A harder search pulls the band pigments away from the slope limit

The object-colour solid is every colour a surface can have under a light, and its boundary is reached only by ideal pigments that switch between reflecting everything and nothing at an instant. The limits assume a pigment that switches instantly asked what a real colorant gives up by not switching instantly, and three models of a real one have since been compared against the ideal: a slope limit, which bounds how fast reflectance may change per nanometre; bands in nanometres, a reflectance made of up to three absorption bands none narrower than forty nanometres; and bands in energy, the same with each band’s width fixed in energy rather than wavelength. For each direction through the solid, each model’s reach is found and set against the ideal’s, and a direction counts as lost when a model reaches less than ninety-nine per cent as far.

The lamp outweighs the pigment model ran that census under daylight, tungsten, a white LED and three narrow emitters, and found the models ranked the same way under every lamp — slope dearest, energy cheapest — but each costing two and a half to three and a half times as much under a broadband lamp as under the emitters. A tube spares a pigment through its phosphors and a mercury lamp’s few losses are deep added five line lamps. Every one of those counts rests on an asymmetry nobody stated in a headline. The slope limit is solved exactly, by a dynamic programme over reflectance levels. The band models are searched: Nelder–Mead from a handful of structured starts per direction, keeping the best.

A pair of lines asks for a cliff, not a gap exposed what that asymmetry might cost. Searching the band model from 680 starts in the mercury lamp’s twenty-five splitting directions, it found the model reaching further than the census had in seventeen of them, by up to three points. Under one lamp, in one kind of direction, the band model had been short-changed. The question here is whether that is true everywhere, and what it does to every comparison built on the census. The expectation, from the mercury search, was that a harder search would lower the band models’ counts modestly under every lamp and leave every ranking intact.

A sixth fewer, and the slope limit left standing #

Searched again by Nelder–Mead from 594 starts in every direction each band model lost by more than one per cent, under each of the nine lamps, both band models reach further than the census found in more than three fifths of those directions everywhere — from 63 to 94 per cent. Summed over the nine lamps, the band model’s count of lost directions falls from 395 to 335 and the energy model’s from 370 to 311, by a sixth each; neither rises under any lamp, and the slope limit’s 463 does not move. The ordering stated for the emitters, daylight, tungsten and the LED holds: slope dearest, energy cheapest. Every model still loses more than 2.3 times as many directions under each broadband lamp as under the emitters. But the slope limit now loses more than a third more than the band model, against a sixth on the census’s search; the coating’s inverted ranking disappears; and the triphosphor tube now costs the band model fewer directions than the three emitters do.

• The expectation holds on rankings: nothing the censuses ranked is re-ranked.
• It understates the size. A sixth of every band-model count, and up to two fifths under one lamp, is not a modest correction.
• Every comparison with the slope limit moves, because only one side of it was searched short.

Read down the figure and the slope limit’s bars are the same as the census’s, because they are the census’s: an exact solution has nothing to search. The band models’ bars are shorter, under nearly every lamp. The largest drops are under daylight — bands in nanometres from 73 to 63, bands in energy from 70 to 54 — and under the halophosphate tube, from 71 and 73 to 61 and 61. The smallest are under the three emitters and the mercury lamp, where the band model’s count does not change at all; those are the two lamps under which the census’s few starts were already closest to the best.

Where the census’s search stopped short #

The harder search improves most of what it is given. Under the triphosphor tube it finds a better band pigment in 24 of the 26 directions the census lost; under the coating alone in 17 of 18 for the energy model; under daylight in 46 of 73 and 56 of 70. It is never below three fifths. Most of the improvements are small — a few tenths of a point of reach — and only some carry a direction back over the line at ninety-nine per cent. But a few are large: the energy model under daylight gains 9.6 points in one direction, and the band model under the mercury lamp 6.5.

Lamp by lamp the picture is the same shape: the slope limit’s pair of bars equal, the two band models’ second bars shorter, barely or not at all under the emitters and the mercury lamp, by a seventh or more under daylight and the halophosphate tube, and by a third or more under the coating. Nowhere does the harder search lengthen a bar, because it keeps the census’s reach whenever it fails to beat it.

The reason the structured starts fall short is the shape of the problem. A band pigment is ten numbers — a baseline, and a centre, width and depth for each of three bands — and the reach it achieves in a direction is a function of them with many separate maxima: a band can sit on either side of an edge the direction asks for, and two bands can share the work of one or split it. A few starts placed by rule find the nearest maximum to the rule’s guess. A sharp edge is bought with depth and the cliff essay both found the best band pigments doing something no rule would place them to do — stacking two bands at one centre to steepen a flank — and the 594 starts here, two bands placed on a grid of centres at half and full depth, are enough to find that kind of arrangement in most directions.

The ordering holds where it was stated #

The census’s headline was that the models rank the same under every lamp: the slope limit loses most, bands in nanometres next, bands in energy least. Under the three emitters, daylight, tungsten and the white LED it still does: 31, 23 and 17 under the emitters; 89, 63 and 54 under daylight; 77, 61 and 59 under tungsten; 83, 54 and 49 under the LED. That claim was made for those four lamps, and the harder search leaves it standing.

Under two of the line lamps it does not quite. Under the triphosphor tube the energy model now loses more than the band model, 22 against 19, where the census had 26 and 25; under the halophosphate tube the two band models tie at 61. The census never stated the ordering for those lamps, and the change is small. What it shows is that the two band models are closer together than the census made them, and that which of them is cheaper depends on the lamp — which is what written in energy, the bands spare the green found about where each model’s bands are narrow and where wide.

A broadband lamp still costs more than twice as much #

The census’s second claim was the one in its title: the lamp outweighs the model. Every model, searched harder, still loses between 2.35 and 3.59 times as many directions under each broadband lamp as under the emitters. The band model’s ratios fall — from 3.17 to 2.74 under daylight, from 2.65 to 2.35 under the LED — because the harder search helps it more under broadband lamps than under the emitters, where its count did not move. The energy model’s rise under tungsten and the LED, because its count under the emitters fell from 20 to 17.

The stronger form of the claim also survives, with less room. Across the emitters, daylight, tungsten and the LED, the widest difference between models under any one lamp is now 35 directions — daylight’s 89 against 54 — and the narrowest difference between those lamps for any one model is 40, the band model’s 23 to 63. On the census’s search the two numbers were 29 and 50. The lamp still matters more than the model, but the margin has gone from 21 directions to 5, because the harder search spread the models apart and drew the lamps together.

Under tungsten the census found the three models “nearly one”, at 77, 69 and 66. Searched harder they are 77, 61 and 59: the two band models nearly one, and the slope limit standing sixteen directions apart from them.

The slope limit’s lead grows #

That is the general shape of what changes. On the census’s search the slope limit lost 1.17 times as many directions as the band model over the nine lamps; searched harder it loses 1.38 times as many. Under daylight, tungsten and the LED its lead over the energy model widens from 19, 11 and 29 directions to 35, 18 and 34. The slope limit was always the most restrictive of the three; it is now more clearly so, and the two band models, which the pigment decides the trade, not the band found cheaper than the slope limit by about a third under a four-emitter lamp, are cheaper still.

This is the right direction for the correction to run. A census that solves one model exactly and searches another will always flatter the exact one, because a search can only fall short. The size of the flattery is now measured: about a sixth of the searched model’s count, varying with the lamp from nothing to two fifths.

The coating’s odd ranking was the search’s #

Under the tube’s phosphor coating alone — its three phosphor bands with the mercury lines taken out — the census had the band model losing 22 directions and the slope limit 14. That was the one place in all nine lamps where the band model came out dearer than the slope limit, and the tube essay reported it without explaining it. Searched harder, the band model loses 13 and the energy model 12. The inversion was the search, and the coating now ranks like every other lamp.

The same correction changes the tube essay’s headline. It found the triphosphor tube sitting with the emitters rather than with daylight: 26 directions lost to bands in nanometres against the emitters’ 23 and daylight’s 73. Searched harder it is 19 against 23 and 63 — the tube now costs the band model fewer directions than the three emitters do. For the energy model it still costs a little more, 22 against 17. The tube essay’s conclusion survives and strengthens: the phosphors are what make the tube spare. Its secondary finding survives too — doubling the mercury lines raises the band model’s count, from 19 to 23 now rather than 26 to 32 — and the halophosphate tube still costs what daylight costs, 61 against 63.

The deep losses fall most where the lamp has lines #

Counted at five per cent instead of one, the band model’s deep losses over the nine lamps fall from 50 to 39 and the energy model’s from 56 to 42, a fifth and a quarter. The mercury lamp’s fall most. The mercury essay found that lamp’s losses unusually deep: a third or more of every model’s losses fell more than five per cent short. For the slope limit that still holds — 11 of its 21 losses are deep. For the band models it no longer does: four of eighteen and four of seventeen, under a quarter each, from six and eight. Every deep loss under the mercury lamp was a direction asking a pigment to reflect one of the lamp’s green and yellow lines and absorb the other, 32 nanometres apart, and the cliff essay showed what a band pigment does with that — stacks two bands on one side of the pair to make a steep flank. The structured starts rarely began there; the grid of starts often does.

Daylight’s worst band-model loss moves from 23.9 per cent to 21.4, and the mercury lamp’s from 10.3 to 7.1, the number the cliff essay already reported for that lamp from its own search.

What changes in the essays that used these counts #

The counts in the earlier essays are the census’s, and each now carries a note. Four findings move:

• The lamp outweighs the pigment model: the band and energy counts under daylight, tungsten and the LED fall to 63, 61 and 54 and to 54, 59 and 49; the broadband ratios become 2.35 to 3.47; tungsten’s three models are no longer nearly one, though its two band models are.
• A tube spares a pigment through its phosphors: the tube’s band-model count is 19, below the emitters’ 23 rather than above; the coating’s 22 becomes 13.
• A mercury lamp’s few losses are deep: the band models’ deep shares fall to four in eighteen and four in seventeen.
• A pair of lines asks for a cliff, not a gap: its own search found five deep losses for the band model; the grid of starts here finds four, one more direction coming under five per cent.

None of the essays’ arguments reverses. Each had a ranking or a mechanism at its centre, and every ranking stated survives. What does not survive is the habit of quoting the band models’ counts as if they were the same kind of number as the slope limit’s.

How the search was run #

Under each lamp, every direction a band model lost by more than one per cent on the census’s search was searched again. A start places two bands on a grid — the first centre every ten nanometres from 380 to 700, the second every forty — at half and at full depth, the third band empty; that is 594 starts. Each runs Nelder–Mead on the model’s ten numbers against the direction’s weights, the best reach is kept, and the census’s own reach is kept if it is better. The band model’s widths are at least forty nanometres and its depths at most twelve; the energy model’s widths are fixed in energy as before. The slope limit is not searched: its dynamic programme over reflectance levels finds the best on its grid by construction.

The directions, lamps, ideal reaches and thresholds are the census’s. Directions the census did not count as lost are not searched again, so a direction a better search might push from just under one per cent to further under is not counted, and no direction can move from reached to lost.

What the search does not reach #

A finite grid of starts. Six hundred starts are many more than a handful and not all of them; under the mercury lamp the cliff essay’s 680 starts and the grid’s 594 disagree about whether one direction is lost by more than five per cent. The counts here are upper bounds that a harder search could lower again, by less.

The directions that were not lost. A direction the census reached within one per cent is not searched again, so the shortfall averages here include the census’s small shortfalls there unchanged.

The slope limit’s own grid. Exact on its grid of reflectance levels is not exact; a finer grid would let it reach slightly further and close a little of the gap that opened here.

Still open: whether a finer slope grid closes the gap #

The slope limit is solved exactly over a fixed set of reflectance levels, and its reach in each direction is the best a staircase on those levels can do. A finer set would let it follow the ideal’s edges more closely, and its counts could fall too — the same correction from the other side.

The calculation is the slope-limit census under the nine lamps with its reflectance levels doubled and doubled again, its lost directions counted as here and set beside the band models searched from 594 starts. The prediction is that the slope limit’s counts fall by under a twentieth, because its limit is on the slope and not on the levels, so its lead over the band models stays above a third. If its counts fall by a sixth as well, the census’s comparison was fair after all and the gap here is the grid’s.

A search can only fall short #

The habit here is about comparing an exact answer with a searched one.

A census that solves one model exactly and searches another has built an asymmetry into every comparison, and it runs one way: the search can miss the best answer, the exact solution cannot. Every ranking the census reports will lean towards the exact model, and the lean is invisible as long as the search is the only search anyone runs.

The failure mode is to read a searched optimum as an optimum. The test is to search the same thing again, harder, in the places that matter, and see which numbers move. Here a sixth of every band-model count moved, the rankings stayed, and every comparison with the exact model shifted the same way — which is what an asymmetry looks like once it is measured.