The gap has to be dark, not the line narrow

Three lines spare a slow pigment ended with a claim it had no way to test: “the advantage is set by line spacing, not by line width or colour temperature.” The lamp it measured had one set of emitter widths, so width was fixed at the moment the sentence was written, and the sentence was an attribution rather than a measurement.

It cannot be right as stated. An emitter broad enough is not a line at all — take the widths up far enough and the three-emitter lamp becomes a smooth lamp with three humps in it, which is what a phosphor LED is, and that essay’s own census put phosphor LEDs with daylight. So width must matter somewhere, and the only question is where and how sharply.

The sweep is cheap: the same three emitters, at the same centres, with every width multiplied by the same factor. Nothing moves, so spacing is held exactly, and whatever changes is width doing it.

Half of it is gone at 1.3 times the nominal width #

At their nominal widths the three emitters leave 62 of the solid’s 312 directions losing more than a per cent of their reach to a forty-nanometre transition limit. Daylight loses 183 of 305. At 1.3 times the nominal widths the lamp loses 144 — over half the distance to daylight, from a change no one would describe as making the lamp broadband.

• The deciding quantity is the light left at the darkest point of a gap, as a share of the lamp’s peak. It goes from 1.0 per cent at the nominal widths to 7.4 at 1.3 times and 44 at twice.
• The two gaps fill at different rates and the narrow one goes first. The blue-to-green gap of 73 nanometres passes a twentieth of the peak at 1.3 times; the green-to-red gap of 97 nanometres not until 1.75.
• At twenty nanometres the sweep does almost nothing — 17 directions at the nominal widths, 22 at 1.5 times, 53 at four times against daylight’s 49 — because there was little rescue to lose.
• At eighty it does nothing at all, 214 through 222 against daylight’s 218, because there was none.
• The count saturates and the depth does not. Past twice the nominal width the share of directions losing stays within four points while the tenth percentile keeps falling, 0.952 to 0.934.

What broadening does to a spectrum that keeps its lines #

The lamp is the one the earlier censuses used: Gaussian emitters at 455, 528 and 625 nanometres, of full widths at half maximum of 20, 33 and 18. The sweep multiplies all three widths by a common factor from one to six and leaves the centres and the relative powers alone, so the lamp’s chromaticity moves only as much as the broadening itself moves it and the spacing is a constant throughout.

The three panels are the whole difficulty. All three read as line spectra — three separated peaks on a dark background, which is what anybody would call a narrow-band lamp — and the census behind them runs 62, 144, 188 out of about 310. The middle panel is the one worth staring at, because the lamp in it is the one that would be described in exactly the same words as the lamp in the top panel and behaves, for this purpose, much more like the bottom one.

What has changed between the panels is not visible as a shape. It is the value at the bottom of the trough between the blue and green emitters, which goes from a hundredth of the peak to a fourteenth to nearly a half. On a linear plot of a lamp’s spectrum a hundredth and a fourteenth of the peak are both indistinguishable from zero, and that is the reason the earlier essay could not have seen this: the quantity it needed was under the resolution of the picture it was looking at.

The same blindness has a measurement half. A grid is not a resolution separated the step a spectrum is tabulated at from the width of the instrument that produced it, and a floor is exactly the quantity that separation matters for: a trough measured through a slit wider than the trough is filled by the slit rather than by the lamp. Every floor quoted here is computed from an analytic spectrum on a five-nanometre grid, so it is what the lamp has rather than what an instrument would report, and the gap between those two is the subject of the open question at the end.

The floor, not the width #

Once the quantity is named the sweep collapses onto one curve.

At forty nanometres the curve falls from 20 per cent of directions to about 60 across two decades of floor and is half spent by 7.4 per cent. By the time the darkest gap carries a fifth of the lamp’s peak, the census has reached nine tenths of daylight’s — the lines have stopped sparing anything worth counting.

Why a floor rather than a width is the mechanism restated. A direction is spared when its optimal reflectance’s switching wavelength sits inside a span the lamp barely measures, and how much a slow switch costs there is set by how much light is in that span to be got wrong. At a floor of a hundredth the span is effectively unmeasured and the transition is free; at a floor of a tenth the lamp is watching, weakly, and a blunt transition starts to cost what it costs under a smooth lamp. The width of the emitter enters only through the floor it produces — which is why a lamp with wide emitters far apart and a lamp with narrow emitters close together can behave identically here, and why neither number alone predicts the other.

This is the same lesson a fourth emitter spends the gap it fills reached from the other direction. There the question was which gap an added emitter takes its room from; here it is how dark the gap has to stay. Both say that the quantity is a property of the span between the lines and not of the lines.

Two gaps, two rates #

The lamp has two gaps and they are not the same size, so they do not fill together.

The blue-to-green gap reaches a twentieth of the peak at 1.3 times the nominal widths and the green-to-red gap at 1.75. The narrow one goes first for two compounding reasons: it is 73 nanometres wide against 97, so the skirts of its two emitters overlap sooner, and the green emitter between them is the broadest of the three to begin with, at 33 nanometres against 20 and 18.

That staggering is why the census does not fall off a cliff. A direction needs one span wide enough for its own transition, so as long as the green-to-red gap stays dark the directions that switch in the amber are still spared, and the earlier histogram of where the census’s edges fall said that is about two fifths of them. The lamp therefore loses its rescue in two instalments — the first between 1.1 and 1.5 times nominal, the second between 1.5 and 2 — and the curve through those points is the sum of two shorter falls rather than one long one.

What is left after the count has stopped #

The census reports two things and it is worth watching them part company.

By twice the nominal widths the count has arrived and stops, at about 61 per cent against daylight’s 60, and it moves by under four points over the remaining factor of three. The tenth percentile has not. It runs 0.981 at the nominal widths, 0.952 at twice, 0.931 at four times — below daylight’s 0.938, which the count never goes below.

Both are true of the same census and they answer different questions. The count asks how many directions are affected at all and it saturates when every direction that can lose something has lost something. The percentile asks how much, and nothing stops it. A lamp broadened past twice its nominal widths is no longer costing new directions; it is costing the directions it already had, more. That the tenth percentile crosses daylight’s while the count does not is the tidiest statement of it: a very broad three-hump lamp has the same number of affected directions as daylight and treats the worst of them worse, because its humps still weight the spectrum unevenly and the unevenness now falls on the switch rather than beside it.

A difference has no rate makes a related point about a colour difference and this one is about a census: a summary that counts occurrences and a summary that measures size are not two views of one number, and a family in which they move together over part of its range will part company over the rest of it.

Where the whole distribution goes #

The sorted census shows the same collapse as a shape rather than as two summaries.

At the nominal widths four fifths of the curve sits above 0.99 and the losses are a short tail. At 1.3 times the middle of the distribution has dropped away and the curve runs close under daylight’s over the worst third. At twice, the two are almost one line from the worst direction to about the eightieth percentile, and the three-emitter lamp keeps an advantage only in the top fifth — the directions whose edges sit in the green-to-red gap, which is still dark at that width.

The shape says something the counts do not. The rescue was never a uniform lift. It was a specific set of directions moved from the middle of the distribution to the top, and the broadening takes them back one group at a time, in the order their gaps fill. The earlier sorted census under daylight and three emitters showed the middle of the distribution lifted; this shows the lift is held by a spectral feature that a modest change in emitter width removes.

Which directions those are is worth naming, because they are not a curiosity of the census. The middle of the distribution is where saturated but not extreme colours sit — the part of the solid a paint range or a brand palette is actually specified in — and the extremes at either end are the near-neutrals, which no limit touches, and the corners, which no pigment reaches under any lamp. No surface can be that colourful measured how far a bound over surfaces can move when the light changes; what moves here is the same bound under a change too small to describe.

How the lamps were swept #

Each lamp is the sum of three Gaussians on the five-nanometre grid with the widths scaled together and the powers unchanged, so the broader lamps carry more total power; every census divides by the lamp’s own luminance, so a common scale cancels and only the shape enters. The floor of a gap is the smallest value the spectrum takes strictly between two adjacent emitter centres, divided by the lamp’s peak.

The constrained reach is the dynamic program used throughout: the reflectance is quantised into 161 levels and may move at most a whole number of levels per band, so each transition width is imposed exactly. Directions whose ideal support is under five units are set aside as pointing into the black corner, which is why the denominators drift between 305 and 312 as the lamp changes — a broader lamp moves a handful of directions across that floor, and it is also why the tenth percentile is compared end to end rather than step by step.

The daylight comparison in every figure is the same census under D65, from the limits assume a pigment that switches instantly.

Where this stops #

The emitters are broadened together and they are Gaussians. A real lamp’s emitters do not scale in step — a green LED broadens with drive current and temperature faster than a red one — and a real emitter has a skirt that is not Gaussian, usually with more energy in its long-wavelength tail. Both would change which gap fills first and neither changes that a floor is the quantity.

The floor is a single number for a gap and a gap is not a point. A span that is dark at one end and lit at the other has the same floor as one that is uniformly dim, and a direction switching at the lit end of the first would not be spared while the floor said it should be. The census would need the floor as a function of wavelength to catch that, and nothing here asks for it because the two gaps of this lamp are close to symmetric.

Colour temperature is held only approximately. Broadening three emitters at fixed powers moves the lamp’s white a little, because a wider green emitter adds more to the middle of the spectrum than a wider red one adds to the end of it, and the census is taken against each lamp’s own ideal solid rather than against a common one. That is the right comparison for the question — how much of what this lamp could show does a blunt pigment reach — and it is the wrong one for a reader asking whether the colours look different, which there is no D65 lamp is the standing reminder about.

And nothing here is a statement about what a lamp looks like. A lamp broadened to 1.3 times these widths renders surfaces slightly better — the same gaps that spare a pigment are the gaps that a screen is a poor lamp prices as bad rendering — so the loss measured here is bought with something, and this census is not the place the purchase is valued.

Still open: what a datasheet would have to say #

A floor is a number a lamp’s own measurement could carry and none of them does. A datasheet for a narrow-emitter lamp states centre wavelengths and full widths at half maximum, both of which describe the peaks, and the quantity that decides this census describes the troughs.

The measurement is not a new instrument, only a new line in the table: the minimum of the measured spectrum between each adjacent pair of peaks, as a fraction of the maximum, at a stated resolution. It is read off a spectrum a lamp maker already takes. The resolution is the part that would need saying, because a floor of one per cent measured through a five-nanometre slit is not the same number as a floor of one per cent measured through a one-nanometre slit, and one slit, two requirements is where the difference between those two instruments was priced.

The prediction worth recording is that measured floors on commercial three-emitter lamps will cluster between one and ten per cent rather than below one, because the phosphor-free emitters that make such lamps are chosen for efficiency rather than for spectral purity and their tails are not designed. If that is right, real narrow-emitter lamps sit on the steep part of the curve here rather than at the top of it, and the rescue reported for the constructed lamp is an upper bound on what a real one delivers.

A quantity that is constant in the example is not a quantity the example measured #

The habit is about which variable an explanation is allowed to name.

The first census had one line lamp. Under it the closest spacing, the widest dark span, the emitter width and the floor between the lines were four different names for one configuration, and any of the four could have been written into the explanation with equal justification — which is to say with none. Spacing was chosen because it had the right units and the threshold landed near it.

The move is to ask, before writing the sentence, which of the coincident quantities the data could have distinguished, and to say plainly that it could not. Then sweep the one that is cheapest to sweep. Widths scale in a line of code and positions do not have to move, so the experiment that separates width from spacing is the easier of the two and it was available from the first day.

The failure mode is an explanation whose variable never varied. It is not falsified by the data it was written from, because the data has one value of it; it is simply unsupported, and it reads exactly like a finding.