A painter's rate of tiring signs every strip but the first

Two painters on their own clocks leave no record of their turns gave each of two painters a creep that grows with their own work rather than with the day: each arrives fresh and slows by a point with every strip they themselves have painted. That made the panel forget how the two had taken turns, and it gave the panel a new thing to say — whether its fatigue runs as one trend through the session or restarts for each group — which, calibrated on panels of known make, named one hand or two painters 59 times in 60 at a tenth of a pixel.

Everything in that essay gave both painters one rate. Real painters tire differently. One works steadily through a long day and slips a point a strip; another is brisk and tires twice as fast. 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 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.

There are three things to find out. Whether eight strips can carry a rate for each painter at all, or whether the fifth unknown buys a fit the noise pays for. How far apart two rates must be before they are read apart. And whether rates read on one panel, carried to another, give that panel’s strips to their hands — which would sort a panel by two painters who share a habit, the case no reading of habits can touch.

Two creeps that grow at their own pace #

The panel is the eight-strip interior of the earlier essays — floor, footstool, table, book, step, lectern, sill, shelf — divided into bays by two painters at habits 1 and 2, four strips each, taking turns. Each strip reports the habit its rows were drawn with, shifted by the painter’s creep in proportion to bays over the logarithm of the strip’s splay — the law a tiring hand draws a different habit found, which makes a creep and a change of habit indistinguishable on any one strip. A painter’s creep on their first strip is the starting creep, one per cent a bay; on their second it has grown by their rate; on their third by twice their rate; on their fourth by three times.

Give the first painter a rate of a point a strip and the second a rate of two, and the model that reads the panel needs five unknowns where it had four: the two habits, the starting creep they share, and a rate for each. It is fitted, as before, over every order of each painter’s own four strips — 576 orders — because a painter’s rate is read through the sequence of that painter’s strips, and the sequence is not known.

The fifth unknown always fits better #

A model with a rate for each painter contains the model with one rate for both, so on any panel it fits at least as well. The question is whether it fits better by more than it does on panels where the two painters tire alike, and that needs the same treatment a camera count needs a tolerance gave a count of cameras: a line set on panels of known make, and a verdict scored on fresh ones.

Two hundred panels by painters who tire alike, read both ways, give the distribution of how much the second rate improves the fit when there is nothing for it to find. The line is set so that one of those panels in twenty is called unequal. Then fresh panels are painted with the second painter faster by a quarter of a point, half a point, a point and two points a strip, sixty of each, and each is called unequal if the second rate improves its fit by more than the line.

At a tenth of a pixel, painters who tire alike are called unequal 8 times in 100 on the fresh panels — the line was set to hold that near five, and sixty panels drawn again land a little above it. A gap of a quarter of a point is read 50 times in 100, half a point 85, a point 97, two points 98. At a quarter of a pixel the same gaps are read 20, 37, 58 and 90 times. So at the fineness the earlier essays found a careful hand works to, a panel says whether its two painters tire alike from a difference of half a point a strip — a creep that grows by four and a half points over one painter’s four strips where the other’s grows by three.

The panel says that much more readily than it says how large the difference is.

At a tenth of a pixel a gap of a point a strip is read as 0.99 points with a standard deviation of 0.76; a gap of two as 1.75 ± 1.12. At a quarter of a pixel a gap of a point is read as 0.49 ± 1.75, which is a reading that says almost nothing about its own size. The reason is that each painter’s rate is fitted from four strips that also have to fix that painter’s habit, with the order of the four searched. Within one painter’s strips a faster rate can be traded against a different order and a slightly different habit, and the search finds the trade. What cannot be traded away is the misfit a wrong single rate leaves across both painters at once, and that is what the detection reads. A panel tells that two painters tire differently long before it tells by how much.

What the fifth unknown does to the verdict #

The earlier essay’s verdict compared the own-work model with the clock model: a fit in which the creep grows with each painter’s own work against one in which it grows with the session, whose ratio, calibrated, separated two painters on their own clocks from one hand who changed habit halfway through. That comparison was fair because the two models had the same four unknowns. With a rate for each painter the own-work side has five, and the question is whether the verdict survives the extra freedom.

It costs the verdict when the painters tire alike, and rescues it when they do not. At a tenth of a pixel, with painters who tire alike, one rate names one hand 95 times in 100 and two painters 100; a rate each names them 70 and 90. The fifth unknown fits noise, as a second lamp fits the shadows of a one-lamp drawing in how many lamps a drawing has, and the noise it fits is noise that also helps a one-hand panel look like two painters, so the calibrated line has to move and blurs as it does. With painters at one and two points a strip the order reverses. One rate now names one hand only 57 times in 100, because a single rate misfits the two painters’ panels until they resemble a clock, the line set to catch them moves up, and it passes the changed hands it ought to leave alone. A rate each names one hand 90 times and two painters 83.

At a quarter of a pixel the one-hand side is poor whichever model is used: with painters who tire alike a rate each names one hand 25 times in 100 against one rate’s 80, and with painters at one and two points a strip 35 against 18. Two painters are named between 60 and 85 times in 100 either way. The earlier essay found a quarter of a pixel the fineness at which the verdict had already lost most of its force, and asking five unknowns of eight strips there spends the rest.

So the fifth unknown is not free and not forbidden. A reader who knows the two painters tire differently — from the detection above, or from a second panel — should give them a rate each. A reader who does not should not, because on painters who tire alike it costs a quarter of the changed hands.

A rate carried to another panel #

The more interesting use of a rate is the one the earlier essay hoped for: as a mark that travels. Suppose the same two painters worked a second panel in the same building, and on that panel they happened to share one habit — both dividing at habit 1, as two members of one workshop might. Nothing in the habits now separates their strips, and every reading in this line of essays that split a panel into hands — one more habit takes two hands is the first — split it by habit. Do the rates, read on the first panel, give the second panel’s strips to their painters?

The reading is a search, as every reading here has been. Every way of splitting the eight strips into four and four, and every order of each four, is fitted with the two rates held at the carried values; with the rates known, what is left to fit is one habit and the starting creep. The split that fits best is the reading.

Strips no rate can assign #

Before the rates can say anything, there are strips about which they can say nothing, and the figure at the top of this essay is a picture of them. Each painter’s creep on their strips has grown from the common start by their rate times the number of strips they have already painted: by 0, 1, 2 and 3 points for the painter who tires at a point a strip, by 0, 2, 4 and 6 for the painter who tires at two. Two strips whose growths coincide report exactly the same habit whichever painter is said to have painted them. A reading holding the rates known cannot choose between giving the first painter’s third strip and the second painter’s second to their own painters or to each other.

The first strips always coincide, at nothing: a painter’s first strip carries the starting creep and no rate at all, so on a panel where the two share a habit their first strips can always be exchanged. That is exact — the fit after the exchange agrees with the fit before it to the last digit of the arithmetic — and it is the reason the rest of this essay scores only the strips after each painter’s first. A rate is a signature of every strip a painter paints after their first, and of their first strip not at all.

Beyond the first strips, coincidences happen when one rate is a small whole-number multiple of the other, or a ratio of two small whole numbers. At rates of one and two the first painter’s third strip ties with the second painter’s second. At one and three, the fourth with the second; at one and one and a half, the fourth with the third. At one and one and a quarter, one and seven-tenths, one and two and a half, nothing ties but the first strips, because no later growth of one painter equals a later growth of the other. Four strips each are few enough that only ratios built from 1, 2 and 3 can tie at all.

What the rates sort #

With those ties understood, the measurement is clean.

Carrying the true rates, with the second panel’s rows read to a tenth of a pixel, the later strips go to their own painter 64 times in 100 when the two tire alike — which is what the search finds when there is nothing to find, since a split into two groups of three always agrees with the truth on some of them. At one and one and a quarter it is 98; at one and seven-tenths every later strip on thirty panels; at one and two and a half, every one again. At the tied ratios the share falls, as it must: 86 at one and a half and at three, 88 at two. Each of those panels has one pair of later strips that no reading can assign, and it is assigned right about half the time.

Carrying rates read on a first panel rather than known costs what the earlier figure said the rates are read to. At a twentieth of a pixel on both panels, the untied ratios give 99, 94 and 99 in 100; at a tenth of a pixel, 84, 88 and 94. A rate read to within three-quarters of a point is a blunt instrument for sorting strips whose rates differ by a quarter. It is a good one for rates that differ by more than the error they were read with, which is the condition every carried measurement has: a third ray is worth what its picture is worth is the same statement about cameras.

What the signature is a signature of #

A rate that sorts strips is a rate that belongs to a painter only if the painter keeps it. That is an assumption about people, and three things about it are worth stating.

A painter’s rate is a rate of one session. The rates here grow with the strips a painter has painted that day. A painter who comes back fresh to a second panel restarts their creep, which the measurement allows, but one whose rate depends on the light, the plaster or the scaffold carries a rate of the conditions rather than of the hand. Two painters working under one set of conditions might then share one rate on one panel and differ on another, and a carried rate would sort the second panel by the first panel’s conditions.

The starting creep is shared. Both painters arrive with the same creep, and that is what makes each painter’s first strip exchangeable. Two painters who start differently — one already a point into fatigue from other work — have first strips that differ by that point, and the first strips would carry a signature of their own: not of the rate but of the start, which a single panel cannot separate from a habit.

The second panel’s painters are known to be the same two. The reading carries two rates and assigns every strip to one of them. A panel by one of the two and a third painter, or by one painter alone, is assigned all the same, and the reading cannot say that it should not have been. A camera count needs a tolerance is the general warning: a reading that is asked how to divide something will divide it.

Where the panel’s evidence stops #

The rows count hands, not cameras began this line of reading with the claim that two habits in one picture mean two hands, and each essay since has found the count to be of a mechanism rather than of a person — a habit, a creep, a clock. Groups in a row are one painter twice read a change of habit from the order of painting; the earlier essay read the number of fatigues from the shape of the creep. This one adds a mechanism that is more nearly personal than any of them: how fast a hand tires. It is the first mark in the series that could sort strips by painter when the painters share a habit.

It is also the first whose blind spots are arithmetic rather than statistical. More strips, finer reading, a better line — none of them assigns a painter’s first strip, and none of them separates painters whose rates stand in the ratio of two small whole numbers at the strips where their growths meet. That is the same kind of blindness a tiring panel keeps its order, not its direction found in a single creep, which carries the order its strips were painted in and not the direction, and the same kind a strip’s scatter points to the end drawn last escaped by reading a second property of the rows. The ties here could be escaped the same way, if a strip carried anything that grows with the number of strips its painter had painted other than the creep.

Still open: whether the scatter breaks the ties the creep cannot #

A painter’s first strip and the other painter’s first strip carry the same creep, and so do two later strips whose growths meet. But a tiring hand does not only creep; a strip’s scatter points to the end drawn last found that its slips grow as it tires, so that a strip painted late in a painter’s sequence is drawn more raggedly than one painted early. Two strips with the same creep need not have the same scatter. A first strip by a painter who tires fast is a fresh hand’s strip, and so is a first strip by a painter who tires slowly; but a third strip by the slow painter and a second by the fast one, tied in creep, were painted at different points in two different hands’ sequences, and if the slips grow with the count of strips rather than with the creep, they differ.

The measurement that settles what that is worth gives each painter’s slips a growth with the number of strips that painter has painted, reads each strip’s scatter alongside its habit, and asks three things: whether the scatter breaks the tie between strips whose creeps meet, at the rates where the creep alone assigns them half the time; whether anything breaks the tie between two first strips, which are both a fresh hand’s and should carry the same scatter as well as the same creep; and how much of the scatter’s evidence is spent identifying the order of a painter’s strips, which the search over orders here had to find from the creep alone.