Two painters on their own clocks leave no record of their turns

Groups in a row are one painter twice; groups in turn say little asked a panel whose strips fall into two habits who the two groups were — two painters, or one painter before and after a change — and read the answer from the order of painting. A tiring hand writes that order into the panel, because each strip’s reported habit is shifted by a creep that grows as the session goes on, and the order that makes the shifts fit one rate of tiring is the order painted. One hand that changed habit leaves its two groups in a row along that order; two hands taking turns leave them alternating. The verdict came out lopsided. A reading of “in a row” convicted one painter twice; a reading of “alternating” acquitted nobody, because a changed hand whose order has been lost is alternating by default.

That essay made one assumption it said it had not tested: that the creep belongs to the clock. Two painters sharing a panel were taken to tire together, as they would if the light were failing or the plaster drying. Real painters mostly tire with their own work. Each arrives fresh and slows with the strips they themselves have painted, so on a panel by two hands the creep is not one trend through the day but two, each starting from nothing and each growing with one painter’s strips alone.

That changes what the panel can say, in two directions at once. It removes something the earlier reading thought it had, and it supplies something the earlier reading never used.

Two clocks keep no common time #

Give two painters at habits 1 and 2 the eight strips of the earlier panel — floor, footstool, table, book, step, lectern, sill, shelf — four strips each, and let each painter’s creep grow by a point with every strip that painter has painted: one per cent a bay on their first strip, two on their second, and so on. Then paint the panel twice. Once the painters take strips in turn. Once one of them paints all four of their strips and then the other paints all four. Each painter keeps the same order of their own strips both times.

The two panels report the same habit on every strip to the last digit, at every fineness of reading. That is not a near agreement found by measurement; it follows from what an own clock is. A strip’s creep depends on how many strips its own painter had painted before it, and nothing else. Interleave the two sequences any way at all — in turn, one after the other, three of one and then four of the other and then the last — and every strip keeps the same count, so every strip keeps the same creep, and the panel is the same panel.

On one shared clock the same two arrangements are different panels. A strip painted seventh in the session carries a creep of seven points whichever painter painted it, and moving a painter’s strips from the alternate places into the last four moves their creep by up to three points; on the steepest strip that shifts the reported habit by a whole habit and a half. The interleaving the earlier essay read was real evidence there. Here it is not evidence of anything, because it is not in the panel.

So the first thing own clocks do is take away the question the order was answering. Two painters on their own clocks have no order of the panel, only two orders of their own; asking whether they took turns is asking the panel something it never recorded.

Alternation is one clock imitating two #

The earlier reading did not know that, and it is worth seeing what it does with panels it was never built for. Given the groups and asked for the one painting order that best fits one creep on one clock, it returns an order every time; the question is what order.

It reads both two-painter arrangements as alternating, and it reads them so almost without exception — at a tenth of a pixel, none of sixty panels painted in turn is put in a row, and one of sixty panels painted one after the other is. A changed hand on one clock is put in a row 52 times in 60 at the same fineness. The panels painted one after the other are the ones that matter. Their groups really were in a row, four strips and then four, and the reading puts them in alternation because that is the only way one clock can carry two creeps that each grow half as fast. A painter’s first strip and the other painter’s first strip carry the same small creep; on one clock that is two strips placed next to each other near the start, then their two second strips next, and so on. The clock reads “two fresh painters” as “two painters taking turns” because it has no other words for it.

That makes the earlier verdict’s weak half weaker than it said. “Alternating” was a hint of two hands at odds of about two to one, set against a changed hand whose order had been lost. Against two painters who tire on their own, it is not a hint of turns at all. It is a description of two fresh starts, and it would be returned if the painters had worked on different days.

A longer search fits better whether or not it is right #

What the own clocks supply in exchange is a second model to fit. Where the clock model says each strip reports its habit shifted by a creep set by its place in the session, the own-work model says the creep is set by its place in its own painter’s sequence. Both models have the same four unknowns — two habits, a starting creep and a rate of tiring — so on the face of it the one with the smaller residual is the better account, and the arrangement it implies is the verdict.

On the face of it is not good enough, and the reason is the search. The clock model is fitted for every painting order of eight strips, 20,160 of them, or 10,080 up to reversal, which a steady rate cannot tell apart. The own-work model is fitted only for every order of each painter’s four strips, 24 times 24, which is 576. Both report the best of what they tried.

On a panel by two painters on their own clocks, read to a quarter of a pixel, the clock model allowed the same 576 tries is clearly worse than the true model: a residual of 0.24 of a habit against 0.12. Allowed a thousand it is at 0.15. Allowed all ten thousand it finds an order that fits at 0.035 — three times better than the model that actually painted the panel. It has not found anything true. It has tried seventeen times as many candidates, and among ten thousand ways of arranging eight noisy numbers along a line there is one that the noise happens to favour.

This is the problem how many lamps a drawing has met with shadows: two centres fit any bundle of shadow lines better than one, on a one-lamp drawing as readily as on a two-lamp one, because the second centre is freedom and freedom always fits. Here the freedom is not a parameter but a search, and it is invisible in a count of unknowns. Four unknowns against four, compared at face value, the clock model wins 49 of 60 two-painter panels at a quarter of a pixel and 59 of 60 at half a pixel. A reader who counted unknowns and compared residuals would conclude that painters share a clock, from panels made by painters who did not.

A line set on panels the verdict never sees #

The repair is the one a camera count needs a tolerance makes for counting cameras: do not ask which model fits better, ask whether the fits differ by more than they differ on panels of known make. Paint sixty panels by one hand who changed habit halfway on one clock and sixty by two painters in turn on their own clocks, read each both ways, and record the ratio of the two best residuals — own-work over clock, so a small ratio favours two clocks. Choose the line on that ratio that misnames the fewest of the hundred and twenty. Then throw those panels away, paint sixty fresh ones of each kind, and score the line on them.

The line lands nowhere near 1. At a tenth of a pixel it sits at 1.90: the own-work model has to fit a two-painter panel nearly twice as badly as the clock model before the panel stops looking like two painters, because that is how much the clock model’s longer search is worth on such panels. At a quarter of a pixel the line is at 2.04, at half a pixel 2.21. The medians show why the face-value line fails. At a quarter of a pixel a changed hand’s ratio has a median of 3.19, and two painters’ of 1.62 — both above 1, both favouring the clock model, but by very different amounts. The difference between the panels is in the size of the clock’s win, and only a calibrated line can read a size.

The calibration needs the slip, which is not free. A reader of a real panel does not know in advance how finely its rows were drawn. But the slip is in the same rows: what each strip leaves over once its own habit is fitted is the hand’s scatter, row by row, and it can be read before any question about painters is asked. And the line moves slowly with it — from 1.69 at a twentieth of a pixel to 2.21 at a half — so a slip read to within a factor of two sets the line to within a tenth.

A tenth of a pixel names both, a quarter names one well #

Scored on the fresh panels, the calibrated line does what the face-value line could not.

At a twentieth and at a tenth of a pixel it names one hand 59 times in 60 and two painters 59 or 60 times in 60. At a quarter of a pixel it names one hand 52 times in 60 and two painters 40 times. At half a pixel it names one hand 46 times and two painters 27 — and 27 in 60 is worse than a coin, which is what a line set at equal error does when the two kinds of panel have stopped differing: it spends its errors where they are cheapest. The face-value line, meanwhile, names one hand every time at every fineness and two painters 59, 43, 11 and 1 times in 60. It is not a test. It is a preference for the model that searched more.

The calibrated verdict is also a different kind of verdict from the order’s. At a quarter of a pixel a reading of “two painters” comes 40 times in 60 from two painters and 8 times in 60 from one hand, odds of five to one; a reading of “one hand” comes 52 times from one hand and 20 from two painters, odds of about two and a half to one. Neither half is decisive at that fineness, but neither is empty, which is the property the order lacked. The order convicted one way and could not acquit; the shape of the fatigue answers both ways, more strongly in one.

What the order could not settle #

The earlier essay’s hardest case was a panel by one hand who changed habit, read as alternating because the change had stolen enough of the rate of tiring to lose the order. That case is not rare. Over a hundred and fifty panels rather than thirty, a changed hand’s groups come out in a row at a quarter of a pixel 37 times in 100 — fewer than the 53 the thirty panels of the earlier essay showed, which was a fortunate draw — and at a tenth of a pixel 90 times rather than 93. So at a quarter of a pixel nearly two changed hands in three are read as alternating, and on the order’s evidence are as likely to be two painters.

The fatigue model rescues most of them. At a quarter of a pixel the order puts 14 of sixty changed hands in a row and reads the other 46 as alternating; of those 46 the calibrated fatigue model names 38 as one clock. At a tenth of a pixel the order misses 8 and the fatigue model recovers 7. At half a pixel the order misses 56 and the fatigue model recovers 43. The reason it can is that it does not need the order to come back. The order is one number per strip read for its place in time, and a change of habit steals from exactly that. The fatigue model asks a coarser question — does the creep run as one trend through the panel or restart for each group — and a coarse question survives a noise that a fine one does not.

That is also the condition on the rescue. It assumes the two painters on the other side of the comparison tired on their own work. If they shared a clock, the alternative to one changed hand is the earlier essay’s two painters in turn, and the fatigue model has nothing to say about them, as the next figure shows.

The verdict counts fatigues, not people #

What the fatigue model actually reads is how many creeps the panel carries: one that runs through the session, or two that start fresh. That is the number of painters only under two conditions, and each can fail.

The first condition is that painters tire on their own work. Two painters taking turns whose creep grows with the session — the failing light, the drying plaster, the arrangement the earlier essay modelled — are named one clock 47 times in 60 at a tenth of a pixel. They are two people with one fatigue, and the verdict counts the fatigue. The second condition is that a single painter does not rest at the change. A hand that changed habit after a break and came back fresh paints its second group with a creep that starts again from nothing — which is, strip for strip, the panel painted by two painters one after the other on their own clocks. The verdict names those panels two clocks 57 times in 60 at a tenth of a pixel, and it would name a rested hand the same, because it is the same panel.

So the two readings now available answer different questions, and neither answers the one first asked. The order asks whether a second habit fell at one moment of one session: in a row says yes. The fatigue asks whether the session’s tiring restarted: two clocks says yes. A painter who changed without resting gives in a row and one clock; two painters on their own clocks give alternating and two clocks; two painters on one clock give alternating and one clock; a painter who rested gives alternating — the clock model’s word for any fresh start — and two clocks. Two of the four combinations name a cause, and two leave a choice between causes that the rows of a panel cannot make.

What the reading leans on #

Both painters tire at the same rate. The own-work model gives the two painters one rate of tiring and one starting creep, which is what makes it four unknowns and comparable with the clock model at all. Two painters whose rates differ would fit it worse, and might be pushed towards a verdict of one clock by a misfit that has nothing to do with the session. A rate for each painter is a fifth unknown from eight numbers, and it was not tried here.

The groups are known. As in the earlier essay, the grouping comes from the rows through the second habit’s split, which one more habit takes two hands found puts 94 strips in 100 with their own hand on eight strips. A misplaced strip gives one painter five strips and the other three, and both models then fit a wrong grouping; how unevenly that falls on the two models was not measured.

The line is set on panels like the panel read. The calibration panels share the panel’s eight strips, its two habits a whole habit apart and its rate of a point a strip. A panel of other strips needs its own line. That is the price of every calibrated count, and it is cheap here because the panels are made, not found.

A strip’s reported habit follows the counterfeit law to first order. Each strip’s habit is shifted by its creep in proportion to bays over the logarithm of its splay, the law a tiring panel keeps its order read the order from. Own clocks keep the creeps small — no painter passes four points — so the law’s own error is smaller here than on one clock, where the last strip carries eight.

Every row is set fresh. A hand that steps its dividers adds to each strip a walk that a stepped hand’s walk is in its creeps found the strip’s own fit takes for a creep of its own. Such a creep belongs to neither clock, and both models would pay for it as they pay for the slip.

The general shape of the answer #

The rows count hands, not cameras began this line of reading with a claim that sounded like counting people: two habits in one picture mean two hands. Each step since has found the count to be of something a little different. A panel’s creeps keep the order it was painted in found that a creep read as time is time only if nothing else grows with the strips. The earlier essay found that a step in habit is read as part of a trend in time. This essay finds that a fatigue counts clocks. The rows record mechanisms — a habit, a creep, a clock — and a person is a bundle of mechanisms that need not travel together.

What carries over from the lamps, and from a walk’s order is on the floor, is the asymmetry of searching. A reading chosen from more candidates is always closer to the data, and the closeness is worth nothing until it has been set against what the same search does on data of known make. Here the size of the search was hidden behind an equal count of unknowns, and it was enough to make the right model lose four times in five.

Still open: whether a painter’s own rate of tiring is a signature #

Everything here gave both painters one rate of tiring. Real painters tire differently, and a rate that belongs to a painter could be more than a nuisance for the model: it could be a mark of the hand. Two panels in one building painted by the same two people would then carry the same pair of rates, and a strip whose creep grows at one painter’s rate would belong to that painter whatever its habit.

The measurement that settles what that is worth gives each painter their own rate — one a point a strip, the other two, say — fits the own-work model with a rate for each group as a fifth unknown, and asks three things. Whether eight strips can carry five unknowns and still leave the calibrated verdict what it is here, or whether the fifth unknown, like the longer search, buys a fit the noise pays for. How far apart two rates must be before they are read apart at a tenth of a pixel. And whether a rate read on one panel, carried to a second panel by the same two painters, assigns that panel’s strips to their hands — which would give the rows a way to find a painter’s strips that does not depend on the habit at all, and so would still work on two painters who share one.