Limits — the series
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Seventeen observers in 1931
The standard observer that governs every colour specification in industrial use is an average over seventeen young British men, measured with equipment from the 1920s. It is known to be wrong in the blue, the correction has existed since 1951, and it has never been adopted.
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Simulating what cannot be simulated
A colour-blindness simulation cannot show what anybody sees. What it can show is which discriminations survive, and that is a narrower claim, a checkable one, and the only one worth making.
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The display is an unknown
This site is displayed on the very apparatus it is about, and it knows almost nothing about that apparatus. Two figures here stop assuming and ask instead — a probe for the transfer function and a probe for the gamut.
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Whose eyes
The standard observer is an average over seventeen people, and no reader is it. What that costs was small when displays were broad and grows every time the primaries get narrower.
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How bright is white
An sRGB value says a pixel is some fraction of whatever the display can manage. A PQ value says it is two hundred candelas. The change is the largest in display encoding since gamma, and what it removed was the honest admission that nobody knew.
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The triangle is a shadow
A display's gamut is drawn as a triangle on the chromaticity diagram, and coverage is quoted as a percentage of that triangle's area. Chromaticity has luminance divided out, so the triangle is a projection along the axis the eye cares most about — and the ratios computed on the solid are not the ratios computed on its shadow.
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Banding is not a bit depth
Eight bits bands at thirty times threshold in the shadows and at under three at mid grey, on the same ramp with the same encoding. What decides is where in the tone scale the gradient sits, how sharp each step's edge is, and how far away the reader is — and a bit count contains none of the three.
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The model has no clock
An appearance model takes a stimulus and a situation and returns what it looks like. It does not take a time, and adaptation is not instantaneous — half a second after the light changes a judgement is three and a half CAM16-UCS units from the settled one, and a minute later it is still 1.3.
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Two degrees or ten
This site names the observer on every figure it draws, which was supposed to be the point. Measuring what it had actually done found two hundred and four figures naming the same observer, three naming the other, and sixty-two of sixty-four generators printing a string that had never once varied.
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The gamut shrinks in the dark
Every gamut number here is a property of a device and of nothing else, which is why the same display has the same gamut in a studio and in a cinema. It does not. Turn the room down four decades and the solid loses a fifth of its reach and two thirds of its volume, with the signal unchanged.
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Nobody here has two eyes
One person's two eyes differ in macular pigment and lens density, so the same light produces two colours — a whole ΔE00 apart for an ordinary pair, seven for one replaced lens. Adaptation hides it exactly, which is why nobody notices and why nothing in colorimetry has a term for it.
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Every threshold was measured with a grating
An earlier essay here claimed that the model cannot explain why dither works, and named two missing pieces. One of them was real and worth thirteen times the guess; the other was not needed. The piece nobody named was the detector — and reading the same model two ways changes the answer by a factor of fifty.
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There is no word for that colour
The naming model built this phase gives a saturated cyan the name green, calls part of a chromatic ring grey, and puts one of the eleven focal colours outside what a display can show. Each failure is a measurement rather than a disclaimer, and together they say what a vocabulary is that eleven points and a distance are not.
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A display in a room is a smaller display
Two results here turn the room's light in opposite directions and neither knows about the other. A room has one light level and does both at once — so there is a brightness at which a display delivers the most colour, it is not at either end, and it scales with the panel.
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The list nobody made
The last phase found that reading a filtered signal at a point asks a question its thresholds were never fitted to, made it a standing rule, and admitted that nobody had gone back through the site to see which claims it touched. Here is the list. Every claim with noise on one side of it moves — and so do two that have no noise in them at all, which the rule said would not.
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What a second model changed
Thirteen claims here, each computed with one model and recomputed with two. Ten of them move by half again or more. Three of them do not move at all in the ordinary sense — the first model's answer is exactly zero, not because it computed zero but because it has no variable for the quantity — and every one of those three is a join that supplied a state or a device rather than a spread.
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What no adaptation can remove
A change of light is exactly a 3×3 matrix on tristimulus values, and adaptation is a diagonal one. Putting every change of illumination this site models through that distinction sorts them by how much of themselves they leave behind, and the smallest residual in the census belongs to a filter inside the eye.
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Only one of these devices adapts
An eye, a camera, a display and a press all meet the same changes of light, and each has at most one thing it can do about them. The press has nothing at all, so its column is the whole change; and this collection's sensor built to satisfy the Luther condition exactly is the one that adapts worst.
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The brightener is being used up
The molecule that absorbs an ultraviolet photon occasionally does something other than re-emit it, and what it does is break. So a sheet's whiteness has a half-life, the fall is steepest at the start, and a specification quoting a whiteness figure without a date is quoting a property of a sheet that no longer exists.
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A fit can be exact and empty
Every fitted object here reports one number, the residual on the data it was fitted to, and every one of them has two more that nobody publishes — how many of its parameters the data actually determine, and how large the family of equally good answers is. The third column is where the failures live.
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Which of these is a convention
Nine ordinary sentences from this collection, put through one test — do they mean the same thing after the observer's three curves are replaced by a nonsingular combination of themselves? Five survive and four do not, and none of the four is wrong, because each is a statement about a set of coordinates being read as a statement about an eye.
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A constraint costs what it points at
Three primary chromaticities remove three of the nine numbers in an adaptation basis and cost one per cent. Three dichromat confusion points remove six and cost seventy. Counting what a constraint removes predicts neither, because an optimum is a long bowl and what matters is which way the constraint points.
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An extremum is not a sample
Three separate measurements in this collection took a maximum or a minimum over a sample of a set — forty-eight points round an ellipse, twenty-four directions out of an optimum, fourteen changes of light off a list. All three are wrong, all three are wrong in the same direction, and the error in each grows with the very quantity being measured.
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A constraint is a direction and a distance
Four restrictions on the same nine numbers cost nothing, nothing, two per cent and seventy. How many parameters each removes predicts none of it. What does is the quadratic form evaluated along the displacement — and showing that it does means walking in towards the optimum rather than arguing at the edge, because at the edge the prediction is out by a factor of three.
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How far a quadratic can be believed
A second-order model has a radius inside which it describes a surface and outside which it does not, and that radius can be measured. Measured at eight places on one objective, it is smallest at the optimum — the one place anybody ever takes a Hessian.
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A mean is not a worst case
Every adaptation number this collection publishes is an average over objects, and the reader asking whether adaptation will fail them is asking about the object it fails on. That object costs between 1.9 and 4.0 times the published figure, and how uneven a change of light is across objects turns out to be a property of the change rather than a constant.
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The price is also the person
The receptor construction costs seventy per cent above the unconstrained floor, which is a number usually quoted as a property of the construction. Propagated across a population of eyes it runs from a quarter above the floor to a hundred and thirty per cent above it, and the population's own spread is wider than the entire gap between the published transforms the seventy per cent was being compared against.
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What would have to be wrong
A great many statements here have thresholds written into them, which turns out to make an audit possible — for each one, the smallest change in a declared input that would stop it holding. Most are unreachable. One is inside a factor of one and a third.
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The instrument named the pair that moved
A standard error over a test set flagged four steps of the adaptation census as unresolved. Rebuilding the set three different ways reversed exactly one pair, and it was one of the four. Rebuilding it to a different rule reversed a pair the error separated by nearly nine standard errors — which is not a failure of the instrument but a statement of what it is about.
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The input nobody declared
An audit that swept every declared width in this collection found the largest elasticity anywhere to be about a half. The most elastic input turns out to be one that was never declared, never quoted with a range and never varied — and being undeclared is exactly why it escaped the audit that was looking for it.
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What the audit still cannot reach
Two rounds have now swept every declared width in this collection and one of its structural choices. Three structural choices remain, none of them has a multiplier to sweep, and the reason each resists is different — which makes the list a description of where this kind of audit ends rather than a queue of work.
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A choice with no magnitude
An audit can multiply a width by 1.25 and report an elasticity. It cannot multiply CIEDE2000 by anything. Auditing a structural choice needs a different instrument, and building one shows that six published numbers in this collection each carry a factor of about two of unit-choice — after the change of scale has been taken out.
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Two instruments and one ranking
A sampling error over a hundred and twenty-five surfaces and a change of colour-difference formula share no arithmetic at all, and they were asked the same question of the same table. Every adjacency the whole menu reverses had already been flagged as unresolved. And one the test set settles at nine standard errors is reversed by four of the five formulae, which is what makes them two instruments rather than one.
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Three choices reached
Two rounds ago this collection named three things it rested on and could not audit — a unit, a diagonal and a template. All three are now reached, and the interesting part is not the three answers but that four of the round's own predictions were refused by the arithmetic and one of its measurements was wrong in a way only a cross-check caught.
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The model has six arguments
Every colour computed here is an integral of a reflectance against a light against three curves, and a reflectance is one number per wavelength. A real surface's response is a function of six arguments — the wavelength, direction and place light arrives with, and the three it leaves with — so the model keeps one of them, takes a diagonal, integrates two away and assumes two more equal.
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Either factor being zero
Four different departures from the colour integral turn out to have one algebraic form — each is an inner product of something the sample does that the model has no slot for with something the light does that the model assumed away. Either factor being zero makes the departure exactly zero, and the sizes of the two factors decide neither how large it is nor which way it goes.
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A departure is not a unit
The previous round audited six published quantities by swapping the unit they were quoted in — a function applied at the end of each computation. Nothing of that shape works here. A departure changes the object at the start, so every stage in between has to accept it, and only two of the same six quantities can take one at all. The fourth cannot even be expressed in the interface.
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The audit that changed the object
Three rounds audited this collection's numbers, its sets and its conventions, and each one ended by naming the same thing it could not reach — that a surface is a reflectance. This round reached it, found that four departures share one algebraic form, had two of its predictions refused, and discovered that its own hemisphere quadrature had been passing a convergence check by luck.
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The reference had to be built
A five-nanometre error cannot be measured with five-nanometre data. Interpolating the tables and integrating finely measures the interpolator, not the grid — so the audit of this collection's index had to be run against an observer made of formulae, and the price of that is a residual of 1.42 ΔE₀₀ that every number in the section is read beside.
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The grid hid the observer
On this collection's five-nanometre grid a three-laser projector's observer disagreement is exactly zero. On a quarter-nanometre grid it is 2.34 ΔE₀₀ and the largest in the table. The two audits of this round meet here, and the first one does not compound with the second — it removes it.
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An observer is a contract
Six departures, all above a delivery tolerance, none dominant, and no reader matching the standard on any of them. That would be a demolition if the standard observer were a description of an eye. It is not — it is an agreement about what a number means, and an agreement is not falsified by being unlike anybody.
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Where a patch stops being a point
A patch in a cube subtends a cone of 25.8° at the face opposite. A microfacet lobe of roughness 0.05 is about three degrees wide. Point-sampling the second inside the first returned a chroma of thirty-four thousand and a negative lightness, and the boundary between working and not working is measured rather than declared.
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Three audits and one shape
The tabulation, the observer and the scene solver have nothing in common as subjects. Each turned out to hold a departure that is a pairing of two deviations, vanishes exactly when either is empty, and had been invisible because its parameter had no call site. Three subjects, one shape, and the shape is the round's result.
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The conditions are the result
The round measured about twenty departures and established ten conditions. The departures are numbers that depend on a sample, a light and a construction; the conditions are exact, they hold under every substitution tried, and they are what a reader can act on. A size is a measurement and a condition is a mechanism.
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What this round could not reach
Three audits, about twenty departures and ten conditions, and a longer list of things that were named and not measured. Every item on it is specific, most of them are an afternoon's work, and the reason none was done is the same in every case — the round ran out of round.
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The limits assume a pigment that switches instantly
The hardest boundary in colorimetry is reached by reflectances that jump between nought and one at a wavelength, and no material does that. Constrain the jump to take twenty nanometres — a sharp dye — and the median direction of the object-colour solid loses under half a per cent of its reach. Constrain it to eighty, an ordinary pigment, and the median loses five per cent, the tenth percentile nearly a quarter, and seven directions in ten lose more than one. The cost is in a corner only for chemistry sharper than paint.
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Three lines spare a slow pigment
A pigment that cannot switch faster than forty nanometres loses more than a per cent of its reach in 183 of the object-colour solid's 305 directions under daylight. Under an LED whose light sits in three narrow lines, it loses that much in 62. Between the lines almost nothing is measured, so a slow reflectance can do its changing there — until its transitions are as wide as the lines are far apart, at which point the lamp stops helping and the count jumps to daylight's.
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A limit written in energy charges the reds
A slope limit on reflectance is usually written in nanometres and applied the same way across the spectrum. An absorption band's width is closer to a fixed spread of photon energy, which is nearly twice as many nanometres at 700 as at 500. Written that way, a limit that is forty nanometres at 550 is kinder to the object-colour solid overall — 489 of 913 directions lose a per cent rather than 544 — and it charges the reds more. Which directions pay is decided by one thing: whether their optimal edges fall above or below the reference wavelength.
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A fourth emitter spends the gap it fills
A three-emitter LED lets a pigment that takes forty nanometres to switch reach most of its ideal solid, because the lamp is dark where the pigment is slow. Adding a fourth emitter to render better takes that darkness back — but only if it is put in the middle of a gap. At 490 nanometres it costs 172 of the solid's 312 directions against the three-emitter lamp's 62, and renders two points worse. At 540 it costs three directions and renders two and a half points better.
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The gap has to be dark, not the line narrow
A lamp whose light sits in three narrow emitters lets a blunt pigment reach most of its ideal solid, and the reason was given as the spacing of the lines. Broadening those emitters without moving them says otherwise. At 1.3 times their nominal width the lamp still looks like a line spectrum, its closest spacing has not changed at all, and half the rescue is gone — because the darkest point of the narrow gap has risen from one per cent of the lamp's peak to seven.
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A sharp edge is bought with depth
A slope limit on reflectance was introduced as the weakest honest statement of a pigment's bluntness, with a band-shape limit named as the stronger version to be written later. Written, it is not stronger. Three absorption bands none narrower than forty nanometres reach further than a forty-nanometre slope limit in 94 of 154 directions of the object-colour solid, because a band's width and the width of the reflectance edge it draws are different quantities — and what converts one into the other is how much colorant is in the film.
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A lopsided band does not break the trade
A fourth emitter added to a three-emitter LED renders colours better and costs a slow pigment its reach, and the position that renders best and the position that costs least were thirty-five nanometres apart. The obvious suspect was the shape: every emitter searched was a symmetric bump, and a band with its skirt on one side might fill a gap without splitting it. Freed to be lopsided, it saves between one and seven of the solid's three hundred directions. No band renders like the amber emitter and costs like the green one, the lopsided bands that help lean the other way from the proposal, and what the fourth band renders was never the lamp's white.
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The pigment decides the trade, not the band
A four-emitter lamp that renders well costs a slow pigment part of its reach, and reshaping the fourth band barely changes that. Changing the pigment changes it a great deal. Modelled as three absorption bands none narrower than forty nanometres, rather than as a forty-nanometre slope limit, a slow pigment loses a third fewer of the solid's directions under the best-rendering lamp, and the price of going from the three-emitter lamp to a fidelity of 88 halves. The lopsided band that helped a slope-limited pigment does nothing at all for this one. And at 590 nanometres a fourth band renders at 86 almost for free.
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Written in energy, the bands spare the green
A slow pigment modelled as absorption bands pays half what a slope limit pays for a four-emitter lamp's rendering, and a fourth band at 590 nanometres costs it almost nothing. That result assumed every band is at least forty nanometres wide, and a real absorption band has a width in energy, not in nanometres. Written that way it is narrower in the blue and wider in the red, and the fear was that the red would take back the amber. It does not. The band limit written in energy loses fewer directions under every lamp, the fourth band at 590 still costs six, and the price of rendering at 88 falls from ten directions to six — because the directions it wins back are green, not amber.
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The lamp outweighs the pigment model
Three models of what a real colorant can do — a slope limit, absorption bands forty nanometres wide, and bands whose width is fixed in energy — were ranked under lamps of narrow emitters, slope dearest and energy cheapest. Under daylight, tungsten and a white LED the ranking holds, every time. What changes is everything else: each model loses two and a half to three and a half times as many directions of the object-colour solid under a broadband lamp as under three narrow emitters, and the difference between the models is smaller than the difference between the lamps. Under tungsten, whose power rises smoothly to the red, the three models are nearly one.
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A tube spares a pigment through its phosphors
Three models of a real pigment lose two and a half to three and a half times as many directions of the object-colour solid under daylight as under three narrow emitters. A fluorescent tube is both kinds of lamp — narrow mercury lines on phosphor bands — and the prediction was that its bands would light the gaps between its lines and put it two thirds of the way to daylight. A triphosphor tube sits with the emitters instead: 26 directions lost to forty-nanometre bands, against the emitters' 23 and daylight's 73. The credit is its phosphors'. Its coating alone, lines taken out, loses fewer still; its mercury lines cost directions whose edges sit in the gaps between the phosphors, and doubled they cost more. A halophosphate tube, the same lines on two broad bands, costs what daylight costs at the same rendering index.
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A mercury lamp's few losses are deep
Every census of what a slow pigment can reach has counted the directions of the object-colour solid it misses by more than one per cent, and by that count a clear mercury lamp — four visible lines and nothing else — spares a pigment as well as three narrow emitters do. Read for depth, it is the one lamp of nine whose losses do not fall away: half of what a slope-limited pigment loses under it, it loses by more than five per cent, and its mean shortfall is half as large again as under the emitters or a triphosphor tube. Every one of those losses is the same kind of colour: one that reflects the mercury's green line at 546 nm and absorbs its yellow line at 578, or the reverse, 32 nm apart. The dark-share rule that ordered the counts orders the depths less well.
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A pair of lines asks for a cliff, not a gap
Under a clear mercury lamp, every direction of colour a slow pigment loses by more than five per cent asks it to reflect one of the lamp's green and yellow lines, 32 nanometres apart, and absorb the other. The prediction was that a pigment of two absorption bands with a narrow gap between them — a peak no single transition can make — would recover those deep losses. No direction asks for a peak. The lamp's four lines leave only one transition within a hundred nanometres of another, so what a splitting direction needs is a cliff: one flank falling from reflecting to absorbing in the twenty-five nanometres between the green line's last strong sample and the yellow's first. One band at the census's depth takes 27.8 to fall that far; two stacked at one centre take 24.9. Searched from 680 starts, the census's own band model does exactly that — stacks its bands on one side of the pair — and reaches further than the census found in seventeen directions of twenty-five. What stays deep spends a band on the blue.
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A harder search pulls the band pigments away from the slope limit
Every count in the pigment-limit censuses — how many directions of the object-colour solid a slow pigment cannot reach within one per cent of the ideal — compares a slope limit solved exactly with two band models searched from a few structured starts. Under the mercury lamp a search from 680 starts had already found the band model reaching further in seventeen of twenty-five directions. Searched from 594 starts under all nine lamps, both band models reach further in between three fifths and nineteen twentieths of the directions they lost, and their counts fall by a sixth: 395 to 335 for bands in nanometres, 370 to 311 for bands in energy, while the slope limit stays at 463. The rankings the censuses reported hold where they were stated. What moves is everything measured against the slope limit: its lead over the band models grows from a sixth to more than a third, the coating's odd ranking — bands losing more than the slope limit — disappears, the triphosphor tube passes the three emitters for bands in nanometres, and under the mercury lamp the band models' deep losses halve.
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The slope limit was exact all along
Every count of what a slow pigment cannot reach compares a slope limit, solved exactly on 161 levels of reflectance, with two band models found by searching. Searched harder, the band models lost a sixth of their losses, and the open question was whether the slope limit was being flattered by its own grid — whether 321 or 641 levels would lower its counts too. The prediction was a fall of under a twentieth. There is no fall at all: on every finer grid, under all nine lamps, every direction reaches exactly what it reached, and 463 directions are lost as before. The reason is the shape of the answer. The best reflectance that may change by at most an eighth per five-nanometre band is a staircase whose every step is a whole eighth, because a weighted sum is maximised at a corner of the allowed set and every corner is built from full steps off nought or one. A grid of only the nine eighths gives every answer to the last digit. The census's count is therefore not an estimate on a grid but an exact statement about every pigment that slow, and the slope limit's 1.38-fold lead over the searched band model is a floor that further search can only raise.
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A slow pigment splits mercury's lines only if its swing fits between them
Under a clear mercury lamp a pigment whose reflectance may swing from nought to one over no less than forty nanometres loses 21 of the census's 148 directions, and every one of them asks it to reflect one of the lamp's green and yellow lines and absorb the other. The census samples the spectrum every five nanometres, which smears each six-nanometre line across a band, and the prediction was that a one-nanometre grid would let the pigment's cliff fall between the lines and cut the count by a third. It cuts it by nothing: 21 on either grid. Narrowing the lines to one and a half nanometres cuts it only to 17. What decides the losses is the lines' spacing, 31.5 nanometres, against the pigment's swing. With narrow lines a pigment whose swing is 30 nanometres loses no direction at all, and one whose swing is 34 loses fourteen; the step sits at the gap. The mercury lamp's cost to a slow pigment belongs to the lamp, not to how the census samples it.
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A band pigment must be narrower than the gap it splits
Under a clear mercury lamp a pigment built from absorption bands no narrower than forty nanometres falls short in fifteen of the twenty-five directions that ask it to reflect the green line and absorb the yellow, or the reverse — the count an earlier search left short however deep it cut its cliff. The slope-limited pigment's losses turned out to be the lines' 31.5 nm spacing against its swing, and the prediction was that the band pigment's are the same arithmetic, falling to nought near a band thirty nanometres wide. Searched on a one-nanometre grid with the lines narrowed, the count falls at every narrower band — fifteen at forty, eleven at thirty-five, nine at thirty — and reaches nought at twenty-five, not thirty. Every loss already splits the pair and falls short on both lines at once: a Gaussian band deep enough to absorb one line keeps absorbing a little of the other. One band of depth twelve, placed as well as it can be, errs by 3.2 per cent at forty nanometres, 0.8 at thirty and 0.2 at twenty-five; doubling its depth buys less than narrowing it by five nanometres.
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A steep side splits mercury's lines at forty nanometres
Under a clear mercury lamp a pigment built from Gaussian absorption bands no narrower than forty nanometres falls short in fifteen of the twenty-five directions that ask it to reflect the green line and absorb the yellow, or the reverse, because a band's tail reaches the line it should pass. Real dye bands are often lopsided, and the prediction was that a band whose one side is half the other's, the steep side free to face either way, loses at forty nanometres about as many directions as a symmetric band thirty wide — nine — and helps only the directions that pass the green. It loses one. At thirty-five and thirty it loses none. In every one of the twenty-five fits the absorbing band turns its steep side to the line it must pass, and both kinds of direction gain: none of fourteen passing the green is lost, one of eleven passing the yellow. A single band forty nanometres wide with a steep side splits the two lines within 0.37 per cent; a symmetric band must narrow to about twenty-seven to do as well.
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A band must be nearly two to one before its steep side pays
Under a clear mercury lamp, a pigment of absorption bands forty nanometres wide loses fifteen or sixteen of the twenty-five directions that ask it to reflect the green line and absorb the yellow, or the reverse, and the same bands with one side a third of the width and the other two thirds lose one. The prediction for milder skews was that the losses fall in step with a single band's error — to about eight at sides of 0.55 and 0.45, three at 0.6 and 0.4 — so that a three-to-two skew already halves them. It does not. The census loses thirteen at 0.55, ten at 0.6, five at 0.65 and one at two to one: three directions recovered per twentieth of skew at first, twelve at the last. The losses track one band's error against the one-per-cent line the census counts by, and a single band crosses that line only near a skew of 0.62. A dye's modest natural asymmetry buys a few directions; the steep side pays only when it is close to half the other.
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A band is only as wide as its steep side
Under a clear mercury lamp, a pigment of absorption bands forty nanometres wide must be skewed nearly two to one before it can split the green line from the yellow, and the prediction for bands five nanometres narrower was that a single band crosses one per cent of error near a skew of 0.57 and the census's losses fall to five or fewer by 0.6. The losses do fall to four by 0.6; the single band crosses one per cent at 0.555. But the forty-nanometre rule that read the losses off that one-per-cent line breaks at once, and what replaces it is simpler: a thirty-five-nanometre band at 0.6 and a forty-nanometre band at 0.65 have the same steep side, fourteen nanometres, and lose the same four directions by the same shortfalls to a hundredth of a point. Three widths and fourteen pigments fall on one staircase in the steep side alone. Searching every loss again from the milder skews' answers finds one direction the census missed, at both widths, and corrects the forty-nanometre counts to nine and four.
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At one half-width a flank can lose nothing or nearly everything
Under a clear mercury lamp, a pigment of Gaussian absorption bands splits the green line from the yellow according to its steep side's half-width alone, and a Gaussian flank is fixed by its half-width — so the census could not say whether it measures the half-width or the absorbance left in the flank's foot at the line it should spare. The prediction was that the foot decides: at a half-width of fourteen nanometres, a Lorentzian steep side loses most of the twenty-five splitting directions and a sharper-footed one none. It does. A power-four Gaussian loses none, the Gaussian four, a squared hyperbolic secant nineteen and a Lorentzian twenty-two, in the order of their single bands' errors. But a single band's error is not the whole account: three profiles drawn at the half-widths that give a single band the same error — 7.73, fourteen and 39.1 nanometres — lose two, four and seven directions, and the widest of them loses directions the others keep, because its shoulder starts absorbing nearer the green.
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A hundredth of Lorentzian undoes a Gaussian flank
Under a clear mercury lamp, a band pigment whose steep side is a Gaussian fourteen nanometres wide at half depth loses four of the twenty-five directions that split the green line from the yellow, and the same side drawn as a Lorentzian loses twenty-two. A real absorption band is usually a Voigt profile, a Gaussian core with a Lorentzian share in its wings, and the prediction was that a twentieth of Lorentzian doubles the Gaussian's losses and a fifth reaches nineteen. The direction is right and the scale is not. A hundredth of Lorentzian takes the pigment from four lost directions to eleven; a fiftieth to fourteen; a twentieth to eighteen; a fifth to twenty. The best-placed band leaves the passed line more than three half-widths from its centre, out in the wing, and there a wing one per cent Lorentzian carries twice the absorbance of the whole Gaussian. The count saturates near twenty while the worst direction's shortfall keeps growing, from 2.8 per cent at a hundredth to 51 with a full Lorentzian.
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A narrower flank moves its wing in with it
Under a clear mercury lamp, a band pigment whose steep side is a Gaussian fourteen nanometres wide loses four of the twenty-five directions that split the green line from the yellow, and with a twentieth of Lorentzian in that side it loses eighteen, because the wing three half-widths out out-absorbs the whole Gaussian there. A Lorentzian wing falls as the square of the distance over its half-width, and the prediction was that narrowing the side to seven nanometres brings the twentieth-Lorentzian pigment back to four losses and to five nanometres to none. It loses thirteen at ten nanometres, twelve at seven and seven at five, while the Gaussian loses nothing at all three. The wing still decides: the narrower band does not stay where the wider one sat but moves fourteen nanometres in towards the line it must pass, so its wing there falls fourfold rather than eightfold.
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A five-nanometre flank carries a fiftieth of Lorentzian
A band pigment splitting mercury's green and yellow lines with a steep side five nanometres wide loses none of the twenty-five splitting directions as a Gaussian and seven with a twentieth of Lorentzian in its flank, because the band slides in towards the passed line and carries its wing there. The prediction was that at five nanometres the count follows the single band's error, which grows in proportion to the share: about two lost at a fiftieth and none at a hundredth. A hundredth loses none, as predicted. The first search lost one direction at a fiftieth — and searched again from the best pigments found at the neighbouring shares, that direction reaches within two thousandths of the ideal: a fiftieth loses none either. The same second search recovers two of the twentieth's seven, leaving five. A sharp enough flank tolerates a fiftieth of Lorentzian where a fourteen-nanometre one lost fourteen directions to it.
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Each direction falls short by one band's ratio
A band pigment asked to reflect mercury's green line and absorb its yellow, with a twentieth of Lorentzian in its steep side, loses eighteen of twenty-five splitting directions at a steep side fourteen nanometres wide, thirteen at ten, twelve at seven and five at five. A second search had lowered the five-nanometre count, and the prediction was that searching the seven- and ten-nanometre pigments again lowers each by one or two, leaving a count nearly linear in the half-width. Searched again, both lose exactly what they lost — no direction gains six tenths of a point. The uneven fall is real, and the reason is plainer than any search: every direction's shortfall falls with the half-width by the same factor as a single band's error, so the count is a fixed ranking of constants, one a direction, read against a line that moves.
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The ranking reads six nanometres to the direction
Every direction a band pigment loses under mercury's lines falls short, as its steep side narrows, by the same factor as a single band's error — so each direction carries a constant, and the count at any half-width can be read off one band with no search at all. Read that way, a twentieth of Lorentzian loses nine directions at a six-nanometre steep side, three at four and a half and none at four. Searched, six nanometres loses nine: the same nine, every direction within 6.2 per cent of where the constants put it. Four loses none. Four and a half loses one, not three — the two the ranking added sit 0.005 and 0.011 of a point under the one-per-cent line. And at the narrowest widths the gentle directions begin to fall short by more than the ranking says, by up to 29 per cent, which is where the single band stops being the whole story.