The ranking reads six nanometres to the direction

Under a clear mercury lamp the hardest colours to reach are the ones that ask a pigment to reflect the green line at 546 nm and absorb the yellow pair thirty-one nanometres away, or the reverse. A pigment built from absorption bands can only make that change through the steep side of one band — a band pigment must be narrower than the gap it splits found a forty-nanometre band losing most of these directions until something in it was sharp — and a hundredth of Lorentzian undoes a Gaussian flank found that a few per cent of Lorentzian shape in that steep side — the slowly falling wing real absorption bands have — eats the gap. At a twentieth of Lorentzian the forty-nanometre pigment loses eighteen of the twenty-five splitting directions with a steep side fourteen nanometres wide at half depth, thirteen at ten, twelve at seven and five at five.

Each direction falls short by one band’s ratio found what lies under those counts. A single band placed as well as it can be between the two lines has an error that falls by 0.62 from ten nanometres to seven and 0.60 from seven to five, and every one of the 25 directions’ shortfalls falls by the same factor within a tenth. Each direction’s shortfall is a constant times the single band’s error — 1.20 for the hardest, under 0.05 for the easiest — and the count at a half-width is how many constants exceed one per cent over the single band’s error there. The constants were read at seven nanometres, and they predicted the counts at ten and five exactly.

A rule that predicts two searched counts from a third is a fit until it predicts a count nobody has searched. The prediction was that the pigment loses nine directions at a six-nanometre steep side, three at four and a half and none at four, and that every direction’s shortfall at each width lies within a sixth of its constant times the single band’s error. If the count at six differed from nine by more than one, the constants would not be fixed between seven and five, and a count between searched widths could not be read off one band.

Six nanometres loses the nine the ranking named #

Searched at a six-nanometre steep side, from the census’s starts and from the best pigments found at five and seven, the pigment loses nine directions: 42, 49, 54, 61, 67, 112, 120, 125 and 131 — the nine whose constants the ranking put above the six-nanometre line.

The prediction holds in the strongest form it could: not the count alone but the identity of every direction counted. Nine of the twenty-five lines in the fan sit above the one-per-cent mark at six nanometres and they are the nine the constants ranked highest. Between seven and five the census had searched nothing; the ranking said three of the twelve directions lost at seven are recovered by six and four more by five, and that is what the search finds.

Every direction short by more than a tenth of a per cent falls within 6.2 per cent of its predicted shortfall; the hardest, direction 61, is predicted 1.661 per cent short and found 1.648. The points lie on the diagonal across the whole range, and the one-per-cent line divides them where the prediction divides them. The nearest call is direction 117, predicted 0.969 per cent short and found 0.971 — kept, as the ranking said, by three hundredths of a point.

Four nanometres loses none #

At a four-nanometre steep side no direction falls short by one per cent. The hardest, direction 61, falls 0.865 per cent short, against a prediction of 0.870. The ranking put every constant below the four-nanometre line, whose height is one per cent over a single band’s error of 0.727 per cent — a constant of 1.38, above even direction 61’s 1.20.

So the pigment with a twentieth of Lorentzian in its steep side reaches every splitting colour under mercury once the steep side is four nanometres wide at half depth, a tenth of the band’s forty. A five-nanometre flank carries a fiftieth of Lorentzian found the same pigment reaching all of them at five nanometres with a fiftieth of Lorentzian in it; at a twentieth it needs one nanometre more. The two together put a rough price on the wing: the three hundredths of Lorentzian between a fiftieth and a twentieth cost about a nanometre of steep side — rough, because how much narrower than five nanometres a fiftieth would allow has not been searched.

Four and a half loses one, not three #

The ranking’s prediction between five and four was the delicate one, and it misses by two.

At four and a half nanometres the ranking names directions 61, 131 and 54 as lost. The search loses only 61, at 1.002 per cent short. Direction 131 is predicted 1.024 per cent short and found 0.989; direction 54 is predicted 1.012 and found 0.995. Both of the directions the ranking added sit under the one-per-cent line by a hundredth of a point or less, and both their predictions sit over it by less than three hundredths.

The miss is not a failure of the proportionality in any useful sense. Direction 131’s shortfall is 3.4 per cent under its prediction and direction 54’s 1.7 per cent under, inside the agreement the six-nanometre directions showed; it is the threshold that converts those percentages into a count of two. At four and a half nanometres the single band’s error, 0.884 per cent, puts the line at a constant of 1.13, and three constants — 1.20, 1.16 and 1.15 — sit just above it. A count read at a width where three constants crowd the line will be off by however many of them a few per cent of scatter carries across, and nothing about the pigment changed to make it so.

The gentle directions drift above the ranking #

The other departure is real, and it grows as the flank narrows.

At six nanometres every direction lies within 6.2 per cent of the ranking. At four the five hardest still lie within 8 per cent, but direction 49 falls 14 per cent short of more than predicted and direction 30, 29 per cent. At four and a half the same two directions lie 11 and 20 per cent above. The large departures are all upward, and all among directions whose constants sit in the middle of the ranking, from 0.59 to 0.86.

An upward departure that grows as the single band’s error shrinks is what a second, fixed contribution to the shortfall would look like. The ranking treats each direction’s shortfall as the splitting band’s wing at the passed line, scaled; anything else the pigment fails at in that direction — the other two bands’ compromises among mercury’s other lines, the base absorbance — would add a floor that does not shrink with the wing, small beside the wing’s part at seven nanometres and a growing share of the total at four. Direction 30 is a natural candidate: it is the direction each direction falls short by one band’s ratio found falling most slowly between widths, by 0.67 and 0.69 a step against the single band’s 0.62 and 0.60, and its Gaussian-only shortfall at five nanometres, 0.15 per cent, is the largest of any direction’s.

The five hardest cross the line within a nanometre #

Direction 125 has crossed the one-per-cent line by five nanometres, at 0.998 per cent short; directions 120, 131 and 54 cross it between five and four and a half, the last two by a hundredth of a point; direction 61 between four and a half and four. The five lines run nearly parallel and nearly straight, each falling by about a third of a point a nanometre over this stretch, and the half-nanometre between five and four and a half carries three of them across. The fifth direction lost at five nanometres, 49, is one of the gentle ones the ranking underrates, and it too is kept by four and a half.

That is the shape behind the staircase. The last five risers of the count lie within one nanometre of half-width, because the five hardest directions’ constants lie within a sixth of one another; a pigment designer who wants every splitting colour under this lamp needs the last nanometre, and gets almost nothing from the one before it.

Over seven searched half-widths the counts lost are 18, 13, 12, 9, 5, 1 and 0, from fourteen nanometres down. Every line of the fan falls with the half-width; none rises; their order is the constants’ order with a handful of swaps among the gentle directions, the ones drifting above the ranking at the narrow end.

How far one band has carried the census #

The single band has been the census’s ruler for most of the run on this flank, and it is worth saying how much more it now carries. A band is only as wide as its steep side found two skewed bands of different widths losing the same directions because they shared a steep side, and read the losses off the point where a single band’s error crosses one per cent. At one half-width a flank can lose nothing or nearly everything found four flank shapes at one half-width losing in the order of their single bands’ errors, from none to twenty-two. A hundredth of Lorentzian undoes a Gaussian flank placed the Voigt pigments on the staircase the Gaussian pigments drew against single-band error, a step or two above it.

Each of those used the single band to order pigments or to place a count. The ranking uses it to place every direction: one search at one width fixes 25 constants, and the single band’s error — computed in a fraction of a second at any width — then predicts which directions are lost at any other width the proportionality holds, and by how much each falls short. Across six searched half-widths from ten nanometres to four, the one it was built at included, the ranking has missed one count, by two, at the one width where three constants crowd the line. That is a much stronger statement than ordering, and it is why its failure at the narrow end is informative: the gentle directions leaving it are the first evidence in this run of a part of the pigment’s shortfall that the splitting band does not set.

It also changes what the run’s remaining questions cost. A count at a new half-width under this lamp no longer needs a census; a count at a new share of Lorentzian might not either, if the constants carry across shares as they carry across widths — and that is the question the next calculation can answer from searches already made.

What a colour scientist should take from this #

A prediction of a count is tested by its identities. Six nanometres losing nine directions would have been a weaker confirmation than six nanometres losing the nine named; the second rules out a ranking that is right in total and wrong in every step.

A count’s error near a crowded threshold is not the model’s error. The ranking missed four and a half nanometres by two because three constants sat within a sixth of each other at the line, and a few per cent of scatter in the shortfall decided how many crossed. The same scatter at six nanometres, where no constant sat near the line, changed nothing.

A proportional model has a range, and its edge is where a fixed term stops being small. The ranking holds within six per cent while the single band’s error is above about one per cent, and the gentler directions leave it below that. The range is set by the size of whatever else the pigment fails at, not by the wing.

How the pigments were searched #

The pigment is up to three absorption bands no narrower than forty nanometres and a base absorbance, each band’s steep side a pseudo-Voigt a twentieth Lorentzian of a stated half-width at a forty-nanometre band, scaled with the band’s width, and its wide side a Gaussian. For each of the 25 directions splitting mercury’s green line from its yellow, the pigment’s reflectance under the lamp, weighted by the direction’s combination of the 1931 matching functions on a one-nanometre grid, is maximised by Nelder–Mead for 2,000 steps from each of the census’s starts and for 4,000 from the census’s best pigments at the neighbouring half-widths — five and seven nanometres for six, five for four and a half and four. A direction is lost when the best reach falls more than one per cent short of the ideal. A direction’s constant is its seven-nanometre shortfall over the single band’s error at seven; its prediction at another width is the constant times the single band’s error there.

What this does not reach #

One share, one lamp. The ranking is tested across half-widths at a twentieth of Lorentzian under mercury; whether a direction’s constant is the same at a hundredth or a fiftieth is the next question, not this one.

The floor is argued, not measured. That the gentler directions’ upward drift is a fixed contribution from the pigment’s other bands is the natural reading of its sign and growth; no direction’s shortfall is decomposed into its bands here.

A search can still stop short. Every count here is the better of two searches, and four and a half nanometres turned on hundredths of a point; a third search could move a direction across the line in either direction.

Still open: whether a direction’s constant belongs to the share #

The ranking was read at a twentieth of Lorentzian and holds across half-widths there. The census also holds the fourteen-nanometre pigment at a hundredth, a fiftieth, a tenth and a fifth, and the five-nanometre pigment at a hundredth and a fiftieth, with no new search needed.

The calculation is every direction’s shortfall at those shares beside its twentieth-of-Lorentzian constant times the single band’s error at that share and width. The prediction is that the constants carry across shares as they carry across widths — each direction within a sixth of its prediction wherever the single band’s error exceeds one per cent — so that the count at fourteen nanometres and a hundredth, eleven, is read off the ranking within one. If directions fall short by half as much again as the ranking says at the smaller shares, the constant is a property of the wing’s shape as well as of the direction, and a count at one share cannot be read from a ranking built at another.

A rule that was asked to name names #

The habit here is about what makes a prediction worth testing.

The ranking of constants was built at one width and checked at two others, and that could have been a description fitted to its own data. The test was a width nobody had searched, and the prediction named which directions would be lost, not only how many. It named them correctly at six nanometres, missed the count at four and a half for a reason the shortfalls themselves showed to be the threshold’s, and at four began to fail for the gentle directions in a way that says where its own assumption — that the wing is the whole of the shortfall — runs out.

The failure mode is to accept a model because it reproduces the counts it was built on, or to reject it because it misses one count by two. The test is to predict the quantities under the count, at a setting the model was not built at, and to read the misses by size and by which cases they fall on.