Contact is not why a jersey stops

The most conspicuous quantitative failure in this collection’s knitted work is an extension ceiling three times too high. The model says a plain jersey can be pulled to three hundred and twenty-two per cent along its courses before it runs out of yarn. Measured jerseys stop at about a hundred.

The diagnosis recorded when that gap was first written down was that the model does not stop adjacent courses passing through one another. This ladder has the machinery to test that, and the test says no.

The geometric ceiling #

At a fixed loop length, the yarn has to reach from one interlacing to the next. Widening a wale therefore costs course height: the two spacings are confined inside an ellipse quadrant, and pulling one out pulls the other in.

The limit is where the course spacing reaches its own minimum. The model takes that minimum to be one yarn diameter, on the grounds that the courses cannot pass through one another, and the result is a course-wise extension of three hundred and twenty-two per cent for an ordinary jersey.

There is nothing elastic in that number. It is geometry: a length of yarn, a fixed pair of endpoints, and a floor on one spacing. No finish, fibre or count moves it — only the tightness factor does.

What contact adds #

The geometric limit puts the floor at the interlacing, where the model placed the two curves one diameter apart. The contact question is different and stricter: at each width, what is the smallest course spacing at which the two adjacent courses stay clear of one another everywhere?

The interlacing is not where the curves come closest, so the answer must be a larger spacing, and the ceiling must come down.

It does. The contact ceiling is two hundred and ninety-nine per cent against the geometric three hundred and twenty-two.

Seven per cent, against a gap that has to close by two thirds.

What that rules out #

The candidate is dead, and it is worth being blunt about it.

Contact between adjacent courses is not what stops a jersey extending. A fabric whose courses are forbidden to overlap can still be pulled to three times what a real one takes.

That is a negative result and it is the point of the rung. Before this, “the model does not stop the courses passing through one another” was a plausible diagnosis with nothing behind it, and it was the diagnosis this collection had recorded. It is now a diagnosis that has been tested and has failed.

What is left standing #

Three candidates survive and they are worth ranking, because the ranking is the useful output.

The loop is not re-solved under load. Every point on the extension curve uses the free shape at that width. A loop that is being pressed against its neighbours takes a different shape — shorter in the direction it is being pulled, because the pressing shortens the free run — and that is a first-order effect nobody has computed.

The interlacing is not threaded. Two loops drawn through one another cannot separate beyond the length of yarn joining them, and that is a much stronger constraint than either ceiling above. It is the omission five other findings share, and it is the strongest candidate by a long way.

And the fabric fails before its geometry runs out. A real jersey at a hundred per cent extension is not at a geometric limit at all; it is at the point where the load required has become large enough to break something or to make the fabric unusable. The “ceiling” a measurement reports is a practical limit and the model’s is a geometric one, and comparing them may be comparing two different quantities.

That last is the possibility the rung cannot rule out and it is the one that would make the whole comparison ill-posed.

Why the seven per cent is so small #

It is worth understanding why the contact constraint bites so little, because the reason is instructive.

As the fabric is pulled along its courses, the loops straighten. A stretched loop is a long shallow arch rather than a tight hairpin, and two long shallow arches one course spacing apart are further from one another than two hairpins are.

So the contact constraint is worst at the relaxed state — where it is a fifth of a diameter of overlap — and gets easier as the fabric extends. By the time the fabric is at two hundred per cent, the courses are nearly clear of one another and the constraint is nearly inactive.

That is exactly backwards from what a reader would expect. Pulling a jersey along its courses separates its courses rather than crowding them, and the crowding it produces is in the other direction.

What crowds instead #

If course-wise extension separates the courses, something else must be crowding, and it is the wales.

Pulling along the courses widens the wale spacing, so the loops in one course move apart. That should relieve crowding within a course, and it does.

Pulling across the wales — the other direction — is where the crowding happens: the course spacing grows, the wale spacing shrinks, and the loops in one course are pressed together side by side.

So a jersey has two extension ceilings and only one of them is a contact problem. The course-wise one is a yarn-length problem, and the wale-wise one is a crowding problem that this rung has not computed.

That is a gap worth naming: the wale-wise ceiling has never been computed with contact at all, and it is the direction where contact should matter.

What the ceiling would have to be #

It is worth asking what a mechanism would have to do to close the gap, because it constrains the candidates sharply.

The model says three hundred and twenty-two per cent and the fabric does a hundred. So the missing mechanism has to remove two thirds of the available extension, and it has to do so at every construction, since the gap is a factor rather than an offset.

That rules out anything that acts only at the extreme. A mechanism that bites at two hundred and fifty per cent and stops the fabric there would leave a ceiling of two hundred and fifty, not a hundred.

It also rules out anything small. A seven per cent correction, a ten per cent correction, or a series of them, do not multiply to a third.

What is needed is a constraint that is active from the beginning and gets tighter, and the threading is exactly that: two loops drawn through one another are limited by the yarn joining them at every extension, not only at the end.

That is a strong argument for the surviving candidate and it comes from the arithmetic of the gap rather than from any model of threading.

Why the model’s ceiling is not simply wrong #

A fair objection is that a model over-predicting by a factor of three might be wrong in some more basic way, and it is worth answering.

The geometric ceiling is not a prediction about a fabric’s behaviour. It is a statement that the yarn runs out at that extension — that beyond it, no arrangement of a fixed length of yarn between fixed interlacings exists at all.

That statement is correct and is not in doubt. What is in doubt is whether a fabric ever gets near it, and the answer is that it does not, because something else stops it first.

So the model is not over-predicting; it is answering a different question from the one a tensile tester asks. The geometric ceiling is an upper bound and it is a true one — no jersey will ever exceed it — and the useful question is what the binding constraint is, which is a different and harder thing.

Framing it that way removes the appearance of failure and leaves the real problem, which is that this collection has an upper bound and no mechanism.

What was counted, and how #

At each of fifteen widths from the relaxed spacing to the geometric limit, the calculation searches for the smallest course spacing at which the two adjacent courses stay at least the demanded clearance apart.

The clearance demanded is the fabric’s own relaxed flattening — 0.780 diameters — rather than a full diameter. Asking for a round yarn returns nothing feasible at any extension, which is not a ceiling: it is the relaxed fabric’s own overlap restated as an empty search, and recognising that took an afternoon.

The search is a bisection on the course spacing, twenty-four steps, with the solve wrapped so that a construction the yarn cannot reach is treated as infeasible rather than as an error.

The check asserts two things. The contact ceiling is below the geometric one, which would fail if the interlacing really were the closest approach — and that is what the model implicitly assumed for its whole life. And the reduction is under twenty per cent, which is the finding stated as a threshold so that it fails if the number ever moves enough to matter.

The wale-wise direction, which nobody has computed #

The rung’s most useful consequence is a question rather than an answer, and it is worth stating clearly because it is the obvious next calculation.

A jersey has two extension directions and this collection has only ever computed one ceiling. Pulling along the courses widens the wales and separates the courses, so the contact constraint relaxes as the fabric extends — which is why the seven per cent is so small.

Pulling across the wales does the opposite. The course spacing grows and the wale spacing shrinks, so the loops within a course are pressed together side by side, and the constraint tightens as the fabric extends.

That is where contact should matter, and nothing in this collection has looked. The geometric wale-wise ceiling is about a hundred and fifty per cent for an ordinary jersey; whether contact brings it down by seven per cent or by half is unknown and is a day’s work with the machinery this ladder has built.

It is also the direction where the discrepancy with measurement is smaller, which makes it the more likely place for a contact explanation to succeed.

Where the model stops #

The loop is the free loop at every width. That is the largest omission and it is the leading surviving candidate, which makes it awkward: the calculation that would test the best remaining hypothesis is the one this rung does not do.

Only the adjacent courses are considered. A loop’s neighbours within its own course are not, and those are the ones that crowd in the other extension direction.

The clearance is a single number. It is the relaxed flattening, held fixed as the fabric extends — and the flattening should itself change with the extension, since the fabric’s own geometry is what demands it.

And the comparison is against a measured number this collection has not measured. “About a hundred per cent” is trade knowledge and a spread rather than a figure.

What the extension curve says meanwhile #

The ceiling is one point on a curve and the rest of the curve is unaffected by any of this, which is worth saying because a reader may otherwise discount the whole calculation.

This collection computes a knitted fabric’s load–extension curve from the loop’s own bending energy: the force at each extension is the derivative of the energy with respect to the width, and the shape of the curve — soft at first, stiffening towards the end — comes from the geometry rather than from any modulus.

That curve is right in its initial modulus, which is where a fabric spends its working life, and it is right in its shape over the range a garment is actually stretched.

What is wrong is where it turns over, which is a region no garment reaches. So the ceiling being three times too high is a defect at the far end of a curve that is otherwise doing its job, and the practical consequence of this rung is smaller than the conceptual one.

That distinction matters for how the collection’s knitted results should be read. A number about a fabric at ordinary extensions is sound; a number about a fabric at its limit is not, and the limit is further away than any fabric goes.

The generalisation #

The rung’s value is entirely in being a negative, and negatives are undervalued in this collection’s own record.

Eighteen phases of ladders have produced a great many findings and very few eliminations. That is not because nothing was eliminated; it is because a candidate that turns out not to explain something rarely gets written up, and the effort spent on it disappears.

A ruled-out mechanism is a result and should be recorded as one. Here the value is precise: the extension gap now has three surviving candidates rather than four, and one of the three has been promoted from “also possible” to “the strongest remaining”, because the field has narrowed.

That is worth more than it looks. The recorded diagnosis was wrong — not vaguely, but specifically, and it was the diagnosis somebody reading this collection would have acted on. Correcting it costs one rung and saves whoever tries next from building a contact solver to close a seven per cent gap.

What a recorded diagnosis is worth when it is wrong #

This rung exists because a previous one wrote down a diagnosis rather than a symptom, and it is worth drawing out what that bought, since the diagnosis turned out to be wrong.

The shortfall could have been recorded as “the extension ceiling is too high”. That is a symptom. Nobody reading it later can do anything with it except measure again.

It was recorded as “the ceiling is too high because the model does not stop adjacent courses passing through one another”. That is a diagnosis, and it is a testable one — which is exactly why it could be shown to be wrong, and why showing it wrong took one rung rather than a research programme.

A wrong diagnosis is more useful than a vague one, because it can be eliminated. A vague note stays open for ever and nothing about it ever gets resolved.

That is an argument for a habit rather than for this result, and it is one of the more transferable things in this work. This collection’s shortfall lists are full of diagnoses, and at least one of them has now been eliminated because it was specific enough to be.

The three surviving candidates, ranked #

Since the field has narrowed, it is worth ranking what is left rather than listing it.

First: the interlacing is not threaded. Two loops drawn through one another are limited by the yarn joining them at every extension. That is a constraint active from the beginning and tightening throughout, which is the shape the gap requires, and it is the omission five findings share.

Second: the loop is not re-solved under load. A loop pressed against its neighbours is shorter in the pulled direction than a free one, and every point on the extension curve here uses the free shape. That is a first-order effect and it is not small.

Third: the two numbers are not the same quantity. A measured extension is where a fabric stops being useful and a computed one is where the yarn runs out. If that is the answer, the gap is not a modelling failure at all and the comparison should never have been made.

The first is the most likely and the third would be the most embarrassing, and neither has been tested.

Who found it, and when #

The geometric ceiling is this collection’s own, from the ladder that first solved a knitted loop as an elastica. The observation that it is three times any measured jersey was recorded in the same ladder, with the contact diagnosis attached.

Measured extensions for plain jersey are in every knitting text and are consistently around a hundred per cent course-wise for an ordinary construction, with a wide spread.

What is this collection’s own is testing its own recorded diagnosis and finding it wrong — which is the sort of thing a collection that writes its shortfalls down carefully can do, and one that does not cannot.

Where the ladder goes next #

The surviving candidates are two, and one of them is a defect this ladder has been circling from several directions: the two half periods that meet at a crest, which come within a fiftieth of a diameter of one another because nothing is drawn between them.

That is what holds a crest apart, and it is not a flattening problem at all.