A second wrong digit is silent only by cancelling the first

The shortest notation has the longest mistakes counted what happens when one symbol of a weaving document is wrong, and found the threading the only notation that can be wrong while the cloth is right. Of 252,968 single-digit threading errors across the catalogue of four-by-four drafts, 576 leave the cloth unchanged, all of them on three-shaft drafts, all belonging to ten cloths.

It ended on the case its census could not reach. A proof-reader does not face one slip; a page of threading has several, and the question that matters is how often a page of them goes unnoticed. The essay predicted a jump. Two changed digits can exchange two shafts, it argued, and an exchange of shafts is exactly the freedom a lifting plan and its threading share — so at two errors the silent share should leap from a fraction of a per cent to something like the redundancy itself.

The census over pairs has now been run. The jump is real and it is not that one.

The census, and what it walks #

A threading on four ends is four digits, each naming the shaft that end is drawn through, and the lifting plan says which shafts rise on which pick. A single error changes one digit to another shaft the draft uses. A double error changes two different ends’ digits, each to any other shaft in use.

Over the whole catalogue that is 1,074,972 double errors, and each is sorted the way the single census sorted: the drawdown shows an end or a pick that never changes face, which refuses it as cloth at all; the drawdown is a different cloth; or it is the same cloth under the relation used throughout for “one fabric” — started a thread along, turned half round or turned over, but not changed.

The walk is complete rather than sampled, for the same reason the single census was. A rate of one in a few hundred is a rate no sampling of a few thousand would pin down, and the claims below are about which drafts and which pairs, which only a complete walk can make.

Five and a half times, not a hundred #

13,816 of the 1,074,972 pairs are silent: 1.285 per cent. The single rate was 0.228, so a second wrong digit multiplies the chance of silence by 5.6.

That is a large factor and a small rate. One slip in 439 goes unnoticed at one error; one in 78 at two. A proof-reader checking a threading against its woven sample is still catching ninety-nine pairs in a hundred, and the prediction that silence would become common — that it would approach the harness’s redundancy, which for a four-column draft on four shafts is twenty-four threadings to one cloth — is off by more than an order of magnitude.

The drawdown’s own check gets better at the same time. A single wrong digit leaves a thread that never changes face 45.0 per cent of the time; two wrong digits do it 59.1 per cent of the time. The share woven as a different cloth — the share a proof-reader actually has to find by looking — falls from 54.8 per cent to 39.6.

Silence spreads to every harness #

The single census found silence only on three-shaft drafts. Two errors find it everywhere.

On two-shaft drafts, which cannot be wrong silently by one digit at all, two in three double errors are silent: 392 of 588. A two-shaft threading is a pattern of ones and twos, and changing two digits in it very often produces the same pattern moved along — the cloth plain weave’s own symmetry makes of it.

On three shafts, where single silence lived, double silence is 5.2 per cent, nearly four times its single rate. On four shafts it is 0.74 per cent — small, but four-shaft drafts are three quarters of the catalogue, and they carry half of all the silent pairs.

The silent pairs belong to 155 cloths of the catalogue’s 426, against the ten a single error could reach. So the second error does not deepen silence where it already was; it spreads it to cloths that had none.

Most silent pairs are swaps, and most swaps are not silent #

The prediction was built on the swap — two ends exchanging shafts — and the census confirms half of it.

Of the 1,074,972 double errors, 131,808 are swaps. 11,728 of the swaps are silent — 85 per cent of all the silent pairs. So the swap is where double silence lives, as predicted.

But only 8.9 per cent of swaps are silent. The other 91 per cent weave a different cloth or none. The prediction said a shaft exchange is silent on every draft, and it is silent on fewer than one draft in ten.

A swap moves ends; the harness’s freedom relabels shafts #

The prediction confused two operations that look alike on paper.

The harness’s freedom is relabelling a shaft. If every end drawn on shaft 2 is moved to shaft 3 and every end on 3 moved to 2, and the lifting plan’s columns for shafts 2 and 3 are exchanged to match, the cloth is identical — that is a lifting plan saying nothing without a threading, the twenty-four threadings of a four-column draft. It is silent on every draft, and it is not a threading error: it changes the lifting plan too.

A swap of two digits moves two ends. With the lifting plan untouched, end 1 now does what end 2 did and end 2 what end 1 did, which exchanges two columns of the drawdown. A cloth with two of its ends exchanged is a different cloth — unless that exchange happens to be one of the cloth’s own symmetries, a shift or a turn that the relation for “the same fabric” already allows.

That is why 8.9 per cent and not a hundred. A swap is silent exactly when the cloth is symmetric under it, and the silent swaps are a census of symmetries, not of the harness’s redundancy.

No silent pair is made of silent slips #

The census found one more thing, and it is the cleanest result in it.

Of the 13,816 silent pairs, not one contains an error that was silent on its own. Every one is two errors each of which, alone, would have woven a different cloth or no cloth — and which together cancel.

The 576 single silent errors never combine into a silent pair at all. A draft on which one digit can be wrong silently is a draft on which a second wrong digit always shows.

So double silence is not accumulation. It is cancellation: the second error undoes the first, exactly, by landing on the one change that restores a symmetry the first broke. That also says what a proof-reader is looking for. A page with two slips that match is a page whose second slip is the first one’s mirror image, and the mirror image of a slip is a thing a careful reader can be taught to look for.

What this does to a page of errors #

The practical question was never about one error or two; it was about a page.

Three things carry over from the counts. Silence is rarer than the guess by more than an order of magnitude — 1.3 per cent at two errors, not something near the harness’s redundancy. It is concentrated in swaps — a proof-reading procedure that checks every pair of ends whose digits look exchanged catches 85 per cent of what two errors can hide. And it comes from symmetry — a highly symmetric cloth, a plain weave or a twill, has more exchanges that are symmetries, so the cloths easiest to weave are also the ones on which a pair of mistakes most easily cancels.

Beyond two errors the count grows too fast to walk exhaustively, and the cancellation argument says why it should stay small: silence needs the errors to compose to a symmetry of the cloth, and the symmetries of a four-by-four cloth are few and fixed while the ways of being wrong multiply. That is an argument, not a count, and it is stated as one.

At two errors the threading is no longer special #

The single census’s headline was that the threading is the one notation that can be wrong silently: of 365,984 single point-paper errors and 344,464 single lifting-plan errors, not one leaves the cloth unchanged. That turns out to be a fact about one error and nothing more.

Two wrong cells of point paper leave the cloth unchanged 30,112 times in 2,744,880 — 1.10 per cent. Two wrong bits of a lifting plan do it 28,768 times in 2,455,928 — 1.17 per cent. The threading’s 1.29 per cent is barely above them. The six-order-of-magnitude difference in reach that the single census measured survives untouched; the difference in silence does not.

The reason is the cancellation found above, seen from the other side. A single wrong cell of point paper changes the cloth by one intersection per repeat, and no shift or turn of a four-by-four cloth maps a cloth onto itself with one cell different. Two wrong cells can: flip a cell and flip its image under one of the cloth’s symmetries, and the result is the original cloth moved. Silence at two errors is a property of the cloth’s symmetries, and every notation reaches them, because every notation can change two cells. The threading reached them at one error only because a single digit already changes a whole column.

So the practical lesson of the single census needs amending. A threading is not uniquely dangerous; it is dangerous one error sooner. A document of any kind with two slips in it has about a one-in-eighty chance of weaving the cloth it meant, and a woven sample cannot tell.

Why two shafts are the loudest place to be quiet #

The two-shaft drafts are the extreme case and worth a closer look, because they are the drafts on which a single slip is always heard and a double slip usually is not.

A draft that needs two shafts divides its four ends into two groups that do opposite things, and counting the shafts a draft needs found only 98 of the 22,874 drafts that simple. Their threadings are patterns like 1212 or 1122. Change one digit and the groups become three-and-one, which is a different cloth or none. Change two and the groups are two-and-two again — and a two-and-two pattern on a four-end repeat is, very often, the original pattern started one end along.

392 of the 588 double errors on two-shaft drafts are silent. The simplest harness is the one on which a proof-reader who finds nothing wrong in the cloth has least reason to believe nothing is wrong in the threading.

A mispick is the same arithmetic run the other way #

The threading’s double errors exchange columns of the drawdown; the lifting plan’s exchange rows. A mispick is one row in the wrong place is the loom’s version of a lifting-plan error, and three mistakes and the shape each one leaves found that each kind of fault draws its own shape in the cloth — a threading fault a line down the piece, a lifting fault a line across it.

The census here says those shapes have a common exception. A pair of faults that happens to compose to a symmetry of the cloth draws no shape at all, whichever notation it came from, and the rate at which that happens is set by the cloth, not by the document.

Where the check belongs #

The single-error census concluded that a threading has to be checked against itself rather than against the cloth, because the cloth cannot report a silent error. The pairs sharpen that. A threading checked by weaving a sample catches 98.7 per cent of double errors outright — 59 by refusing to make a cloth, 40 by making the wrong one — and the 1.3 per cent it cannot catch are swaps and near-swaps between ends the cloth treats as interchangeable.

So the one check the sample cannot do is exactly the one a written threading makes easy: read the digits in pairs, and ask of any two ends whose digits look exchanged whether they should be. It is the only check whose target the cloth hides.

The same holds, amended, for the other documents. Four ways to write a weave down are four places a pair of slips can compose to a symmetry, and none of them is safe against it; what differs is how often a single slip is heard first. And a specification, which names a cloth by counts rather than drawing it, cannot be proof-read against a sample for the opposite reason — most of its slips name a cloth that is not there, and the sample shows a cloth, so nothing matches and nothing is learned.

What was counted, and how #

Every draft in the four-by-four sweep, 22,874 of them, written as a matrix and factorised into a threading on the fewest shafts it needs and a lifting plan. For each, every single threading error — each end moved to each other shaft in use, 252,968 in all, which recounts the single census’s 576 silent errors exactly and is required to — and every double error: each unordered pair of ends, each moved to each other shaft in use, 1,074,972 in all.

Each changed threading is drawn down against the unchanged lifting plan and sorted three ways. A drawdown with an end or a pick that never changes face is refused. Otherwise its canonical form under the same-cloth relation is compared with the draft’s own: equal is silent, different is a defect. Every pair is required to land in exactly one of the three. Each silent pair is further marked by whether it is a swap and by whether either of its single errors was silent alone. The same sort is run on every pair of point-paper cells, 2,744,880 of them, and every pair of lifting-plan bits, 2,455,928. The whole walk takes a few seconds.

What the count cannot say #

It is four ends by four picks. Larger repeats have more ends to exchange and more symmetries to exchange them under, and whether the silent share rises or falls with the repeat is not something this sweep can say.

A real error is not uniform over shafts. A copying slip is likelier to write a neighbouring digit than a distant one, and a threading error on a loom is likelier between adjacent shafts. Weighting the census by any such model would move every rate here; none of the counts of which pairs are silent would change.

And “the same cloth” is this collection’s relation. A cloth turned over or turned half round is counted as unchanged. A weaver who needs the face up — a twill whose direction matters for the yarn’s twist, as twill direction and its name found — would count some of these silent pairs as defects, and the silent share under that stricter relation is smaller.

Who found it, and when #

That a threading and a lifting plan can be traded against each other is as old as the harness loom, and the count of equivalent threadings is this collection’s. The single-error census is this collection’s too. The double census, and the finding that silence at two errors is cancellation rather than accumulation, are new here — and they correct the collection’s own prediction, which took a relabelling of shafts for an exchange of ends.

Still open: whether the silence grows with the repeat #

Every count here is for four ends. An eight-end draft has twenty-eight pairs of ends to exchange instead of six, and a cloth with a longer repeat has more symmetries for an exchange to land on — or fewer, if the longer repeat is less regular.

The two effects pull opposite ways, and which wins decides whether a proof-reader’s task gets harder or easier on the cloths a mill actually weaves. A census of double errors over a sample of eight-end drafts, with every pair of ends exchanged, would say; the four-by-four walk cannot.