Spectral power distribution — where it appears
Named by 63 essays across 7 fields — each of them below, with the objects they name alongside it.
A spectrum is not a colour
What arrives at the eye is a function of wavelength. What the eye reports is three numbers. Keeping the two apart is the single most useful habit in the subject, and almost every confusion in applied colour comes from letting them merge.
Three numbers
A spectrum has as many degrees of freedom as anyone cares to give it. The eye reports three. Everything colour science can do, and every way it fails, follows from that one collapse.
The sun is not one illuminant
Daylight is not a blackbody and is not one spectrum. It is a one-parameter family reconstructed from three measured basis functions, its chromaticities lie on a curve that is near the Planckian locus and not on it, and the difference is the reason a colour temperature needs a second number beside it.
A bounce is a multiplication
Colorimetry multiplies an illuminant by a reflectance once and integrates. A surface in a room is lit by every other surface the lamp reached first, so the spectrum arriving at the eye has been multiplied several times — and the second multiplication is where the whole apparatus of matching starts to come apart.
A shadow has its own illuminant
Outdoors there are two lights, not one. The direct beam is a radiator reddened by the atmosphere; the sky is precisely the power that reddening removed. A shadow is lit by the second alone, so shadowed ground and sunlit ground sit under illuminants 21 units of ΔE apart — before any surface, any eye or any opinion is involved.
Why blue and yellow make green
The oldest fact in colour, and the usual explanations are wrong. It is not because green sits between blue and yellow, and it is not a fact about the eye at all — it is that the only band both pigments return is their overlap, and the overlap of a blue and a yellow reflectance is green. Computed, the mixing path bows away from the straight line by a measurable amount.
Two lamps do not average
Light adds, band by band, and every number a lamp is sold by is a projection of the sum rather than a sum of the projections. Two radiators sitting exactly on the Planckian locus mix into a light that is measurably pink; two poor lamps mix into a better one than either.
There is no D65 lamp
D65 is standardised by three numbers, and three numbers do not pin a spectrum. A source built to hit them exactly — matched to a part in a billion, with a third of the visible band nearly empty — separates pairs that D65 says are identical by up to eight and a half units of colour difference.
The eye has a shutter
An isoluminant flicker fuses at fifteen hertz and a luminance one at sixty, so a light whose colour changes forty times a second is a steady light of a colour it never emits. And the frequency at which flicker stops being visible is not a property of the eye — it moves twelve and a half hertz for every decade of light.
A lamp has a waveform
A lamp modulating a thousand times a second is a hundred times past the frequency at which flicker fuses, and it is plainly visible — as a dotted trail, during any glance across the room. The reason is not a new measurement; it is the spatial contrast sensitivity function, arriving from an unfamiliar direction.
A screen is a poor lamp
A display's white is a light. Shone on a surface it renders colour worse than a fluorescent tube — and the wider the gamut, the worse it gets, because the narrow primaries that buy a large triangle are exactly the ones that leave holes in the spectrum.
A lamp has a direction
A white LED is a blue die under a converter, and light leaving at an angle has travelled further through the converter than light leaving straight up. So the mixture is different in every direction, the beam is bluer in the middle than at its edge, and the number on the box is one direction's worth of a device that has no single colour.
Three ways to dim a lamp
"Dimmed to ten per cent" names three different colours. Switching the lamp on and off faster than anybody can see changes the chromaticity by nothing at all — exactly, to floating point. Reducing the current moves it a few units. Reducing the power to a filament moves it eighteen hundred kelvin down the Planckian locus, and stays on the locus exactly the whole way.
A lamp switched on is not the lamp measured
A luminaire takes about seven minutes to reach nine tenths of its working temperature, loses a quarter of its output on the way, and moves ΔE00 2.6 while it does. That is slower than every clock in the eye — so for the first minutes after a switch is thrown, both ends of the measurement are moving, and every appearance claim here has assumed one of them was still.
The index is one observer's opinion
A colour rendering index is printed to a tenth of a point and computed for a single set of colour-matching functions that no standard names as a choice. Scored by two hundred eyes instead, one lamp's spread is wider than the whole range the standard observer puts five lamps in — and two lamps the standard separates by four tenths of a point come out the other way round for every member of the population.
The shutter samples the lamp
A lamp switched between two drive currents at a hundred hertz is one steady colour to a person and two spectra to a camera. A thousandth-of-a-second exposure catches whichever phase the shutter opened at — two and a third stops of exposure and five units of colour, decided by nothing but timing — and a rolling shutter writes the difference across the frame as bands.
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.
Which lamp changes are free
The changes of light that existed before electricity commute with one another to a couple of parts in a thousand, so one set of axes handles all of them. The lights the lighting industry invented do not, and the worst pair in the census is seventy-six times further from commuting than the best.
Only one dimmer is invisible
An earlier essay separated the three ways to dim a lamp by the chromaticity each arrives at. Asked instead what an adapted observer is left with, the ordering is different and one method comes out at exactly zero — a duty cycle is a scaling, a scaling is a gain in every basis, and adaptation removes all of it at every depth.
The lamp that stopped emitting ultraviolet
A blue-pumped white LED has a die at 450 nanometres and emits nothing shorter. Between about 2005 and 2020 that lamp replaced almost every other indoor source, which removed the excitation supply from a great many rooms — so brightened materials stopped glowing indoors without a single one of them being reformulated.
Three numbers cannot see a line
An instrument that returns three filtered readings of a spectrum determines a three-dimensional projection of it and is exactly blind to the other seventy-eight. On daylight that costs almost nothing; on a fluorescent tube, three quarters of the lamp lies in the part no reading reaches, and adding filters recovers it slowly.
The colour is right first
A reconstruction of a lamp from twelve filtered readings gets its colour right to half a unit while two thirds of its spectrum is still unmeasured. That combination is not a partial success — it is the exact condition under which a spectral prediction made from the reconstruction will be confidently wrong.
Which changes of light pay for it
A fitted adaptation basis beats the receptors by 0.68 units on average, and an average is a poor description of what it does. On six of fourteen changes of light it is worse, and the whole of its advantage comes from four — a tungsten lamp and three coloured walls.
A notch a pigment cannot cut
The worst change of light a painted room can produce has no maximum inside the box the search was given, which the previous round reported as a family with no worst case. Bounded by what a molecule can actually do, it has one — at a band six nanometres wide, narrower than any pigment and narrower than the box.
A room bounds its own bounces
The adaptation census has one bounce and two bounces as separate rows, and a search over the family treats the count as a free integer it always takes to the largest value offered. A room offers no integer at all — it applies a geometric mixture of every number of bounces, and that mixture is bounded by the walls reflecting less than everything.
The darkest wall anybody sells
Asked which property of a paint decides the worst change of light a room can produce, anybody would answer how saturated it is allowed to be. A ceiling on saturation never comes near binding, because the worst wall is dark rather than colourful — and the constraint that does bind is one nobody would nominate.
A tolerance in the wrong coordinates
A display primary's tolerance is written as a region in chromaticity, because that is where the colorimetry lives. Nobody has a knob for chromaticity. What a maker of an emitter sets is a peak wavelength and a bandwidth, and the map between the two is so anisotropic that on the red primary its condition number is over eleven thousand.
The index is a choice too
Every colour in this collection is a sum over eighty-one numbers running from 380 to 780 nanometres in steps of five. That index is not a property of the eye, the light or the sample — it is a tabulation, and it holds three separable decisions that behave completely differently from one another.
Two ends and one is empty
Extending this collection's wavelength range down to 300 nanometres moves a red pigment under daylight by 0.502 ΔE₀₀. Extending it up to 830 moves the same colour by 0.00015. A fifth of a thermal source's power lies outside the range and almost none of its colour does, and confusing those two shares is how a range gets argued about instead of measured.
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.
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.
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.
The rods' route is priced by the lamp
A rod signal in a dim room disturbs a colour match, and how much depends on which of the cone pathways it reaches — a weight the physiology leaves uncertain, especially for the blue–yellow pathway. Under daylight the uncertainty is nearly free: a rod signal that skips the S pathway costs 0.90 at the median surface against 1.03 for one that enters all three. Under a phosphor white LED it is worth a factor of 2.75, 0.45 against 1.24. What decides it is one number per lamp: how large the rod signal is compared with each cone class's own catch of the light.
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.
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.
The reference lamp must not move
To measure an uncertain weight, use the condition in which the answer depends on it most. That is right about half of an asymmetric colour match and exactly wrong about the other half: a match measures a difference of two displacements, and a reference field that also moves with the weight cancels the signal the test field carries. Daylight is the least sensitive of five lamps and belongs in every one of the three best pairs — 75 settings against a phosphor LED, 2,804 against the pair of lamps the principle as stated would have chosen.
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.
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.
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.
The lamps two channels miss are smooth to the camera
A narrow channel at 450 nm tells a phone which lamps are structured, and a white LED pumped in the violet might hide from it; a channel at 500 nm does the same job, and an LED with a cyan emitter should blind it. Reading both and calling a lamp structured if either departs was supposed to cover both families. It covers the cyan family only because the 450 nm channel already did, and half the violet-pumped LEDs escape both channels. Every one that escapes leaves the camera's colours as accurate as a smooth lamp does. What the classifier misses is what the camera does not need to be warned about.
Two channels estimate what one can only sort
A narrow sensor channel was fitted to call lamps smooth or structured, and the label turned out to be a proxy for the thing a camera cares about: its own colour error under the lamp. Scored against that error directly, the best channel is still the one at 450 nm — the band where camera and observer differ most comes close and no closer. Read as an estimate rather than a verdict, a second channel at 500 nm, useless as a vote, earns its place: together they predict the camera's error to within 0.14 ΔE00 over fifty-four lamps, and deciding when to repair from that estimate raises a quarter of the false alarms the label raises.
A straightened channel repeats the one beside it
A phone's narrow sensor channel at 570 nm ranks a camera's colour error under a lamp almost as well as the one at 450, and estimates it worse than guessing the mean, because one lamp reads ten times higher than any other. Read through a logarithm, the channel is rescued: its estimate improves from ±0.57 to ±0.23. It still does not reach the 450 nm channel's ±0.16, and put beside that channel it adds nothing at all, where a 500 nm channel cuts the error to ±0.14. Straightened, 570 nm sorts the lamps as 450 nm already does; and the two lamps it reads highest have the most different errors, which no transform of one reading can tell apart.
One ratio reads a peak and a trough
Two narrow sensor channels, at 450 and 500 nm, estimate a camera's colour error under a lamp to ±0.14 ΔE₀₀; the proposal was to read their ratio as one number instead. Taken as one channel's departure over the other's, the ratio is worthless — no better than the mean, because for smooth lamps both departures are nearly nothing. Taken as the departure of the two readings' ratio from what the camera predicts for it, one number estimates the error to ±0.141, as well as the two channels as two. The reason is a sign: under every structured lamp the channels depart in opposite directions, a peak at 450 and a trough at 500, and the ratio adds them; where a cyan fill lifts 500, it subtracts them, as the error falls. And no five per cent calibration error moves any of these estimates by a hundredth.
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.
The tube a third channel misses has no family
Two narrow sensor channels read as one ratio estimate a camera's colour error under fifty-four lamps to ±0.141 ΔE₀₀, and their worst miss is the halophosphate tube, whose trouble lies in its long-wave phosphor. A third narrow channel in the long-wave gap was predicted to halve that miss near 610 nm. No channel does, at any centre or width: the best buys under a tenth of the whole error and leaves the tube's miss above three quarters of what it was. At 610 nm the camera already predicts the tube's share almost exactly. The tube's error follows where its main phosphor band sits, and a 620 or 640 nm channel does read that — inside the tube's own family. The census has one halophosphate tube, and a lamp left out of a census that holds nothing like it is a lamp no channel can be fitted to.
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.
A second lamp pair finds a lens band's centre
A band of extra absorption in an older lens is found by a colour-matching fit only if the fit has a parameter for it, and a band parameter at a fixed wavelength finds only a band near that wavelength. Freeing the band's centre as a fifth number asks whether the matches can place it. From one daylight-against-LED session they place a band above 430 nm to within six nanometres and one below it only to eight to twelve, where moving the band and ageing the lens change the matches together — and there they barely see the band at all. A second session, the LED against a three-emitter lamp, places every centre within five and a half nanometres. The fluorescent tube's line at 436 nm helps where it sits and nowhere else.
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.
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.
A stricter fit declares a wider line
A two-nanometre table of a mercury lamp whose every line is self-reversed admits every reversed line as a Gaussian within half a per cent, and declares a width its own ceiling cannot stand on from a depth of two and a half. The obvious repair is to fit harder. The prediction was that at a tenth of a per cent the table refuses the reversed blue and green lines, admits the least-reversed yellow, declares unsafely from it between — and that a threshold that strict would refuse even unreversed lines on a real instrument's noisy table. The refusals come where predicted: the blue from a depth of three, the green from four. But the stricter threshold makes the declaration less safe, not more. It narrows the range of widths each line admits from below, so the yellow, never refused, declares a wider width at every depth, and the ceiling built on it falls below the true peak from a depth of two instead of two and a half. And a tenth of a per cent survives a quiet instrument: no unreversed line is refused until the table's noise reaches half a per cent — and then the first line refused is the yellow, the one that declares.
Two nanometres hide the yellow's notch from any fit
A plain Gaussian fit admits the reversed tube's 578 nm yellow line at every threshold on a two-nanometre table, and the ceiling built on its width fails from a column depth of two and a half. A fit with the notch in it — an emitted Gaussian with an absorbed one taken out of its centre — might see what the plain fit cannot. The prediction was that two nanometres leave too few entries to determine the notch, and one nanometre finds it from a depth of two. Two nanometres hide it completely: the best notch fit carries a negative notch, a spike, at every depth, and the readings a two-standard-error rule calls reversed are the same noise draws at every depth, each fitting a notch that erases the whole line. One nanometre finds it sooner than predicted — in every reading from a depth of one and half of them at a half — and its fitted depth climbs with the column. But its standard error cannot be trusted with seven entries and four numbers: the rule calls a third or more of unreversed readings reversed. The notch's fitted depth itself separates them cleanly.
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.
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.
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.
A sensor calibrated on lamps cannot read an LED
A display that declares its room's light in the light's own chromaticity keeps a dark picture's shadows untinted, and the chromaticity has to come from a sensor. The prediction was that a small three-channel sensor calibrated on a 2,700 K and a 6,500 K lamp reads mixtures of a lamp and daylight within 0.001, because they are mixtures of its calibration lights, and misreads phosphor LEDs by 0.003 to 0.006. The mixtures nearly hold: misread by up to 0.0023, past 0.001 from half daylight, because daylight is not a 6,500 K lamp — and at a tenth of a candela no mixture tints the moon picture by a unit. The LEDs are three times worse than predicted. Every one of thirty white LEDs built onto the Planckian locus is misread by 0.007 to 0.017, and every one tints the shadows by 1.7 to 3.6 units. The divide is not lamp against daylight but smooth against spiky: every Planckian lamp and mixture under 0.0035, every LED and fluorescent tube over 0.007.
No calibration reads the lamps and the LEDs together
A three-channel ambient sensor pinned on two Planckian lamps reads every lamp and every lamp-and-daylight mixture well enough to keep a dark picture's shadows under a ΔEITP unit, and misreads every white LED badly enough to tint them by two to four. Its errors on the LEDs all point one way, and the prediction was that a matrix fitted in least squares to every light in a census would bring every LED under 0.004 while the lamps rose only towards 0.003. It does not. Fitted to the census, each light held out, the sensor's worst LED error falls from 0.017 to 0.0066 and its worst lamp error rises from 0.0033 to 0.0126; thirteen of forty-seven lights still pass a unit, among them the 2,700 K lamp the original calibration read exactly. A matrix exact on the lamp, daylight and one 4,000 K LED reads every mixture and every 4,000 K LED nearly perfectly and fails the 2,700 K LEDs, none read better than 0.0063. A fourth channel leaves six lights beyond a unit. Three broad channels have one matrix, and it can be right about one kind of spectrum.
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.
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.
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.
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.
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.
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.
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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The objects these essays reach for when they reach for this one.
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