cover. It would give a plausible number for every construction in this field, it would never be obviously wrong, and it would be impossible to say what any of its numbers meant. Its failures would not be legible because it makes no falsifiable claim about anything.

The cost is that a narrow invariant will be misread. A number that is exact and easy to compute gets quoted, and quoting strips the question it answers. Nothing about “two separable layers” carries with it the fact that separability is topological — it has to be said, every time, by somebody who knows.

That is why this essay exists and why it is a rung on the integrity ladder rather than a footnote. The invariant has been in use since the site’s foundation, it has been quoted in every field, and the sentence saying what it does not measure has never been written down in one place until now.

What was counted, and how #

Nothing new is computed here; this essay assembles results the two ladders established.

The layer counts come from the same stronglyConnected implementation used by every weave, braid and warp knit on the site, run on graphs built by hand for the three-system and crossed-warp constructions. The anchorage figures come from the capstan applied to wrap angles read off those same graphs — so the number and the verdict are computed from one description of one object, rather than from a picture and a separate account of it.

The assertions worth repeating are the ones that would fail if this essay’s claim were wrong. The V and W constructions are both asserted to be one cloth, and a pile bound to nothing is asserted to be two. The leno and its control are both asserted to be one cloth, and the two are asserted to differ in crossings and in nothing the criterion reads. Those are assertions that the criterion cannot distinguish the cases — an unusual thing to assert, and the honest way to make a claim about a blind spot rather than merely describing one.

Where the model stops #

Friction is not the only thing the criterion misses. It also has no notion of yarn strength, of bending stiffness, of how a fabric behaves after washing, or of anything that happens outside the plane of the contacts. Friction is the one this field needed; it is not the whole of the blind spot.

The capstan is not the only way to fill the gap. A finite-element model of yarn contact would give a much better answer and would need geometry, material properties and a great deal of computation. The capstan is chosen because it needs one angle and one coefficient, and because the angle comes free from the construction.

And nothing here bounds how large the gap is in general. Two constructions have been shown to sit in it. Whether there are fabrics that pass the criterion and fail catastrophically in some third way is an open question, and the honest answer is that this site has no method for finding them other than looking.

A margin that is still an integer #

The claim above is that there is no continuum: a separation exists or it does not, and any “how firmly” question is a different kind of question with a different kind of answer. That is right about force, and it has been allowed to stand for something slightly stronger than it supports — because there is a margin available from the same digraph, it is exact, it is an integer, and this collection has never computed it.

Strong connectivity is a yes or no. How many arcs have to be removed before it stops being yes is a number, and it is the standard notion of a digraph’s arc connectivity. In the fabric’s own terms:

the smallest number of crossings whose loss splits the cloth.

A draft in which every crossing is load-bearing — where losing any one of them disconnects the graph — is a genuinely more precarious construction than one in which the connectivity survives the loss of any five, and the criterion as used here reports both as one cloth with nothing to distinguish them.

That is a second exact invariant rather than a softening of the first. It makes no modelled claim, it has no fitted coefficient, and it cannot be quoted as a quality score, because it answers a stated question: how many contacts is the integrity redundant in? An answer of one means the cloth is one cloth by a single crossing per repeat.

Which half of the gap it closes #

It matters that this does not repair the essay’s main argument, and seeing exactly where it stops is more instructive than the invariant itself.

It would separate the leno from its control. A gauze and an open plain weave at the same sett differ in how many crossings they have, and a construction with more contacts has more ways for its connectivity to survive losing one. Whatever the two arc connectivities turn out to be, they are being asked about a difference that genuinely exists in the graph.

It would not separate a V-fastened tuft from a W-fastened one. Those two differ in wrap angle at the binding point, not in how many binding points there are — one half-turn against three, at the same contacts — so the digraphs are the same shape and the arc connectivity is the same number. The capstan is still the only thing that tells them apart, and it is still a model with a coefficient in an exponent.

So the honest division is that the criterion’s blind spot has two halves. The half about how many contacts there are is topological and is recoverable exactly. The half about what each contact is doing is not topological at all, and no graph invariant reaches it.

That is a better statement than the essay’s original one, and it is a smaller retraction than it looks: the two constructions the field actually ran into are one of each kind, which is presumably why the distinction was never forced.

What it would cost, and what it would connect to #

The computation is not free but it is not expensive. Arc connectivity is a sequence of maximum-flow problems, so on graphs of a few dozen nodes it is a few milliseconds, and the four-by-four census — 22,874 drafts — would be a matter of seconds rather than of a redesign.

Two things it would connect to immediately, and the second is the one worth having.

It would rank the drafts the census already sorts into one cloth and several. The 144 separable drafts have arc connectivity zero by construction; what nobody knows is how the other 22,730 are distributed, and whether the ones sitting at one are recognisable weaves or curiosities.

And it is exactly the quantity a missing end is asking about. A dropped end removes every crossing that end made, so a draft whose connectivity depends on few enough arcs is a draft that a single fault turns into two cloths. That is a failure mode the fault ladder found by enumerating drops and the integrity ladder has no invariant for — and arc connectivity is the invariant, computed on the sound cloth, that would predict it before the fault occurred.

Neither has been run here. Both are stated because a limit that turns out to be half a limit should be recorded as half a limit, and because naming a computation nobody has done is more useful than naming a quantity nobody can have.

Two gaps, and only one of them is the criterion’s #

It is worth separating the criterion’s blind spot from a second gap that sits beside it and is often confused with it, because the remedies differ.

The criterion’s gap is about the question. It asks whether a separation exists. It does not ask what force anything holds at. No amount of better computation closes it, because the two questions have different kinds of answer — an integer and a force — and there is no function from the first to the second.

The second gap is about the encoding. A pile fabric has no matrix and a leno has no fixed column order, so the site’s measurements — float, interlacing, cover, plane group — either fail or answer about the wrong system. That gap is about what can be described, not about what can be decided, and the criterion sails straight through it.

The two are independent, which is the point. A construction can be inside the encoding and in the criterion’s blind spot: an ordinary open plain weave has a perfectly good matrix, every measurement applies, and it still slips. And a construction can be outside the encoding and entirely clear of the blind spot: a double plush’s one-to-two step has no matrix and is exactly what the criterion decides.

Keeping them apart matters because the wrong remedy is tempting for each. The encoding gap tempts one to extend the matrix, which produces a notation that describes more and decides less. The criterion gap tempts one to soften the criterion into a score, which produces a number that describes nothing and decides nothing.

Who found it, and when #

The criterion in the form used here — a separation as a closed set under an above-and-below relation, decided by strong connectivity — is the site’s own statement of an idea that appears in the textile-geometry literature in various forms, and its exactness is not in dispute anywhere.

The observation that it is blind to friction is not a discovery either; any weaver knows a gauze needs its crossings. What is worth having is the pairing: the exact statement of what the criterion decides, set beside the exact statement of what it does not, with a second model named and quarantined to fill the gap. That pairing is what lets this field describe constructions the site’s central encoding cannot express, without at any point pretending the encoding reached further than it does.

Where this goes next #

This field has two more essays and both are about the encoding rather than about the criterion. How many cloths there actually are counts the four-by-four census as fabrics rather than as notations, and finds the site’s own separation rate more than doubling. And what the matrix cannot say collects the whole boundary in one place, across every construction this collection has met.