Colour Science & Perception
what the eye settles, and what it cannot
Concept

Bound — where it appears

A stated limit on what a parameter may be, which turns an unbounded search into an answerable one. A bound that the answer sits on is doing the work; a bound the answer never approaches is indistinguishable from no bound at all unless somebody checks.

Named by 28 essays across 5 fields — each of them below, with the objects they name alongside it.

A published residual is a mean, and the worst object in the room costs twice it. Three bars for each of the 14 changes of light in the adaptation census, ordered by how uneven the change is across surfaces. The first bar is the published mean residual. The second is the worst single surface in the audit's published test set. The third is the worst surface anywhere in the region that set is drawn from, found by search rather than by reading a maximum off a lattice. The mean-to-worst ratio runs from 1.90 to 4.02 and averages 2.43, so every published adaptation number has a worst case about twice it that no essay had ever quoted. The gap between the second and third bars is the other finding: a maximum over 125 sampled points understates the region's own maximum by up to 34 per cent.

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.

limits · Limits A published residual is a mean, and the worst object in the room costs twice it. Three bars for each of the 14 changes of light in the adaptation census, ordered by how uneven the change is across surfaces. The first bar is the published mean residual. The second is the worst single surface in the audit's published test set. The third is the worst surface anywhere in the region that set is drawn from, found by search rather than by reading a maximum off a lattice. The mean-to-worst ratio runs from 1.90 to 4.02 and averages 2.43, so every published adaptation number has a worst case about twice it that no essay had ever quoted. The gap between the second and third bars is the other finding: a maximum over 125 sampled points understates the region's own maximum by up to 34 per cent.

An extremum is still not a sample

Two rounds ago three measurements turned up that took a maximum over a sample of a set and were short by up to a factor of two. The same error was live in a fourth place the whole time, on the set of surfaces every adaptation number is averaged over, and it is short by up to a third.

matching · Gamut Every worst surface sits on a number somebody typed. The region the test surfaces are drawn from, in its own two modulation coordinates: a square of allowed depths with a diamond inscribed in it, the diamond being the requirement that the two depths sum to no more than 0.7. The 14 marked points are the worst surface for each change of light in the adaptation census, found by search over the whole region. Every one of them lies exactly on the diamond, and every one is also at the brightest level the region allows — both declared constraints active, on all 14 rows, with no interior maximum anywhere. That is the opposite of what bounding the wall gave: there the worst case turned over at a band width of six nanometres because a narrow band returns too little light, which is physics. Here the worst case is a reading of two numbers. The one constraint that is about the world — a paint's excitation purity may not exceed 0.6 — is slack everywhere: the most saturated surface the region admits reaches 0.459.

Every worst surface sits on a declaration

Bounding the wall in a painted room produced a real worst case — the residual turns over at a band six nanometres wide because a narrower band returns too little light. Bounding the surfaces the residual is averaged over produces nothing of the kind, because all fourteen answers sit exactly on two numbers somebody typed and the one constraint that comes from the world never binds at all.

scene · Scene What a code lattice costs, and where. Twelve thousand colours quantised to 8 bits per channel through the sRGB transfer function and read back, with lightness across the bottom and the colour difference the rounding cost up the side. The mean is 0.191 and the worst case is 1.15, a factor of 6.0. The bars are band means, and they rise: the encoding spends its codes in the shadows, so the top of the ramp is where the lattice is coarsest against a metric that does not compress as hard.

A lattice has no derivative

Every departure priced here was priced by perturbing something and reading the answer, which requires the thing being perturbed to have a derivative. A file written on a code lattice does not have one — its output is flat almost everywhere and jumps on a set of measure zero — so quantisation can be bounded and never propagated. The bound is 1.15 colour differences at eight bits per channel against a mean of 0.19, and it is worst where the encoding spends fewest codes.

matching · Gamut What a slope limit costs the object-colour solid, direction by direction. For each transition width, how much of its ideal reach the solid keeps: the median direction, the tenth percentile, and the worst. At twenty nanometres the median keeps 0.997 and the tenth percentile 0.985; at eighty they keep 0.946 and 0.766, and at 160 0.826 and 0.417. The worst direction falls from 1.00 at five nanometres to 0.20 at 160, with directions reaching under five units beyond black set aside. The cost is in a corner only at widths sharper than an ordinary pigment's.

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.

limits · Limits How many directions a slope limit costs, under four lamps. For each transition width and each lamp, the share of the solid's directions that lose more than one per cent of their reach — each lamp's solid against its own ideal. Daylight, a tungsten lamp and a phosphor LED run close together. The three-emitter LED, whose power sits in lines at 455, 530, 625 nanometres, costs 62 directions at forty nanometres where daylight costs 183, and by eighty — about the spacing of its lines — it costs 214 against 218.

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.

limits · Limits A 40-nanometre limit written in nanometres and written in energy. The transition width a reflectance is allowed, across the spectrum, for two ways of stating the same sharpness. Written in nanometres it is 40 everywhere. Written as a fixed spread of photon energy, which is how an absorption band's width is set, it is 40 at 550 nanometres and grows as the square of the wavelength: 21 at 400 and 67 at 700. The steps are the quantisation the calculation actually imposes.

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.

limits · Limits Two tables, two directions of error: 5-node ramps of four inks. For each ink, the mean signed lightness error between the nodes: of the ordinary forward table, which errs light, and of four ways to fill the inverse table, all of which err dark except the one refitted against its own objective. On cyan the forward table errs by +0.206; an inverse filled from the press by -0.317; one inverted from the ordinary table by -0.554, because it inherits the forward table's error on top of its own; one inverted from the refitted table by -0.367; and one refitted in its own right by -0.002.

The inverse table errs dark

A profile's forward table predicts a print lighter than the press makes, and refitting its nodes removes the bias. The table a colour engine actually uses to separate an image is the other one — the inverse, from colour to ink — and it errs the opposite way: filled exactly from the press it asks for too much ink and prints dark, and built by inverting the ordinary forward table it prints darker still. Inverting the refitted forward table removes the inherited part and leaves the inverse's own. The round trip through both tables improves only when each is refitted against its own error, and then the two are no longer each other's inverse.

applied · Delivery What a fourth emitter costs, by where it is put. A three-emitter LED with one more emitter added at each position from 470 to 610 nanometres, and for each lamp the number of the object-colour solid's directions that lose more than a per cent of their reach to a 40-nanometre transition limit. The three-emitter lamp itself costs 62 of 312. A fourth emitter in the middle of the blue-to-green gap, at 490 nanometres, costs 172; one at 540, beside the green emitter, costs 65. The two dips sit on the existing lines and the two peaks sit between them.

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.

limits · Limits The rescue is spent on light between the lines, not on width. The three emitters of a narrow-band LED broadened together, from their nominal widths up to six times them, plotted against the light left in the darkest of the lamp's two gaps as a share of its peak. Up is the share of the object-colour solid's directions that lose more than a per cent of their reach, at three transition limits, with each limit's cost under daylight marked at the right. At forty nanometres half of the rescue is gone by a floor of 7.4 per cent — emitters only 1.30 times their nominal width — and all of it by about a fifth. At twenty nanometres and at eighty there is little to lose either way.

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.

limits · Limits How steep an edge a band draws, and what it costs. A Gaussian absorption band of stated width produces a reflectance edge whose own width depends on how deep the band is, because the exponential saturates: where the absorbance is large the reflectance is already nought and the edge is over. A forty-nanometre band at an absorbance of 3 draws an edge 32 nanometres wide; at 12 it draws one 21 nanometres wide. Below an absorbance of 2.3 the band never reaches a reflectance of a tenth at all and has no edge in this sense. The dashed lines are each band's own width, which is the number a slope limit would have been given.

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.

limits · Limits What declaring a narrowest feature buys, and where it stops being true. The median looseness of a Cauchy–Schwarz bound whose lamp variance is bounded by a declared narrowest feature, against the width declared, for a fluorescent tube and a three-laser projector. Each lamp's own Bhatia–Davis bound — the peak declared and nothing else — is the upper dashed line, and the bound with the true variances is the lower one. The marks are the width each lamp's lines actually have. Declaring it truly takes the tube from ×196 to ×86 and the projector from ×30 to ×14. The open circles are declarations the lamp does not meet, where the bound falls below the error.

A declared width buys a factor of two

A colour engine given two separately blurred spectral tables cannot bound its own error from them, and the bound that always holds — the peak declared and nothing else — sits a median 196 times above the error under a fluorescent tube. Adding one number, the width of the lamp's narrowest feature, brings that to 86. It never fails on any declaration the lamp truly meets, it fails on 47 of 68 notches on one it does not, and its rank correlation with the error it bounds is 0.27.

light · Light Narrowing the slit makes the declared-width bound looser, not tighter. The median ratio of the declared-width bound to the error it bounds, over sixty-eight notches under each lamp, with the slit and the grid narrowed together from five nanometres to one and each lamp declaring its true line width; the dotted lines are the bound with both variances known. Under the tube the declared bound goes from ×86 at five nanometres to ×253 at one, while the true-variance bound stays between ×12 and ×15. Under the projector the declared bound lies on its true-variance bound at every slit.

A narrow slit shrinks the error, not the bound

A colour engine given two separately blurred spectral tables can bound its own error if the lamp declares its narrowest feature, and under a fluorescent tube that bound sits 86 times above the error at a five-nanometre slit. Narrowing the slit was supposed to close the gap, because a comb of narrow features stops fitting in a narrow window. It does the opposite. At one nanometre the error falls 27-fold and the bound only sevenfold, so the bound is 253 times loose and certifies no notch within a colour difference. The looseness was never the comb. It is the lamp's one declared peak, charged to every window.

light · Light The ceiling a table reads for each window, against the lamp it came from. The fluorescent tube between 525 and 590 nanometres, with the green and yellow mercury lines, and for each window of a one- and a five-nanometre slit the highest the lamp could be there according to the table's own entries. The dashed line is the tube's peak, which the global declaration charges to every window. At one nanometre the ceiling follows the lamp, a median 2.05 times each window's true peak; at five it sits on the global peak for twenty nanometres either side of each line.

A narrow table can vouch for its own peaks

The bound a colour engine can put on its bandpass error is loose because a lamp declares one peak for its whole spectrum and the bound charges it to every window. A table measured through a slit about as narrow as the lamp's lines can do better: no feature of the declared width can be blurred below a computable fraction of its height, so the table's own entries cap how tall the lamp can be in each window. At one nanometre that ceiling is never below the truth and sits twice above it, where the global peak sat twenty times above. The bound halves, and for the first time it certifies notches within a colour difference. At five nanometres it buys nothing at all.

light · Light How well the table's entries around the 546 nm line fit each width of line. For the tube's green mercury line, 1.2 nm wide, the seven table entries around it fitted by a line of each width from 0.2 to 12 nm through the same slit on a sloping background, centre, height and background free: the best fit's miss. Through a one-nanometre slit the miss has a sharp minimum at the true width and only widths near it pass half a per cent; through two, a broader dip; through three and five the curve is flat from the narrowest width tried up to about two, and any of them passes.

A narrow table declares its own lines

A bound on a colour table's bandpass error needs one number from the lamp's maker: its narrowest feature, 1.2 nm for a fluorescent tube. Through a slit about as narrow as that feature, the table's entries around each line carry its width. Measured to half a per cent, a one-nanometre table's entries admit only line widths within four per cent of the truth, and a bound built on the narrowest of them is within two per cent of the bound on the maker's number. Through two nanometres the table still declares a usable width. Through three it cannot rule out a line five times narrower, and the bound it supports is the global one. The failure is as sharp as predicted, and never unsafe.

light · Light Wings raise the width a table declares, and it follows the line's half maximum. The width the tube's table declares for itself at half a per cent — the narrowest width any of its peaks admits — as every mercury line is given Lorentzian wings from nothing to half its core's width, through one and two nanometre slits, beside the Voigt line's own full width at half maximum. With no wings both slits declare under 1.2 nm; with wings a quarter of the core's, 1.36 through 1 nm and 1.41 through 2 nm, against a half-maximum width of 1.37. Through one nanometre the table declares nothing at the widest wings: every line is refused.

A line's wings widen the width a table declares

A narrow table declares its lamp's narrowest line width by fitting each peak with a Gaussian, and the worry was a line whose profile is not Gaussian: a mercury line broadened by collisions has Lorentzian wings on its core, every line shares them, and a Gaussian fit can only take the wings in by widening. It does. With wings a quarter of the core's width, the table declares 1.36 nm for a 1.2 nm core — close to the winged line's own half-maximum width, 1.37. The prediction called that the unsafe direction. It is not: the ceiling and the variance the earlier bound builds on the declared width never fall below the winged lamp's own, because through a slit a winged line behaves like the wider line the table says it is.

light · Light Where the ceiling on a self-absorbed line holds, and where it does not. In the windows within four nanometres of the 546 nm line, the smallest ratio of the ceiling the tables put on a window's peak to the absorbed line's true peak, built on the narrowest width the table admits for the line and on the emitted 1.2 nm, through each slit. On the admitted width the ceiling holds to a depth of three and falls below the peak at 3.5 (×0.981) and 4 (×0.885) through two nanometres. On 1.2 nm it holds through two nanometres at every depth.

A reversed line is read as one line

A strong mercury line in a dense discharge absorbs its own centre on the way out: at an optical depth of one its top is flat, and beyond that it dips, until at four the line is two humps with a trough eighty per cent deep between them. The prediction was that a one-nanometre table refuses such a line from a depth of about one and a two-nanometre table reads it as a Gaussian about the width of its flat top, safely. Both hold, and the second only up to a point. A two-nanometre table admits the line at every depth to four, and from three and a half the width it admits is too wide for the humps: the ceiling built on it falls eleven per cent below the line's true peak. The only line that could make a declared width unsafe was one narrower at its peak than its entries suggest — and a reversed line is exactly that.

light · Light Where a two-nanometre table's ceiling on a notched line fails. In the windows within four nanometres of the 546 nm line, the ceiling built on the narrowest width a two-nanometre table admits for the line, as a multiple of the absorbed line's true peak, behind columns a whole, a half and a third as wide as the line. The uniform column's ceiling crosses below one at a depth of 3.42 and ends where the table refuses the line; the half and third columns' cross at 2.34 and 2.35 and are never refused, falling to ×0.73 and ×0.80 at a depth of five.

A notch in a line is never refused

A dense discharge's green mercury line loses its centre to the atoms between the glow and the tube wall, and those atoms are cooler than the glow, so their absorption is narrower than the line it cuts. With the absorbing layer half or a third as wide as the line, a one-nanometre table still refuses the line from an optical depth of one, exactly where it refused the broad dip. A two-nanometre table never refuses it at all: to a depth of five, with ninety-eight per cent of the centre gone, one Gaussian fits its entries within a seventh of a per cent. And the ceiling built on the width it reads fails from a depth of about two and a third — a whole unit sooner than behind a column as wide as the line — and never recovers, falling to three quarters of the true peak.

light · Light Where a reversed tube's table declares a width its ceiling cannot stand on. With every mercury line behind its own column, in proportion to its strength, the ceiling built on the width the whole table declares — the narrowest any of its peaks admits — as a multiple of the reversed lamp's true peak in the worst window, through one and two nanometres behind columns as wide as the lines and half as wide. The one-nanometre lines stop where the table refuses every peak and declares nothing. Through two nanometres behind the cooler column the table never stops declaring, and its declaration is unsafe from a depth of two and a half, falling to ×0.69.

A reversed tube declares from its least reversed line

A spectral table declares its lamp's line width as the narrowest width any of its peaks admits, and that declaration survived every reversed line so far because the tube's other lines were left unreversed to set it. A real discharge reverses its strong lines together. With every mercury line behind its own column, in proportion to its strength, the prediction was that a one-nanometre table would refuse every strong line and declare from the weak violet line, and a two-nanometre table would declare an unsafe width from a green-line depth of two and a half. The violet line is never read as a peak at all. The yellow line, the least reversed of the three that are, sets the declaration at every depth and widens it as it reverses. Through one nanometre the table refuses the blue and green lines and goes on declaring from the yellow — unsafely, in the violet's window, the one line the table never checked. Through two nanometres behind a cooler layer it refuses nothing, and from two and a half its declaration fails in the blue line's window, down to seven tenths of the true peak.

light · Light The best slow pigment is a staircase of eighths. Under daylight D65, the direction the slope-limited pigment reaches least well — 76.1 per cent of the ideal: its weights across the spectrum, scaled; the ideal pigment, reflecting fully wherever the weight is positive; and the best reflectance changing by at most an eighth per five nanometres, solved on 641 levels and on nine. The two solutions coincide wherever the lamp gives light: every band sits on a whole number of eighths, climbing and falling eight steps at a time where the ideal jumps. Where the lamp gives no light any reflectance serves, and the two may differ there.

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.

limits · Limits Declared at the notch fit's width, the ceiling holds to two. For the reversed tube — every line behind a cool column half its width — the ceiling a declaration builds over the lamp's true peak, in the window where it is lowest, against the column's depth. On the one-nanometre table: the plain declaration, which falls below the true peak from a depth of a half and stops at two, where the table refuses every line; the ceiling on the notch fit's width; and the ceiling on the tube's true 1.2 nm core. The notch fit's ceiling lies on the true core's, safe to a depth of two and unsafe from two and a half; on the two-nanometre table the true core's is safe to three.

The notch fit declares the true width, and the true width runs out

A lamp's table declares the narrowest line width its peaks admit, and builds on it a ceiling for how high any line can rise between the entries. A reversed tube breaks both: its lines are notched by cooler gas, a plain fit reads each notched line as a wider one, and on a one-nanometre table the ceiling falls below the true peak from a column depth of a half, until at two the table refuses every line and declares nothing. A fit with the notch in it reads the yellow's notch on that table, and the proposal was to declare the line it finds underneath — notch taken back out — with the prediction that the ceiling then holds to a depth of three. It holds to two. The notch fit returns the tube's own 1.2 nm core within two hundredths of a nanometre at every depth, clean and in noisy readings, and the ceiling built on it is safe to two and fails from two and a half, in the window beside the reversed blue line. So does the ceiling built on the true width itself, at the same depths to within a hundredth. The repair is perfect, and what fails past two is the ceiling's rule, which assumes the lines it bounds are not notched.

light · Light With the notch in it, the ceiling holds to the deepest column. For the reversed tube on the one-nanometre table, the ceiling over the lamp's true peak in its lowest window against the column's depth: built on the 1.2 nm core with no notch, and with each window's factors taken for its line's notch — from the true depths, from the notch fit's depth read as an optical depth, and from the fitted depth used directly. At a depth of five the four ceilings sit at 0.776, 1.110, 1.295 and 0.972 times the true peak.

A ceiling that knows the notch holds to the deepest column

A lamp's table promises a ceiling for how high any line can rise between its entries, and for a reversed tube — every line notched by a cooler column — that ceiling fails on a one-nanometre table from a column depth of two and a half, because its rule assumes the line it bounds has no notch. The notch fit already reads the yellow line's notch, and the proposal was to build the ceiling for a notched line instead: in each window, the factor for a slit-blurred line carrying that window's notch. The prediction was that it holds to a depth of four. It holds to five, the deepest column the census runs, at every depth and in every one of 440 noisy readings. The fit's depth must be read as an optical depth, −ln(1 − d); taken as the first-order depth it fits as, the ceiling fails from four and a half. Part of the safety is the fit's own error: read that way, a deep notch comes out too deep — 4.25 for a true 2.29 at depth five — and an overstated notch only raises the ceiling.

light · Light A coarse table holds to the deepest column on a borrowed notch. For the reversed tube on the two-nanometre table, the ceiling over the lamp's true peak in its lowest window against the column's depth: with no notch, with the true notch, and with the notch borrowed from the one-nanometre fit; beside the one-nanometre table's ceiling on its own fitted notch. Without a notch the coarse ceiling fails from three and a half; with the borrowed notch it stands 8.4 to 42.4 per cent above the true peak at every depth.

A coarse table can borrow a notch, but not a phantom

A one-nanometre table can bound a reversed lamp by building its ceiling for the notch its own fit reads. A two-nanometre table cannot read the notch, and its ceiling, built for an unnotched line, fails from a column depth of three and a half. The proposal was to borrow the notch from the finer table, and the prediction that the coarse ceiling then holds to five with a wider margin. It does: 8 to 42 per cent above the reversed lamp's true peak at every depth, against the fine table's 1 to 29. But borrowed from the fine table's noisy readings, one notch in 440 sinks the coarse ceiling below the true peak — a notch of 0.13 the fit read in noise where there was no column at all. Under a wide slit a shallow notch flattens a line's top, two flanking entries see more of it, and a ceiling that assumes the notch is set lower; the phantom set it too low. Borrowing only notches the depth rule calls — deeper than 0.2 — keeps all 440 safe; borrowing those more than twice their error does not.

light · Light From three nanometres a table needs no notch at all. For the reversed tube read through slits of two to five nanometres, the ceiling built on the line's unnotched core width, over the lamp's true peak in its lowest window, against the column's depth. The two-nanometre ceiling fails from three and a half; the three-, four- and five-nanometre ceilings stay above the true peak at every depth, the least by 4.7 per cent.

A wide table needs no notch to borrow

A two-nanometre table of a reversed tube cannot read the notch a cooler column cuts in its lines, and its ceiling, built for an unnotched line, fails from a column of three and a half; borrowing the notch a one-nanometre table read rescues it at every depth. The prediction for coarser tables was that a shallow notch lifts their flanking-pair factor further — to a fifth at five nanometres — and that the five-nanometre ceiling on a borrowed notch still holds to five. Both hold: the rise is 15, 18 and 20 per cent at three, four and five nanometres, and every borrowed ceiling stays above the true peak. But a table three nanometres wide or wider never needed the notch. Its unnotched ceiling stands at least 4.7 per cent above the true peak at every column depth, because each of its entries carries so little of any line's peak that the ceiling built on them sits high above every line, reversed or not — and borrowing the notch takes up to twenty points of that margin away.

light · Light The ratio of slit to core nearly decides when a notch matters. For the reversed tube with line cores of 1.2, 2.4 and 3.6 nm, read through slits of two to five nanometres: the first column depth at which the ceiling built for an unnotched line falls below the true peak, against the slit over the core, with "never" for tables that hold at every depth. 0.56: 0.5, 0.83: 1, 0.83: 1.5, 1.11: 2.5, 1.25: 2, 1.39: 3, 1.67: 3.5, 1.67: 4, 2.08: 3.5, 2.50: never, 3.33: never, 4.17: never — every ratio of two and a half or more never fails, every ratio up to 2.08 fails, the threshold a little above the predicted two.

A broad line brings the notch back to a wide table

A table of a reversed tube read through a slit three nanometres wide or wider never needs the notch a cooler column cuts in its lines: its ceiling, built for an unnotched line 1.2 nanometres wide, stays above the true peak at every column depth. The reason offered was that a wide entry carries too small a share of a narrow line's peak to be fooled, and the prediction was that broader lines undo it — that at a core of 3.6 nanometres the five-nanometre table fails from a column of about three, and that a table needs the notch wherever its slit is less than about twice the line's core. The five-nanometre table fails from a column of three, exactly. Across twelve tables of three cores and four slits, the column at which the unnotched ceiling first fails rises with the slit-to-core ratio — from a half at 0.56 to three and a half or four near 1.7 — and every table at a ratio of 2.5 or more holds; but a ratio of 2.08 still fails, so the threshold sits a little above two. With the true notch, every table holds.

light · Light Wings raise every core-built ceiling's margin over the peak. The reversed tube with every line a 2.4 nm Gaussian core convolved with Lorentzian wings, read through slits of four, five and six nanometres, each window's ceiling built for an unnotched line of the core's width: the least ceiling over the true peak at any column depth to five, against the wings' width. 4 nm: −5.7%, +2.2%, +13.0%; 5 nm: −11.6%, −3.3%, +8.0%; 6 nm: +7.3%, +15.4%, +27.8%.

Wings make a broad line easier to bound from its core

A spectral table read through a wide slit can still bound a reversed mercury tube's true peaks with a ceiling built for an unnotched line, so long as the slit is more than about two and a half times the lines' Gaussian core; at a ratio of 2.08 a five-nanometre table fails from a column depth of 3.5. A real discharge line has Lorentzian wings, which widen the width a table declares, and the prediction was that wings move that threshold out in proportion to the declared width, while a ceiling built on the declared width keeps it where it was. Both halves fail, the first in direction. With wings a tenth of a 2.4 nm core's width the five-nanometre table holds to 4.5, and with a quarter it holds at every depth, by eight per cent: wings take power from a line's peak and give it to the table's entry. A ceiling built on the wider declared width fails three of six winged tables that the core-built ceiling serves.

light · Light Even half a core of wings leaves the three-nanometre table short at the deepest column. The three-nanometre table's worst ceiling over the true peak at every column depth, built for the core's width, with the lines bare and with wings a tenth, a quarter and a half of the core's width. It fails from depths of 2, 2, 2.5, 5 as the wings widen; with wings a half it holds to 4.5 and falls 1.1 per cent short at five.

Wings rescue a three-nanometre table only to its deepest column

Lorentzian wings on a reversed tube's lines make every four-, five- and six-nanometre table's ceiling, built for the lines' 2.4 nm core, safer by eight to twelve points of margin a step, so that wings a quarter of the core's width carry all three to every column depth. The three-nanometre table, at a slit-to-core ratio of 1.25, is the one a bare broad line failed earliest, from a depth of two. The prediction was that it still fails with a quarter's wings, from about three, and holds at every depth only with a half's, because a narrow slit averages less of a line's skirt. The first half holds, half a step early: a quarter's wings fail it from 2.5. The second fails by a hair: with wings half the core's width the three-nanometre table holds to a depth of 4.5 and falls 1.1 per cent short at five. Its margin climbs by 6.6, 9.0 and 13.2 points a step — the smallest lifts of any table, from the lowest start.

light · Light Built for the notch, the three-nanometre table holds at every depth. The three-nanometre table's worst ceiling over the true peak at every column depth, built for the unnotched core and for each window's notched line, with the lines bare and with wings half the core's width. Built for the notch, the bare table's least margin is +0.3% at a depth of 1.5 and +8.1% at five; built for the core it was −29.9% at five.

A ceiling built for the notch holds the three-nanometre table

A table of a reversed discharge lamp read through a three-nanometre slit cannot see the notch a cool column of vapour cuts into each line's core, and a ceiling built for an unnotched core — the bound on how high the lamp can peak inside each window — fails from a column depth of two, and even with broad wings at five. The prediction was that a ceiling built for each window's notched line holds at every depth with any wing, by more than two per cent. It holds at every depth with any wing. With bare lines it holds by three tenths of a per cent at its tightest, at a moderate column, beside mercury's violet lines. Knowing the notch costs the shallow columns and pays the deep ones, because a shallow notch flattens a line's top and a deep one splits it.

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