The workshop that throws drawings away

What survives being copied measured two copyists and left one mechanism out on purpose. A mark copyist redraws what is in front of them and accumulates a hand error every generation, which is a random walk: the spread across lineages grows as the square root of the generation count, fitted at 0.5001 against a predicted one half. A method copyist redraws from a remembered recipe and forgets the sheet’s own error at once, which is not a walk at all — its spread does not grow. That essay named a third mechanism, a workshop that discards a copy which looks wrong, and set it aside as the extension “most likely to matter in practice and is not modelled here at all.” This essay models it.

Adding one rule — reject a proposal outside a band of acceptable appearance, keep the old sheet instead — changes the mark copyist’s law rather than its size. The random walk becomes a stationary process, indistinguishable in its rate of spreading from the method copyist it was never confused with before, and the two mechanisms then separate only by where they settle rather than by how fast they move.

A third lineage on the same panel #

Both of the original copyists share one hand: the same generator, the same error at every step, applied to the same starting sheet. A workshop that also rejects bad copies is given that identical hand, so any difference between the three lines is a consequence of the rule and not of a noisier draughtsman.

The rejection is not a redraw with a smaller error. It is a refusal: the workshop’s own last sheet stands in for the generation that would otherwise have been accepted, error and all, so a rejected generation costs the lineage nothing beyond the time it took to look. That is the whole mechanism, and it is one rule against the mark copyist’s own — measure, compare against a taste, keep or discard.

The band the workshop compares against is what this essay calls an acceptance region: a range of the tracked ratio the workshop is willing to let stand, centred on whatever it currently regards as correct. Nothing about the mechanism requires the acceptance region to be narrow, or centred on the truth, or even centred on the taught rule — it is centred, in every figure here, on 0.7, a value chosen only because it sits between the two established endpoints and lets both a wide and a narrow band be tried without either swallowing the other. What the acceptance region does, at any width, is turn a rule that was purely local — does this one proposal look acceptable — into a fact about the whole lineage, because every kept generation becomes the standard the next proposal is measured against.

Why a bounded region produces a bounded spread #

The saturation figure below states the arithmetic plainly, but the reason for it is worth having first, because it is a general fact about confinement and not a fact peculiar to copying.

A quantity drawn uniformly across a band of half-width $a$ has variance $a^2/3$ — the standard result for a uniform distribution — so its standard deviation is $a/\sqrt{3}$. A random walk that is reflected, or refused, whenever it would leave a band of that half-width does not stay uniform inside the band, but it is confined by the same boundary, and a process confined to a fixed region has nowhere left to grow into: its spread is set by the size of the region rather than by how long it has been running. That is the entire content of “the spread saturates” restated without a single lineage in it, and it is why the acceptance region’s own half-width, rather than the hand error or the generation count, is what the saturated spread should track.

The established law, recalled rather than re-derived #

Before asking what the third line does, it is worth having the first two on the same page in their own established form, because the claim about the third line is a claim about how far it has moved from one of them and how close it has moved to the other.

Nothing about that figure is recomputed for this essay — it is the earlier essay’s own instrument, called again, and the numbers agree with what what survives being copied already reported to four figures. That agreement matters more here than it did there: a claim that selection moves a mark copyist’s exponent from near one half toward near zero is only interesting if “near one half” is a number this collection has already pinned down independently, rather than a fresh fit invented to make the new claim look dramatic.

The three regimes together #

Run the same three lineages further, over five hundred repetitions each, and fit an exponent to each family’s spread against the generation count.

The walk’s own fitted exponent here, 0.542, sits a little high of the 0.5001 the nine-hundred-lineage figure above reports, and the gap is sampling: five hundred lineages fitted for a three-way comparison is not the nine hundred a single dedicated measurement can afford, and the earlier figure is the one to trust for the walk’s own value. What the three-way panel is for is not re-measuring the walk — it is putting all three regimes under one hand error so that the comparison between them is not begging any question about whether they were run under different conditions.

Read that way, the finding is not “selection slows the walk down.” A slower walk would still be a walk: its spread would still grow, merely at a smaller rate, and the exponent would still be one half of something. What selection produces is an exponent of 0.022, statistically indistinguishable from zero and from the method copyist’s own 0.033. Selection does not shrink the growth. It removes it.

A law with no fitted parameter #

That the selected process stops growing invites the next question: stops growing at what value, and does that value depend on anything other than how wide the workshop’s taste is.

A confined random walk — one that cannot leave a bounded region because anything that would take it outside is refused — has a stationary spread fixed by the width of its confinement and nothing else, and √3 is the specific divisor a uniform confining band gives. That the fitted exponent lands at 0.993, a percentage point off a predicted one, is what separates “the spread saturates” from “the spread saturates at a curve somebody chose because it looked right.” The 5.8% worst departure is the width of the disagreement between an idealised uniform-band calculation and lineages that are still finite, run for a finite number of generations, and it does not grow with the band — it is the largest of six comparably-sized errors, not a trend.

The control: opening the band until it stops mattering #

A measurement that only ever shows the effect present is not a test of the effect. The band in every figure so far has width 0.0233 or wider; the honest question is what happens as that width is opened up until the workshop is barely rejecting anything at all.

Four per cent is not zero, and the honest form of the control goes further than any single picture here can show: widen the band enough and there is eventually nothing left for the rule to refuse, at which point a workshop that “discards a bad copy” is discarding none, and the selected lineage has to be the mark copyist’s own random walk under a different name. That is not a numerical claim about where the crossover happens — it follows from what the rule does. A rejection rule that rejects nothing has removed itself from the mechanism, and a mechanism with the rejection removed is definitionally the walk already measured above. The saturation law drawn a moment ago is consistent with exactly this limit: its predicted spread grows with the band’s own half-width, without any ceiling written into the prediction, so a band wide enough to enclose the walk’s own unconstrained excursions imposes no constraint at all. A control that must recover an already-established case, rather than merely look plausible, is the kind this collection asks for everywhere, and this is what it looks like here: not a new number, but the old law surviving a limit that removes the very thing being tested.

Not the same mechanism as blending in the method #

What survives being copied already has a mechanism that stops a spread from growing without limit, and it is worth being precise about why the acceptance region is a different one rather than a second name for it. That essay’s own extension lets a fraction $\alpha$ of every generation come from redrawing the method rather than measuring the sheet, which also produces a stationary spread — settling, by its own reckoning, at seven times the hand’s own error when $\alpha$ is one in a hundred, at 2.3 times when it is one in ten, and at 1.15 times at $\alpha = 0.5$ — and a half-life for the memory of the original that shortens from sixty-nine generations to seven to one across that same range.

That mechanism stops growth by continuously injecting a little of the method into every generation, whether or not any particular proposal looks wrong. The acceptance region stops growth by refusing outright whenever a proposal looks wrong, and otherwise doing nothing different from the plain mark copyist at all — no method is ever redrawn, no recipe is ever consulted, and a lineage that never once produces a bad proposal in a given run behaves exactly like the unselected walk from the earlier essay, indistinguishable from it not merely in law but in every sheet it produces. The two mechanisms converge on the same symptom — a spread that stops growing — from opposite mechanical premises: one is continuous and mixes in the answer, the other is a threshold and only ever discards. A workshop could in principle run both at once, and nothing in either model would say which was responsible for an observed stationary spread; that is a real limitation of reading a settled population back to a cause, and it is recorded here rather than resolved, because resolving it needs a further discriminator neither model currently supplies.

What survives when the spread cannot separate them #

If selection makes the mark copyist’s spread indistinguishable from the method copyist’s, a reader holding only that one statistic has lost the ability to tell the two traditions apart. Something else has to carry the distinction, or the finding above would mean the two mechanisms had actually become the same mechanism, which they plainly have not.

Two statistics survive where the growth rate does not. The location: a selected mark copyist settles on the workshop’s own taste, here centred at 0.7, because that is the value the acceptance rule is built around and confinement pulls everything toward the middle of what it permits. A method copyist settles on the taught recipe, 0.8, because that is what the recipe says and nothing about redrawing from it ever samples anywhere else. Those two numbers are 7.5 times the (shared) spread apart, which is a separation a reader could find with nothing more than a straightedge on a histogram, no exponent required. And the shape: a process confined by a hard rejection boundary piles up density near the edges of what it is allowed to be, which is what an excess kurtosis of −1.26 describes — flatter than a bell curve, with more of its mass away from the centre than a Gaussian of the same spread would carry. A recipe redrawn with independent Gaussian hand error at every generation is, unsurprisingly, closer to an actual Gaussian: excess kurtosis 0.37.

So the statistic that finds the earlier essay’s own warning — “the two readings are exclusive, since the same mechanism that preserves one destroys the other” — is not the spread once selection is present. It is where the population sits and what shape it has piled into, and both require nothing more exotic than the same set of copies that measured the exponent in the first place.

What a selected tradition looks like from one sheet #

The distinction above is a statement about a population of drawings, which is not usually what survives. A reader ordinarily has one late sheet, not fifteen hundred lineages to histogram.

One sheet cannot report a kurtosis, but it can report where it sits relative to two candidate centres — the rule that draws another room’s own taught ratio and, if independently known, a workshop’s own house style. A single late copy from a selected mark-copying tradition and a single late copy from a method-copying tradition are, taken alone, both consistent with “settled near some number”; what separates the two readings is knowing which number a candidate centre actually is. The slip that leaves no trace already established that a single hand’s own slip moves a panel’s implied horizon by an amount an unaided eye cannot catch, and the logic here is the same shape one level up: the population statistic that would resolve the ambiguity — a spread growing or not growing — is exactly the one thing a lone sheet cannot supply, because a spread needs more than one sheet to compute. Four marks before anything is said sets out what a reader can extract before any interpretation begins, and a settled centre with no accompanying population is one honest answer to add to that discipline: it says where a tradition landed, and says nothing at all about how it got there.

The hand error itself is not a free assumption either. One hand step each prices a single unsteady step at the scale of one panel, and stepped, or measured from the zero shows that two ways of laying out the same spacing accumulate that step differently even before any copying enters the picture — which matters here because every lineage in every figure above is built from that same one-step error, repeated and, in two of the three regimes, either accumulated or refused. A workshop whose own hand is steadier or shakier than the value used here would saturate at a different spread, by exactly the acceptance region’s own half-width divided by √3, but it would saturate — the law is about the shape of the confinement, not the size of the step that is being confined. What a panel says about its maker asks the individual-hand version of this question; a selected lineage’s settled centre is the same question asked of a workshop across generations rather than of one sheet.

The honest limit #

This model has the same scope as the one it extends, with one addition and the same omissions. The addition is a single new parameter: the width of the band a workshop will accept. Everything else — one hand error, one ratio being tracked, two pure copying strategies feeding a shared refusal rule — is exactly as thin as what survives being copied already flagged its own model to be, and the thinness is deliberate rather than an oversight: a model with one new knob that changes the law rather than merely the size of an effect is a stronger result than a model with ten knobs that changes nothing qualitative.

What is not modelled is a workshop whose taste itself drifts — the acceptance band’s own centre wandering across generations rather than sitting fixed at 0.7, which would put a random walk back inside the mechanism at one remove, riding on the boundary rather than on the marks. Nor is there a model of a workshop that rejects for reasons uncorrelated with the ratio under study — rejecting on colour or on cost of materials rather than on this one construction’s resemblance to a taught appearance — which would leave the spread untouched and give a different kind of selection than the one measured here. Both are real workshops; neither is this one.

What this is an instance of #

A tradition is a system for moving a picture through time, and what survives being copied already read that as this field’s own version of the trade every drawing system makes — record one thing, spend another. Selection sharpens rather than overturns that reading: what a selected tradition preserves is not the original and not quite the recipe either, but the workshop’s own standard of acceptable appearance, indefinitely, at the price of ever converging on anything the recipe or the first sheet actually said. What perspective gave up makes the same point about a drawing system rather than a lineage of copies — a station point bought at the price of a ratio, a plan bought at the price of a height — and a workshop’s taste is a station point of its own kind: fixed, informative, and not free.

The three-procedure comparison in three procedures, one panel asked what a single surviving sheet says about the hand that made it, before any question of a lineage arises at all. This essay’s own answer to the equivalent question about a lineage is the pairing above: a growing spread names a mark copyist with no selection at work; a spread that has stopped growing names either a method copyist or a selected mark copyist, and the population’s centre — 0.7 against 0.8, here — is what tells the second pair apart, because the rate of spreading, once selection is present, no longer can.