A fourth emitter spends the gap it fills

Three lines spare a slow pigment found that a limit on how fast a reflectance may change costs almost nothing under a lamp whose light sits in three narrow emitters. A pigment that takes forty nanometres to go from absorbing to reflecting loses more than a per cent of its reach in 183 of daylight’s 305 boundary directions and in only 62 of the three-emitter lamp’s 312, because the lamp measures almost nothing between its lines and a slow reflectance can do its changing there.

That essay named the threshold as the spacing of the closest pair of lines, and made a prediction from it. A fourth emitter halfway between blue and green would bring the closest spacing under forty nanometres, so the rescue should collapse at about forty instead of about eighty. It is a clean prediction and it is easy to run: the same census, with one more Gaussian emitter added at each position across the lamp’s span.

The prediction is right about the fourth emitter and wrong about the quantity. Where the emitter goes changes the cost by a factor of nearly three, and the closest spacing does not order it.

The cost is set by which gap loses its room, not by which spacing shortens #

A fourth emitter at 490 nanometres raises the count of directions a forty-nanometre limit costs from 62 to 172 of 312. One at 540 raises it to 65. One at 470, whose closest spacing is fifteen nanometres — the shortest in the whole sweep — costs 123.

• The curve has two peaks and they sit in the middle of the two gaps, at 490 and at about 575; it has three dips and they sit on the three existing emitters.
• The closest spacing ranks the positions at a rank correlation of 0.54, which is a relation rather than a rule. An emitter tucked beside an existing line shortens a spacing to almost nothing and costs almost nothing.
• What a direction needs is one span of darkness wide enough for its own transition, and an emitter beside an existing line leaves every span it did not sit in intact.
• The blue-to-green gap is where the rescued directions keep their edges: 51 per cent of the spared directions’ switching wavelengths fall between 485 and 515 nanometres, against 38 per cent of the unrescued ones’.
• Rendering and the census do not peak in the same place. The best-rendering position is 580 nanometres, at a fidelity index of 88.1 against the three-emitter lamp’s 78.6; the cheapest position for the solid is 540, at 81.0.

One emitter, twenty-nine places to put it #

The lamp is the one the earlier census used — Gaussian emitters at 455, 528 and 625 nanometres, of full widths at half maximum of 20, 33 and 18 — and the fourth is a Gaussian of width 24 and about half the power of the others, moved in five-nanometre steps from 470 to 610. Everything else is unchanged: the same 2,400-direction Fibonacci sweep taken every sixth, the same boundary floor, and each lamp’s constrained reach measured against its own ideal solid rather than against anybody else’s.

The shape is the finding. The curve dips to 67, 65 and 66 at 530, 540 and 545 — on and beside the green emitter — and to 66 at 610, beside the red one. It rises to 172 at 490 and 110 at 575, which are the midpoints of the two gaps. A three-emitter lamp costs 62, so the best fourth positions cost three or four directions and the worst costs a hundred and ten more than that.

The prediction from spacing would have made 470 the worst position in the sweep, since a fourth emitter fifteen nanometres from the blue one leaves a closest spacing of fifteen. It costs 123, which is well below 490’s 172 and above 540’s 65 — in the middle, and for a reason spacing cannot express.

That middle value is worth a sentence of its own, because it is neither of the two clean cases. An emitter at 470 does not split the blue-to-green gap in the middle; it moves the gap’s left edge fifteen nanometres to the right, leaving one span of 58 nanometres where there had been 73. Fifty-eight will hold a forty-nanometre transition and 73 held it more comfortably, so the directions whose switching wavelength happens to fall in the fifteen nanometres that were absorbed lose their room and the rest keep it. The cost is proportional to how much of the span was taken, and taking a slice off one end takes less than cutting it in two.

What a direction actually needs #

An optimal reflectance in a given direction is one where the weighting is positive and nought where it is negative, and the weighting is the lamp’s spectrum times a combination of the matching functions. The places where a reflectance has to switch are the places where that combination changes sign — and under a line lamp the sign between two lines is decided by almost no light, so a slow reflectance may take as long as it likes crossing it, provided it has finished before the next line.

So a direction is spared when one span of darkness at least as wide as its own transition contains the wavelength where it must switch. That is a statement about a span, not about a spacing, and the two come apart the moment more than two lines are in play.

The two panels are the mechanism. At 490 the blue-to-green gap of 73 nanometres becomes two spans of 35 and 38, neither of which will hold a forty-nanometre transition, and every direction that was switching there loses its room at once. At 540 the fourth emitter sits twelve nanometres from the green one; the blue-to-green span is shortened to 85 and the green-to-red span to 85, both still ample, and almost nothing changes. The gap that was going to be split is the whole story, and the spacing the emitter creates with its nearest neighbour is beside the point.

This is the same geometry a narrow primary buys a disagreement described from the observer’s side: what a narrow-line lamp leaves undetermined are the wavelengths between the lines, and different eyes and blunt pigments both take shelter in exactly that undetermined region.

Where the rescued directions keep their edges #

If a fourth emitter is expensive when it takes room from a gap, then how expensive depends on how many directions were using that gap. That is measurable directly.

The spared directions pile their edges into two ranges, and both are gaps. One hundred and nine of their edges sit between 485 and 515 nanometres and eighty-four between 560 and 610 — the two dark spans — while the lost directions’ edges scatter across the middle of the spectrum with a concentration near the green emitter itself, where the lamp’s weight sits on both sides of the switch and a slow transition costs what it costs under daylight.

That histogram predicts the cost curve before the cost curve is computed. The blue-to-green gap holds more of the spared edges than the green-to-red gap, and so a fourth emitter in the first costs more than one in the second: 172 against 110. It also explains why the second peak is broader and lower. The green-to-red gap is 97 nanometres wide, so an emitter near its centre still leaves two spans of about 48, which will hold a forty-nanometre transition — awkwardly, with nothing to spare, but it will hold it.

Why the edges gather where they do is a fact about optimal colours rather than about lamps. The directions that reach furthest into the saturated part of the solid are pass-bands and stop-bands — a reflectance that is one across a range of wavelengths and nought outside it, or the reverse — and the ranges that produce a colourful result are the ones that isolate one part of the matching functions from the rest. Those boundaries land near 490 and near 580 because that is where the long- and middle-wavelength functions cross and where the middle one falls away. A limit written in energy charges the reds found the same two places from the other direction: the census is most sensitive to how a limit is stated exactly where the most edges are, and the census here is most sensitive to where a lamp puts light for the same reason.

Only one width cares #

The whole argument is about a transition fitting in a span, so it should stop mattering when the transition is much smaller than every span and when it is larger than all of them. It does.

At twenty nanometres the whole sweep moves by ten points of share, from 15 directions to 46 against the three-emitter lamp’s 17. A twenty-nanometre transition fits in a 35-nanometre span, so splitting a gap in two does not take a direction’s room away; it only crowds it. At eighty the sweep moves by seven points and the whole curve sits near four fifths, 207 to 225 against the three-emitter lamp’s 214 — because an eighty-nanometre transition already fits in no gap the lamp has, so there is nothing for the fourth emitter to spend.

Forty nanometres is where the position matters, and it swings by thirty-four points of share. That is the width at which the lamp’s own gaps are the same size as the transition, which is to say the width at which the original rescue was found. The interesting version of this is not that a fourth emitter is expensive but that its cost is concentrated at exactly the pigment sharpness a designer would be relying on the lamp to forgive.

What the emitter was added for #

A fourth emitter is not added to a lamp to be kind to pigments. It is added because three narrow lines render surfaces badly — a screen is a poor lamp is the same fact from the display side — and the honest output is the pair of quantities, not either alone.

The two curves are single-peaked and their peaks are forty nanometres apart. Rendering climbs from the three-emitter lamp’s 78.6 to 88.1 at 580 nanometres, which is the amber position a lamp maker would actually reach for, and falls to 76.5 at 490 — a cyan emitter makes the lamp render worse as well as costing the solid most, because it pushes the correlated colour temperature from 7,500 K to 8,800 and leaves the deficient amber region where it was. So one end of the sweep is bad at both things and there is no case for it.

The census is cheapest at 540 and rendering there is 81.0, two and a half points better than three emitters. That is the shape of the trade: the first points of rendering are nearly free and the last are not.

The result has an obvious cousin in display design, and the two are not the same problem. A fourth primary is a design priced the fourth channel a display adds, and there the fourth primary’s job is to enlarge the set of colours the display can make; four primaries have a choice is the part where a four-primary display has more than one way to make a given colour and has to pick. A lamp has neither property. Its fourth emitter does not enlarge a gamut, because a lamp is not asked to reproduce anything, and it has no choice to make, because a lamp emits its whole spectrum at once. What it has instead is this: the same power that makes surfaces look right where the lamp was dark is the power that stops a blunt pigment hiding there.

Eleven positions are not beaten on both counts, and the three-emitter lamp is one of them. Reading up the frontier: the first 2.4 points of fidelity cost 3 directions, and the remaining 7.1 points cost 44. Between 540 and 565 the rate is about six directions a point; above 565 it is about four, because the far amber positions are buying their last points of rendering by moving the white rather than by filling the gap.

The question the earlier essay asked was whether the two curves cross, so that a lamp could render acceptably and still spare slow pigments. They do not cross — the frontier is monotone — but the rate along it is not constant, and that is the more useful answer. A designer who wants three points of rendering can have them for three directions; one who wants ten cannot have them at all with one emitter, and one who wants nine pays a hundred and ten.

How the lamps and the census were built #

Each lamp is a sum of Gaussians on the five-nanometre grid, normalised nowhere — the census divides by the lamp’s own luminance, so a common scale cancels. The fourth emitter’s width of 24 nanometres and power of 0.30 relative to the others are stated rather than fitted; a narrower or weaker one moves the peaks’ heights but not their positions, because what a position does is decided by where the emitter’s power sits and not by how much of it there is.

The constrained reach in each direction is the dynamic program the earlier essays use: the reflectance is quantised into 161 levels and may move at most a whole number of levels per five-nanometre band, so a transition width is imposed exactly rather than rounded. Directions whose ideal support is under five units are set aside, which is why the counts are out of 312 rather than out of 400.

The fidelity index is this site’s own, computed in CAM16-UCS against a reference of the lamp’s own correlated colour temperature, on twelve constructed reflectances. Rendering in three numbers is where its scale and its limits are set out, and the scale factor in it is a presentational choice; what is being compared here is one lamp family against itself, so the factor cancels out of every difference quoted.

What this does not settle #

One emitter, one width, one power. A designer adding a fourth emitter would choose its width and its power as well as its centre, and the sweep holds two of those three fixed. The direction of the effect is set by geometry and will not change, but the height of the peak at 490 would fall for a narrower emitter and rise for a broader one.

The fidelity index is an average over twelve samples, and rendering in three numbers is about exactly why that is the wrong summary for a failure concentrated in one part of the hue circle. The worst-sample shift under the three-emitter lamp is 6.17 in CAM16-UCS, and the frontier drawn on worst-sample shift rather than on the index would not be the same frontier.

The position that renders best is where a lamp maker would already put it. Amber emitters near 580 nanometres are the standard fourth channel in a four-channel white LED, sold for exactly the rendering the index here reports, so the expensive corner of this trade is the one that is commercially occupied. The lamp in the shop decides is the reminder that what is on sale is what a specification will be judged under, and nothing in that essay or this one suggests a lamp maker should choose otherwise — the amber emitter is bought with rendering that a reader can see, and the reach it spends is a bound a reader cannot.

And the census is a bound rather than a forecast. It says how much of each lamp’s own ideal solid a pigment of stated bluntness can reach, and the solids differ between the lamps: no surface can be that colourful is the measurement of how much a bound over surfaces moves when the light changes. A reader standing under either of these lamps sees colours, not fractions.

Still open: whether a gap can be filled without being split #

The two things the designer wants are in conflict only because an emitter is a bump. A gap is spent when something is put in the middle of it, and rendering is bought by having power where there was none — but not necessarily in the middle.

The computation worth running is a fourth emitter constrained to sit at the edge of a gap rather than anywhere in it: a wide, low emitter whose skirt fills the near side of the amber span and leaves the far side dark. Its centre would be near 560, where the frontier already says the rate is best, but its shape would be different — asymmetric, which a Gaussian cannot be. A skewed band with two parameters, an offset and an asymmetry, would say whether the frontier here is a property of the trade or a property of the family of shapes it was searched over.

The prediction is that the frontier moves and does not disappear, because the rendering that the far amber positions buy is partly a shift of the white point, and a shift of the white point cannot be had without power where the white needs it. If it does disappear — if some asymmetric band renders like the 580 emitter and costs like the 540 one — then the whole conflict was an artefact of insisting the emitter be symmetric, which is a thing about the model and not about lamps.

A threshold named from one example is a coincidence until it is swept #

The habit is about the quantity a threshold is attributed to.

The earlier census had one line lamp, and its rescue ended at about the width of that lamp’s closest spacing. Closest spacing is a natural quantity, it had the right units, and it fitted. It was also the only quantity available with one lamp, because with three lines in fixed positions the closest spacing, the widest span and the total dark room are all the same number wearing three hats.

The move is to sweep the lamp before believing the attribution, and to sweep it in the direction that separates the candidate quantities rather than in the direction that is easiest. Adding a fourth emitter separates them: a position beside an existing line makes the closest spacing tiny and leaves every span intact, and a position in the middle of a gap does the opposite. One sweep, and the two hats come off.

The failure mode is a mechanism read off a single configuration, where several quantities coincide. It is not wrong so much as untested, and the test is a configuration in which the candidates disagree — which is usually a configuration nobody would build, chosen for exactly that reason.