The pigment decides the trade, not the band

A lopsided band does not break the trade freed a four-emitter lamp’s fourth band in every way the earlier search had fixed it — width, power, and a split shape with one side longer than the other — and found the trade between colour rendering and a slow pigment’s reach moving by a few directions of the object-colour solid at most. The conflict was about the lamp’s light in the amber, not about the band’s shape, and it survived every shape tried.

That census held one thing fixed on the other side. Its slow pigment was a slope limit: a reflectance allowed to change from nought to one over no less than forty nanometres, anywhere, in any pattern. The limits assume a pigment that switches instantly introduced it as the weakest honest statement of a colorant’s bluntness, and a sharp edge is bought with depth found that a truer model — reflectance as Beer–Lambert absorption through a few Gaussian bands, none narrower than forty nanometres — is not a stronger version of it but a different constraint. A deep band draws an edge sharper than its own width, because the exponential saturates, and in most directions of the solid under daylight three such bands reach further than a slope limit of the same nominal width.

The closing section of the lopsided-band essay predicted what that would do to the trade: “the band-shape pigment moves the frontier further than the lopsided lamp did — by twenty or thirty directions at a high rendering rather than six — because the conflict is between the lamp’s light in the amber and the pigment’s ability to switch there.”

A third fewer directions, and half the price #

Over the 153 directions of the census, the cheapest lamp rendering at a fidelity of 88 costs a slope-limited pigment 50 directions and a band-shape pigment 33. Raising the fidelity from the three-emitter lamp’s 78.6 to 88 adds 19 directions to a slope limit’s losses and 10 to a band-shape pigment’s. A lopsided fourth band saves a slope-limited pigment one direction at that rendering and a band-shape pigment none.

• The prediction holds, and it holds by about the amount predicted. On the full sweep of the solid, twice as dense as this census, seventeen directions are about thirty-four.
• Under every lamp scored, a band-shape pigment loses fewer directions than a slope limit — by six to twenty, fewest with a fourth band at 540 and most with one at 590.
• The shape of the fourth band stops mattering. Symmetric or lopsided, the lamps rendering at 88 cost a band-shape pigment the same 33 directions.
• At 590 nanometres a fourth band renders at 86 for six directions more than the three emitters cost a band-shape pigment. A slope-limited pigment pays eighteen for the same lamp.

The same lamps, charged twice #

The census is ten lamps. The three-emitter LED, with Gaussian emitters at 455, 528 and 625 nanometres; the same with a fourth Gaussian band twenty-four nanometres wide at power 0.3 — the earlier sweep’s band — at seven positions from 520 to 600; and the cheapest symmetric and lopsided lamps the lopsided-band census found rendering at 88, both with a band at 585 nanometres. Each lamp is scored for its fidelity index, and for how many directions of the object-colour solid a slow pigment loses more than a per cent of its ideal reach in — once with a forty-nanometre slope limit, and once with three absorption bands none narrower than forty nanometres and none deeper than an absorbance of twelve.

The directions are every twelfth of the Fibonacci sweep of the sphere, 152 to 157 of them depending on how many clear the boundary floor under each lamp. That is half the density the slope-only censuses used, because finding the best band-shape reflectance in a direction is a search over ten parameters rather than a dynamic program, and costs about a thousand times as much.

Every lamp’s two points are joined by a line that runs left, towards fewer losses, and the lines get longer towards the top of the figure. The three emitters lose 31 directions to a slope limit and 23 to absorption bands, eight apart. The lamps rendering at 88 and better lose 50 to 53 and 33 to 38, fifteen to twenty apart. The lamps that render best are the ones a band-shape pigment is most spared by, which is exactly the shape of result that moves a frontier: the rendering end of the trade gets cheaper and the other end stays where it was.

Read as a frontier, the open circles — the slope limit — climb from 31 directions at a fidelity of 78.6 to 50 at 88.9. The filled ones — the bands — climb from 23 to 33. The trade is still there. It costs half as much.

Why a band-shape pigment is spared #

Under the lamp that renders best, bands reach further than the slope limit in 100 of 153 directions, and the slope limit reaches further in 12. The rest tie, most of them at the ideal. The worst direction under the slope limit reaches 92.5 per cent of its ideal; under bands, 95.0. And the directions where bands gain most are the ones the slope limit loses most in — the cloud leans away from the diagonal at the bottom left and hugs it at the top right.

The reason is the one a sharp edge is bought with depth gave, applied to a lamp with a fourth band. A slope limit can change a reflectance at no faster than one unit per forty nanometres, anywhere, so a reflectance edge placed in the amber has to be forty nanometres wide, and a fourth band there lights all forty. A deep absorption band forty nanometres wide produces an edge much narrower than forty, because the band’s absorbance falls from twelve to nought and the reflectance only changes over the part of that fall between about three and nought. A band-shape pigment’s edges are sharp and its bands are wide, and a lamp that lights the amber punishes wide edges, not wide bands.

In this direction the ideal reflectance steps from nought to one at 597 nanometres, just past the fourth band’s peak. The slope limit spreads the step from 580 to 620 in a straight ramp, and the fourth band at 590 lights the lower half of the ramp, where the reflectance is still low: that direction reaches 89.6 per cent of its ideal. The best band-shape reflectance puts an absorption band forty nanometres wide at 560 with an absorbance of twelve, and its red flank rises from nought to one between about 590 and 615 — steeper than the ramp, and placed to rise just past where the lamp’s fourth band sits. It reaches 94.2 per cent. The second band, 77 nanometres wide at 488, darkens the blue and green where the ideal is nought. It cannot reach the violet with enough absorbance to darken it too, so the band-shape reflectance climbs back towards 0.75 at 400 nanometres where the slope limit stays at nought — a cost it barely pays, because the lamp has almost no light there.

Which directions the amber takes #

A count of lost directions hides which ones they are, and the two pigments differ in which ones as well as how many. Going from the three emitters to the best symmetric lamp rendering at 88, the slope limit newly loses 26 directions and regains 4; the band-shape pigment newly loses 17 and regains 6. Both regain some, because a fourth band that lights the amber also changes where the lamp’s light is balanced, and a few directions whose edges the three emitters placed badly are placed better.

The directions newly lost are located by where their ideal reflectance switches. For the slope limit, 22 of their edges fall between 560 and 600 nanometres, where the fourth band sits; for the band-shape pigment, 14. Both pigments lose about the same number with edges in the blue-green, around 480 to 500 nanometres — ten edges and nine — where the fourth band does nothing directly and the loss comes from the lamp’s rebalanced white moving the weighting. So the whole of the difference between the two pigments is in the amber, where the fourth band is, and it is the difference between a slow ramp and a sharp flank placed where the lamp’s light is.

That locates the result more tightly than the counts do. A band-shape pigment is not generally hardier than a slope limit under a four-emitter lamp; it is hardier exactly where a fourth band lights it, because that is where a slope limit’s edges are forced to be wide and a band’s are not.

Where the fourth band costs least #

The two curves have the same shape from 520 to 580 nanometres, the band-shape one sitting eight to fifteen directions below: a fourth band near 540 is cheap for both, and one near 570 or 580, where rendering is best, is dear for both. At 590 they part. The slope limit’s cost falls from 53 to 49; the band-shape pigment’s falls from 38 to 29, six above the three emitters alone. The lamp there renders at 86.

So the best amber position is the same for both pigments. What differs is the price, and it differs by three: six directions for a band-shape pigment, eighteen for a slope limit. A fourth emitter spends the gap it fills framed the trade as renders two and a half points better at 540 and costs a lot at 580; for a real colorant, the more useful statement is that 590 renders seven and a half points better than the three emitters and costs almost nothing.

The reason 590 is kind to bands is visible in the example. A fourth band at 590 sits where a band-shape pigment’s red-facing edges fall: those edges rise over the last few nanometres before the band’s centre gives way, and a band centred at 560 or 570 — where the solid’s orange directions want their absorption — has its flank crossing 590. A lamp band at 590 lights the top of that flank, where the reflectance is already near one. Lamp bands at 570 or 580 light its middle.

What lopsidedness bought, and does not #

The lopsided band the earlier census preferred saves a slope-limited pigment one direction here — nineteen added against twenty — and a band-shape pigment none: ten added whichever shape the band has. That band has a sharp blue side at 575 and a long red skirt into the red emitter, and it was kind to a slope limit because it kept the stretch between 540 and 575 dark, where the slope limit’s slow ramps want to sit. A band-shape pigment has no slow ramps. Its edges are sharp wherever it puts them, so it gains nothing from a stretch being kept dark for it.

That is the clearest way to state what this census found. The lamp’s shape and the pigment’s model both move the trade, and they are not independent. A lopsided band is a remedy for a slope-limited pigment’s specific weakness, and a pigment without that weakness has no use for the remedy.

What a lamp designer should take from it #

Know the pigments the lamp will light. Two models of a slow pigment, both defensible and both built on the same forty-nanometre figure, disagree about a lamp’s cost by a factor of two. A lamp designed to be kind to one of them — by lopsiding its fourth band, or by placing it at 540 — is optimising against the wrong model if the room’s pigments are real colorants. The limits assume a pigment that switches instantly introduced the slope limit as a floor on bluntness, and as a floor it is right: no real pigment switches faster. As a model of where a real pigment’s edges fall, it charges the amber too much.

Put the fourth band at 590 if the pigments are real. It renders at 86, seven and a half points above the three emitters, for six more lost directions under an absorption-band model. Three lines spare a slow pigment found that three narrow lines spare a slow pigment by keeping dark where it changes; for a band-shape pigment a fourth line at 590 keeps most of that sparing, because the pigment’s red-facing edges have already finished changing by the time they reach it.

And the rendering index is still the price. Nothing here removes the trade. Rendering at 88 costs a band-shape pigment ten directions it would not otherwise lose — fewer than a slope limit’s nineteen, but not none. A designer who wants both an index of 90 and a pigment’s full reach cannot have both with any of the lamps tried.

How the reach was computed #

The band-shape reflectance is $\rho(\lambda) = \exp\!\big(-k_0 - \sum_j k_j \exp(-4 \ln 2\,((\lambda - \mu_j)/w_j)^2)\big)$ with up to three bands, each $w_j$ at least forty nanometres and each $k_j$ at most twelve, on the collection’s five-nanometre table. For each direction of the solid, its reach under a lamp is the largest weighted sum of that reflectance against the direction’s colour-matching combination and the lamp, found by the same Nelder–Mead search from structured starts that a sharp edge is bought with depth used under daylight — starts built from where the direction’s weighting changes sign, with the two-band optimum included as a start for three so that more bands can never reach less. The slope limit’s reach is the dynamic program of the limits assume a pigment that switches instantly, quantised to 161 levels. A direction is lost when its reach is below 99 per cent of the ideal — the reach of a reflectance allowed to switch instantly — under the same lamp.

What this leaves out #

The search is a search. A Nelder–Mead search over ten parameters can stop short of the best reflectance, and under four narrow emitters a direction’s weighting changes sign between every pair of lines, which makes the structured starts’ job harder than under daylight. A long search from forty extra random starts was run on six directions under the lopsided lamp, including its three worst. It improved the worst by 1.9 per cent of the ideal reach, from 91.9 to 93.8, and the others by a tenth of a per cent or nothing, and no direction crossed the 99 per cent line either way — so the counts stand, and the census’s worst reaches are slight underestimates.

Half the directions. The census uses every twelfth direction of the sweep rather than every sixth, so its counts are about half the slope-only censuses’ and its differences should be doubled to compare with them.

And the bands are Gaussian. A real dye’s absorption band is closer to a Gaussian in energy than in wavelength, broader in nanometres at the red end, which a limit written in energy charges the reds found makes a slope limit charge the reds more. The same correction applied to the band widths here would make the band-shape pigment’s red-facing edges broader, and the 590 position’s advantage somewhat smaller.

Still open: whether a band written in energy keeps the amber free #

The 590-nanometre result turns on one thing: that a band-shape pigment’s red-facing edges have finished rising by 590. A band forty nanometres wide at 560 is about 0.16 electronvolts wide; the same width in energy centred at 600 would be about 46 nanometres, and the edge it draws on its red side broader by about a sixth.

The calculation is this census with each absorption band’s minimum width fixed in energy rather than in nanometres, at 0.16 electronvolts, and the question is whether 590 stays cheap. The prediction is that it does, with the six directions becoming about ten, because the edges that matter at 590 belong to bands centred between 540 and 570, where a width of 0.16 electronvolts is 37 to 41 nanometres — within a few per cent of forty either way. If the cost at 590 rises past fifteen — most of the way to a slope limit’s eighteen — then the result here belongs to the choice to write widths in nanometres, and a lamp designer would need to know which way a real colorant’s bands are written before trusting it.

The model of the thing is part of the trade #

The habit is about asking which side of a trade a finding is really about.

Two censuses in a row changed the lamp and held the pigment, and found the trade robust. The robust part was real — no lamp escapes it — but the size of the trade belonged as much to the pigment’s model as to the lamp. Changing the model changed the price by half, made the preferred band shape irrelevant, and moved the cheapest good lamp from 540 to 590.

The failure mode is to optimise one side of a trade against a model of the other side that was chosen for convenience. The slope limit was chosen because it has a clean dynamic program. Its answers about lamps were sound as bounds and misleading as advice, and the advice changed once the other side was modelled as what it is.