Groups in a row are one painter twice; groups in turn say little

One more habit takes two hands, and only part of a walk gave a divergent picture one more number than a single steady hand needs — a second habit — and found that the share of the strips’ disagreement it takes separates two painters from one hand that steps its dividers. The split it chooses is a map: these strips with one habit, those with the other. What the map cannot say is who the two groups were. Two painters sharing a panel draw two habits; so does one painter who changed habit partway through — after a break, after being shown a better way, after moving from the floor to the ceiling. Both are two groups that agree within themselves, and both are taken by one extra number.

There is one thing that differs, and it is when. A painter who changed habit once painted every strip of the first group before every strip of the second. Two painters working one panel through a day take strips in turn, or nearly so. A tiring panel keeps its order, not its direction found that the painting order of a panel is written into it, up to reversal: the creep that grows as a hand tires shifts each strip’s reported habit by an amount that depends on the strip’s splay and on how far into the session it was painted, and the order that makes those shifts fit one rate of tiring is the order painted. So the arrangement of the two groups along the recovered order is the evidence. Groups in a row say one painter twice. Groups that alternate say two.

The question with a number in it is whether the order survives being asked to carry a second habit, and the answer has a shape that a symmetric test would miss.

The panels, and what the reader is given #

The panels are the six strips of the earlier order essay — floor, footstool, table and book, with a step at a splay of 1.08 and a lectern at 1.26 — and an eight-strip panel that adds a sill at 1.16 and a shelf at 1.35. Every strip is painted with a creep — the steady widening of the bays that a tiring hand draws as a different habit on any one strip — that grows with the session: one per cent a bay on the first strip painted, a point more on each strip after. The creep belongs to the clock, not to a person, which is what makes two painters comparable with one. Two painters working one panel through one day tire together, and the only difference between the two arrangements is where the second habit falls in time.

One hand paints at habit 1 and changes to habit 2 after half the strips. Two hands at habits 1 and 2 take strips in turn. Every interior row is drawn with a slip of a stated size, from a tenth of a pixel to a whole one. Each strip is then read alone for the habit it reports, which is the one number per strip the earlier order essay used.

The reader is given the groups. That is a deliberate division of labour, argued at the end: the groups come from the rows through the second habit’s split, which the essay before this one found puts 94 per cent of strips with their own hand on eight strips, and the order comes from the reported habits once the groups are fixed. For every one of the 20,160 orders of eight strips, up to reversal, the reader fits two habits, a starting creep and a rate of tiring to the eight reported habits; the order that fits best is the reading.

On this pair of panels, read to a quarter of a pixel, the reading does exactly what the argument wants. Both orders come back as painted. Set out along them, with the counterfeit of the fitted creep taken off each strip’s habit, the one hand’s strips sit at one level for four places and the other level for four; the two hands’ strips change level at every place. The habits fitted are 0.92 and 1.90 for one panel and 0.93 and 1.90 for the other — a little under 1 and 2, the first-order law the creep’s counterfeit obeys — so the levels are the two habits, and the arrangement along the order is the whole difference.

A change at one moment is most of a trend #

The slider says something less comfortable. Coarsen the rows to half a pixel and the two hands’ panel still alternates, but the one hand’s breaks into four runs. That is not bad luck with one panel. It is built into what a change of habit is.

A painter’s fatigue enters the reported habits as a trend: the later a strip was painted, the further its habit is shifted, in proportion to its bays over the logarithm of its splay. A change of habit at one moment enters as a step: every strip after the moment is shifted by the gap between the habits. A step halfway along a sequence and a straight trend along it are nearly the same pattern — both put the first half low and the second half high — and the fit, given both, can trade one for the other. The rate of tiring is the only number the order is read from, and it is the number the step steals from.

This can be measured before a single row is slipped, from the arrangement alone. For the true order and equal scatter in every strip’s habit, the standard error of the fitted rate is set by how far the column it multiplies stands from the other columns. Over two hundred random painting orders, a second habit that alternates widens it by 5 per cent on eight strips and 7 on six: an alternating pattern is nearly orthogonal to any trend in time, so the habit is read without touching the rate. A second habit that changes halfway widens it by 73 per cent on eight strips and 79 on six.

So the arrangement the reading is trying to detect — groups in a row — is the arrangement that most damages the reading. That is the asymmetry the rest of the measurement follows from.

In a row convicts; in turn acquits nobody #

With the groups known, panels of each kind were painted in thirty random orders at each fineness of reading, and the question asked of each was whether its groups came out in two runs along the order read.

On eight strips two hands taking turns were never read as one hand twice — none of thirty panels at any fineness, from a tenth of a pixel to a whole one. Chance alone would put two groups of four in two runs 2.9 times in a hundred, and the reading does better than chance because an alternating second habit leaves the rate of tiring nearly untouched: even where the order read misplaces some strips, it keeps the two hands’ strips alternating. One hand that changed habit was read with its groups in a row 93 times in 100 at a tenth of a pixel, 53 at a quarter, 17 at a half and 7 at a whole pixel.

Read the other way, those numbers say what each verdict is worth. At a quarter of a pixel, every panel read in a row came from one hand twice, and none from two hands: a reading of “in a row” is as near decisive as thirty panels can show. But every panel by two hands was read as alternating, and so were 47 in 100 by one hand who changed habit — so “alternating” is evidence of two hands at odds of about two to one, which is a hint and not a verdict. The test convicts one way. It can find one painter twice; it cannot clear a picture of being one.

On six strips both halves weaken. Two hands are read in a row 3 to 10 times in a hundred, which is chance for groups of three; one hand who changed is read in a row 60 times at a tenth of a pixel and 20 to 27 at coarser readings, not far above that chance. The two strips beyond six are what make the test work, for the same reason the earlier order essay found four strips a guess and six a reading: the order is one number per strip, and every number the fit takes for itself is one strip fewer to place.

What the second habit costs the order #

The contiguity test is only as good as the order beneath it, and the order is what the second habit charges.

With one steady habit — the panel the order essay read — eight strips give back their order 100 times in 100 at a tenth of a pixel, 90 at a quarter, 63 at a half and 13 at a whole pixel. With two hands taking turns the same figures are 100, 80, 30 and 10: the extra habit is one number more fitted from eight, and alternating it costs the order a little at fine readings and a good deal at coarse ones. With one hand who changed habit halfway they are 93, 50, 13 and 3. Past the finest reading the changed hand keeps about half of what the steady hand keeps, or less, which is the 73 per cent wider rate seen from the side of the result.

That ranking is also the explanation of the asymmetry above. A changed hand whose order is read wrongly does not come out in a row by chance: its groups are scattered through the wrong order, and it is read as alternating. So every loss of order turns a “one painter twice” into a false “two painters”, while two hands whose order is lost stay alternating whatever order is read. The errors all run one way.

Where the change falls #

A painter does not have to change habit halfway. A lesson after the first strip, or a new brush before the last, is a step one strip long.

From the arrangement alone, a short step should be the better case. A step one strip long is nearly orthogonal to a trend, and the rate of tiring’s standard error comes out at 1.31 times the steady hand’s for a change after the first strip and 1.34 before the last, against 1.73 halfway. The measurement only half agrees. Read in a row, sixty panels a point, the changed hand scores between 38 and 55 per cent wherever the change falls — no clear advantage for the short step. And the whole order comes back 22 times in 100 for a change after the first strip and 27 before the last, against 47 halfway.

The reason is the lone strip. A change after the first strip makes that strip a group of one, and a habit fitted from a single strip takes everything the strip says: its reported habit is matched exactly by its own habit, and nothing is left over to say when it was painted. Its place in the order is set by the other seven strips’ fit, which has no information about it at all, and it lands in the wrong place more often than any other strip. It is still often read as being in a row, because a lone strip placed at either end of the order read is a run by position alone. A painter who changed habit after one strip has, in effect, painted one strip that the order cannot see.

One number a strip cannot carry three things #

The reader so far has been given the groups. It is fair to ask what happens without them — whether the reported habits, which carry the order, can also carry the grouping, so that one search over every grouping and every order finds both.

They cannot. On six strips, searching every grouping of at least two strips a side together with every order, the groups come back for two hands in turn every time from exact rows, 23 times in 100 at a tenth of a pixel and 17 at a quarter; for one hand twice, 57 times from exact rows, 10 at a tenth and none at a quarter. With the groups given, the same panels’ orders come back 80 and 57 times in 100 at a tenth of a pixel. A strip’s reported habit is one number, and it is being asked to say which habit drew the strip, how tired the hand was, and when — three unknowns per strip with one number to read them from, held together only by the assumption that the fatigue is a straight line in time.

That is why the division of labour at the start is not a convenience. The groups have to come from what the rows say beyond their habit: the second habit’s split, which the essay before this one read from the full rows of every strip, not from one summary number of each. The rows count hands, not cameras is where that richer reading begins. Given the groups, the reported habits have only two things left to say, the fatigue and the order, and they say them.

What the verdict rests on #

The creep grows with the clock and is shared. Two painters working one panel are taken to tire together. If each tires with their own work instead, two hands taking turns leave two slower trends, one per group, and the reading’s single rate is the wrong model for them; a hand who changed habit still leaves one. That would make the test sharper in one direction — two hands would fit a single trend badly — and it was not measured here.

The groups are known. The map from the rows puts 94 per cent of strips with their own hand on eight strips. A misplaced strip is an alternation inside a run, and against groups in a row it would break the run; how often the map’s mistakes turn a correct “in a row” into “alternating” has not been measured, but it can only push the errors further in the direction they already run.

The order essay’s linear law. Each strip’s reported habit is shifted by its creep in proportion to bays over the logarithm of splay, to first order. At a point of growth a strip over eight strips the creep reaches 8 per cent, and the law’s own error is part of why the habits come back at 0.92 and 1.90 rather than 1 and 2. It is shared by every reading here and does not favour either arrangement.

Every hand sets each row fresh. A hand that steps its dividers adds to every strip a walk that a stepped hand’s walk is in its creeps found the strip’s own fit absorbs as a creep of its own. The reported habit would then carry a fourth thing that varies from strip to strip with no relation to time, and the order would pay for it as it pays for the slip.

Two habits, a whole habit apart. At half a habit apart the essay before this one found two hands harder to tell from a walk than from one hand; here a smaller gap makes the step smaller, which steals less from the rate and also gives the reading less to place. Both arrangements would be read more like one steady hand.

The general lesson, which the order essays keep arriving at #

A panel’s creeps keep the order it was painted in, while the hand is steady found, for a panel of painted pavements, that the creeps rank the pavements in the order painted — and that a creep set by something other than time, each pavement’s length, is read as a confident order that has nothing to do with the painting. This is the same finding seen from the other side. The order of painting is read from a drift in time, and anything else that drifts in time — a change of habit is the cleanest example — is read from the same drift. Where the two are orthogonal, as alternating hands are, both are read. Where they are not, as a single change is, the reading charges one to pay for the other, and the charge falls on whichever was less sharply determined to begin with.

That is also why the verdict is one-sided. A camera count needs a tolerance is the collection’s general statement that a reading earns a number only against what chance would give; here chance gives two hands in a row less than three times in a hundred, and a reading of “in a row” exceeds it overwhelmingly. Nothing comparable protects “alternating”, because a changed hand whose order has been lost is alternating by default. The same lopsidedness turns up wherever an order is read from something that changes along it: a walk’s order is on the floor, not the page reads a narrative’s sequence from where its copies stand, and it too returns the order only while nothing else — there, a walk that doubles back — puts neighbours in space out of step with neighbours in time. Of the two questions the earlier map left — two painters, or one painter twice — the order can answer the second with confidence and the first only with odds.

Still open: whether a hand’s own fatigue tells two painters from a shared clock #

Here both arrangements tire with the clock, so two painters and one differ only in where the second habit falls. Two real painters tire with their own work: each starts fresh and slows with the strips they themselves have painted, so a panel by two hands in turn carries two creeps growing at half the rate each, one per group, where a hand who changed habit carries one creep growing through the whole panel.

The measurement that settles what that is worth paints a panel twice more — two hands in turn whose creeps each grow with their own strips, and one hand who changed habit with one creep growing with all of them — and fits each with both models: one shared rate of tiring for the panel, and a rate for each group. It asks how often the better-fitting model names the arrangement correctly, at what fineness of reading, and whether the two-rate model rescues the case the order cannot settle — a panel read as alternating, which on the evidence here might be either. If two hands’ separate fatigue shows as two slopes where one hand’s shows as one, the question the order could only answer one way would be answered both ways, by a different number.