The shortest notation has the longest mistakes

Four essays have measured what each notation can say. Four ways to write one weave down, a profile draft, a threading with its lifting plan, and a specification — each asked for an image, a redundancy and a failure mode, and each answered.

None of them asked what happens when a document is wrong. A notation is written by somebody, copied by somebody else and read by a third, and every one of those steps can put a symbol in the wrong place. What that costs is a property of the notation, it is computable, and across these four it varies by six orders of magnitude.

It also turns out not to be a design choice at all.

Counting over a piece, not over a repeat #

The reach has to be counted over cloth rather than over a repeat, because the whole point of a repeat notation is that it is written once and used many times.

The piece is a square metre of ordinary shirting: 1,800 ends and 1,800 picks, which is eighteen threads a centimetre each way, woven from a four-by-four repeat. That is 450 repeats across and 450 down, and 3,240,000 intersections.

Now take each notation in turn and ask what one wrong symbol does to that piece.

A point-paper cell is one intersection of the repeat, and the repeat is everywhere: 450 × 450 = 202,500 intersections wrong, scattered on a lattice across the whole metre.

A threading digit is one end of the repeat, drawn on the wrong shaft. That end recurs 450 times across the width and it is wrong on every pick: 450 × 1,800 = 810,000.

A lifting-plan bit is one shaft on one pick. Every end on that shaft is wrong, on every repeat of the plan down the length — on a four-shaft straight draw, 450 ends by 450 picks = 202,500.

A profile draft’s block is a square of the repeat. At two blocks each way that is four intersections, everywhere: 4 × 202,500 = 810,000.

A jacquard card’s hole, in a design that does not repeat, is one.

The product does not move, and that is the result #

Set the symbol counts beside the reaches and the right-hand column is constant:

Every product is the piece. It is not an empirical regularity and it is not a coincidence of these five: each of these sets of symbols partitions the cloth — every intersection is decided by exactly one of them — so the count times the reach is the number of intersections by definition.

Which says something that ought to have been obvious and is not usually stated. A notation cannot be both short and local. Shortness is exactly coarseness; a document with a quarter as many symbols has symbols that reach four times as far, and no amount of design changes the product because the product is the cloth.

So the account’s whole economy inverts here. A jacquard card set is the most expensive document in weaving — one hole per end per pick, with no compression at all — and it is the only one in which a mistake is a mistake rather than a fault line. A threading is eight bits and a mistake in it takes a quarter of the cloth.

A count of intersections is not a shape #

The product settles how much cloth an error touches and says nothing about what it looks like, and the three notations make three different-looking faults out of comparable counts.

A threading error is a stripe. The end it belongs to recurs across the whole width and is wrong on every pick, so the fault is a set of continuous lines down the piece, one every four ends. That is the classic mill defect and it has a name in every fault list.

A lifting error is a bar. The pick it belongs to is wrong on every end drawn on that shaft, so the fault runs across the cloth and stops — it is one pick in four, over the whole width.

A point-paper error is neither. It is a lattice of isolated intersections, one per repeat in both directions, and no two of them touch. At 202,500 intersections it is the same order of magnitude as the other two and it is not a mark: it is a change to the weave’s texture, distributed perfectly evenly, which reads as a different cloth rather than as a flaw in this one.

Which is the practical inversion of the product. The notation whose errors reach furthest makes the faults easiest to see, because a coarse symbol’s reach is contiguous; and the notation whose errors reach least far in the repeat makes the fault hardest to notice, because a fine symbol’s reach is spread. A stripe down a piece is found by a person walking past the loom. A weave that is uniformly the wrong weave is found by nobody until somebody compares it with the sample.

Three things a wrong symbol can do #

Reach is not the whole of a defect, because not every changed intersection is a defect that anybody sees. Change one symbol and the drawdown does one of three things.

It stops being a cloth. The interlacing condition is this account’s own minimum: an end that never goes under anything is not woven in, and a pick that never goes over anything is not either. A drawdown that fails it is refused by the drawdown itself — the check a mill already makes when it draws the design out, and it costs nothing.

It becomes a different cloth. The error survives to the loom and appears in the piece.

It leaves the same cloth. Nothing anywhere will ever report it, because there is nothing wrong with the fabric. A silent error is not a fault; it is a fact about where a check can be placed, because a notation with silent errors cannot be proof-read against its own output.

Running every single-symbol change over every one of the 22,874 drafts in the sweep gives all three shares.

The threading is the exception twice over #

The census puts the threading in a class of its own in two opposite directions, and both follow from the same property.

It is caught most often. 44.99 per cent of single-digit threading errors leave a matrix that is not a cloth, against 28.60 per cent for a lifting-plan bit and 26.03 per cent for a point-paper cell. The reason is mechanical: moving an end to another shaft replaces its whole column with a copy of another column, and two identical columns are much likelier to leave some pick with nothing on one face than one flipped intersection is. A coarse symbol is a symbol whose errors are gross, and gross errors are the easy ones to catch.

And it is the only symbol that can be wrong silently. Of 252,968 single-digit threading errors, 576 leave the cloth exactly as it was — 0.228 per cent. Every point-paper error and every lifting-plan error changes the cloth or destroys it; none of the 365,984 and none of the 344,464 is invisible.

Which is worth holding still for. The most robust document in the account is point paper, the one nobody works from. It is also the longest of the repeat notations, which is the product again.

Where the silence lives, exactly #

The 576 silent errors are not scattered. They fall on 576 of the 22,874 drafts, they belong to ten cloths of the 426, and every one of those drafts is a three-shaft draft.

The mechanism is visible once the shaft count is known. Four ends on three shafts means one shaft carries two of them. Move one of the remaining ends onto a shaft that is already in use and the drawdown gets a duplicated column and loses a distinct one — and in a three-shaft draft whose columns happen to be translates of one another, the result is a translate of the original. A translate is the same cloth, because a repeat has no origin.

Two shafts cannot do it, because with two columns on four ends a change either leaves the column classes alone or reduces them to one, and one column is not a cloth. Four shafts cannot do it, because with four distinct columns any merge loses a column the cloth needs.

So the silence is a property of the middle of the shaft range, and it is the sparse case: three shafts is the count almost nothing is woven on — 5,184 drafts need exactly three, and the ten cloths that admit a silent error are a tenth of a per cent of the catalogue.

That is a comfortable answer and it comes with a warning. The census is over a four-end repeat and the silence is a shaft-count phenomenon, so a sixteen-end repeat on eight shafts has far more room for one end to land on a shaft that reproduces it. The number here is small; the claim that it is small in general is not made.

What this says about where a check belongs #

The three shares together decide where proof-reading is worth doing, and the answer is different for each document in a mill.

A threading is worth checking against the drawdown, because 45 per cent of its errors are refused there for free and it is the document whose errors reach furthest. The remaining 55 per cent reach the loom, and a quarter of a piece is a great deal of cloth to weave before finding out.

A lifting plan is worth checking at the loom, because its errors are local in the warp — one shaft’s ends on one pick — and a lifted-wrong pick is visible in the first centimetre of weaving. Its reach is large in the piece and small in the first repeat, which is the distinction that matters for when a fault is found rather than for how much cloth it spoils.

And a card set cannot usefully be proof-read at all. Three and a quarter million holes, each deciding one intersection, means an error rate per hole of one in ten thousand puts 324 wrong intersections in a square metre — scattered singletons, each a one-intersection fault, none of them repeating and none of them catchable by any structural condition. The notation with no amplification has no self-check either, and the two are the same property seen twice: a symbol that decides nothing but itself cannot be contradicted by anything.

That is the honest form of the trade’s own practice. A jacquard’s cards are verified mechanically, hole by hole, by a machine built for it; a threading is verified by drawing the design out and looking; and nobody verifies point paper, because point paper is the verification.

The one document that is checked by the cloth itself #

There is a fourth place a check can sit, and this account has already measured it without calling it that.

A specification is checked by arithmetic on its own fields — the setts against the jam for its own count — and that check needs no cloth and no loom. It is the only one of the four notations with an internal consistency condition strong enough to reject a document on its own, and the reason is exactly that it names a physical object rather than a combinatorial one.

The repeat notations have no such condition beyond interlacing, and interlacing is weak: it rejects a quarter of point paper’s errors and nothing else about them. So the four notations are checked in four different places — a specification by arithmetic, a threading by the drawdown, a lifting plan by the first centimetre of weaving, and a card set by a machine built to read holes — and the place is decided by the notation’s reach rather than by anybody’s preference.

That is a better account of mill practice than “experience”. A mill that checks its threadings and not its point paper is not being careless about point paper; it is checking the document whose errors are worth the walk, and the one that is hardest to weave from is not the one that is hardest to get right.

The same number, seen from the other side #

A lifting plan says nothing without a threading found that a draft with c distinct columns on S shafts has S!/(S − c)! threadings, all producing the identical cloth, and read that redundancy as the weaver’s freedom to balance the harness. That is the same quantity as the silent-error count and it is a good illustration of what a redundancy is for.

A redundancy is a set of documents mapping to one cloth. Moving between two of them is, by definition, a change that the cloth does not see. A silent error is a redundancy entered by accident, and a notation’s silent-error rate is its redundancy measured locally rather than globally: the global figure counts how many documents share a cloth, and the local one counts how many of them are one symbol apart.

The two numbers are not proportional, which is why both are worth having. The harness redundancy is large — twenty-four threadings for a four-column draft on four shafts — and the silent-error rate is 0.228 per cent, because almost none of those twenty-four is reachable by changing one digit. Most of a redundancy is far away, and the part of it that is dangerous is the part within one mistake.

What was counted, and how #

The sweep is this account’s own: every four-by-four draft in which each end and each pick reaches both faces, 22,874 of them.

The factorisation is canonical rather than chosen. Each draft’s columns are grouped in order of first appearance, which fixes one threading out of the redundancy class, and the lifting plan follows. A different choice of representative changes which digit is which and changes none of the counts, because the census runs over every digit of every draft.

“The same cloth” is same-cloth relation, this account’s own relation — the operations that leave a fabric the fabric it is. A silent error is one whose drawdown has the same canonical form, which is a stronger statement than “looks similar” and is the only one this account is willing to make.

The reach arithmetic is exact and has no model in it. It is a count of intersections in a stated piece, and the check behind the table is that each notation’s symbols account for the piece exactly once — a violation would mean a reach had been derived wrongly rather than that some notation was unusual.

Two symbol kinds are in the reach table and not in the fate census, and the omission is deliberate. A jacquard hole and a profile block have no four-by-four census to run over: the card is the drawdown, so its fate census is point paper’s; and a profile’s errors depend on which two weaves the blocks carry, which is a different enumeration and belongs with A profile draft is a notation whose alphabet is weaves.

What the count cannot say #

Nothing here weights an error by how likely it is. A drawer-in misreads a 3 as a 2 far more often than as a 4, and a card punch’s failures are not uniform over the holes. Every share above treats all single-symbol changes as equally probable, which is the right null assumption for a property of a notation and the wrong one for a mill’s actual defect rate.

And nothing here is about more than one error. Two threading digits wrong is not twice one digit wrong: the second can cancel the first, and the cancelling pairs are exactly the shaft permutations the redundancy is made of. A two-error census would have a much larger silent share and it has not been run.

The reach is intersections, not visibility. A float’s length decides what the eye sees, and 202,500 scattered single intersections on a lattice is a different-looking fault from 810,000 in 450 continuous stripes down the cloth — which is a threading error’s actual appearance and is the classic mill defect with its own name. The count says how much cloth is affected; it does not say how much of it a buyer rejects.

Who found it, and when #

That a wrong threading shows as a stripe down the piece and a wrong lift as a bar across it is as old as the shaft loom and is in every mill’s fault list. What is not in any of them is the product, and the reason is that a fault list is written per notation while the product only appears when the notations are set side by side and counted over the same piece.

The identity itself is elementary once stated — it is the observation that a partition of a set has size times block size equal to the set — and its interest is entirely in what it forbids. Every proposal to make weaving documents shorter is a proposal to make their errors bigger, in an exactly computable ratio, and the trade’s most expensive notation buys with that expense the only error behaviour in the account that is not amplified.

The silent-error census is this account’s, and it needed the complete sweep: 576 in 252,968 is a rate no sampling would have distinguished from zero, and the fact that all of them sit at three shafts is invisible without the shaft count attached to every draft.

Still open: what a second error does to the silence #

Every number here is for one wrong symbol, and the one place the single-error assumption is doing real work is the silent share.

Two threading digits changed can exchange two shafts, and a shaft exchange is silent on every draft rather than on ten cloths. So the silent share does not grow smoothly with the error count — it jumps, from 0.228 per cent at one error to something of the order of the redundancy itself at two, and the jump’s size is S!/(S − c)! per draft rather than a rate.

The measurement is a census over pairs, which is 252,968 × 3 changes deep and perfectly feasible, and the number it would give is the one a proof-reading procedure actually needs: not how often one slip goes unnoticed, but how often a page of them does.