A band is only as wide as its steep side

A clear mercury lamp gives a surface four visible lines to reflect and nothing between them, and the hardest colours it can show are the ones that ask a pigment to reflect its green line at 546.5 nanometres and absorb its yellow at 578, or the reverse. A pair of lines asks for a cliff, not a gap found that those twenty-five directions are what decide the lamp’s whole account, and a band pigment must be narrower than the gap it splits found a pigment of symmetric absorption bands forty nanometres wide losing most of them: a band deep enough to absorb one line leaves its tail across the other.

Two essays then made the band lopsided. A steep side splits mercury’s lines at forty nanometres gave each band one side a third of its width and the other two thirds, and the losses fell from sixteen to one. A band must be nearly two to one before its steep side pays asked how little skew would do and found the gain coming late: sixteen, thirteen, ten, five and one directions lost at skews of 0.5, 0.55, 0.6, 0.65 and two to one, the count tracking a single band’s error against the census’s one-per-cent line.

A dye chemist has two handles on a band, its width and its asymmetry, and the obvious next question was whether they trade. The prediction was that a band five nanometres narrower needs a skew about a twentieth milder: a single thirty-five-nanometre band would cross one per cent of error near a skew of 0.57, and the census’s count would fall to five or fewer by 0.6, where the forty-nanometre count needed 0.65. If instead the narrower count still needed 0.65, width and skew would not trade, and the steep side would have to be steep whatever the width.

Fourteen nanometres, whatever the width #

Searched on the one-nanometre grid with bands thirty-five nanometres wide, the band pigment loses ten of the 25 splitting directions symmetric, nine at a skew of 0.55, eight at 0.575, four at 0.6, one at 0.625 and none at 0.65 or at two to one. A single band of that width crosses one per cent of error at a skew of 0.555. And a thirty-five-nanometre pigment at 0.6, whose steep side is fourteen nanometres wide at half depth, loses exactly the directions a forty-nanometre pigment at 0.65 loses — whose steep side is also fourteen — each by the same shortfall to within a hundredth of a percentage point. Placed by the steep side’s half-width alone, fourteen pigments at thirty, thirty-five and forty nanometres fall on one staircase.

• The prediction’s count holds: four lost at 0.6, under its five.
• Its crossing does not: one band crosses one per cent at 0.555, earlier than 0.57.
• The trade is not a trade. A narrower band does not substitute for skew at some exchange rate; the two only ever mattered through the half-width of the side that faces the line to be spared.

Drawn against skew, the three series are three separate curves, each falling from its symmetric count to nothing; drawn against the steep side, they lie on top of one another. The thirty-nanometre symmetric band, whose sides are each fifteen nanometres, loses eight directions, and so does the thirty-five-nanometre band at 0.575, whose steep side is 14.9. The thirty-five-nanometre band at 0.55, with a steep side of 15.75, loses nine, and the forty-nanometre band at 0.6, at sixteen, loses nine. At fourteen nanometres both series lose four, at thirteen and a fraction both lose one, at twelve and a quarter and below nothing is lost. The only pair that does not sit together is the symmetric thirty-five, at seventeen and a half, against the forty at 0.55, at eighteen: three directions apart, every one of them lost by the wider steep side.

The staircase is steep where it matters. Between sixteen nanometres and fourteen it drops from nine to four, and below thirteen it is empty. A pigment designer reading it needs one number about a proposed band — how wide its steeper side is — and not two.

Where the narrower band reaches five #

Read against skew, as the prediction was stated, the thirty-five-nanometre series does what was asked of it. It reaches five or fewer at 0.6, a twentieth of skew before the forty-nanometre series reaches the same count at 0.65, and it reaches nothing at 0.65, where the wider band still loses four. Its first steps are slow, as the wider band’s were: one direction recovered from symmetric to 0.55, one more at 0.575, then four at once between 0.575 and 0.6 and three more by 0.625.

The prediction’s arithmetic was right about the exchange rate and right for a reason it did not name. A twentieth of skew at forty nanometres moves the steep side by two nanometres; five nanometres of width at a skew of 0.6 moves it by two as well. A band thirty-five nanometres wide with its wide side 0.6 of the width has a steep side of 0.4 × 35 = 14; a band forty wide at 0.65 has 0.35 × 40 = 14. The two series were always going to cross the same count at those two skews, because they are the same steep side.

That is also why the exchange rate is not constant. Along one width’s series the steep side is (1 − r) × w, so a twentieth of skew is worth w/20 nanometres of steep side — two at forty, one and three quarters at thirty-five. Narrowing a band by a nanometre buys 1 − r of a nanometre of steep side, less as the band grows more lopsided. There is no fixed number of nanometres of width per twentieth of skew; there is one quantity both are spending.

The single band says the same #

The forty-nanometre essay set the census beside a single band, swept through every placement in fiftieths of a nanometre, and found the census’s count following that band’s error. Drawn against skew, the single band’s error is three curves, one for each width. Drawn against the steep side, it is one curve, to a part in a million: a band thirty-five wide at 0.6 and a band forty wide at 0.65 err by 0.522 per cent each, and a symmetric thirty-nanometre band errs exactly as a thirty-five-nanometre band does once its steep side is fifteen.

The reason is in what the band has to do. Twelve deep at its centre, it absorbs the line it sits on to nothing whatever its shape; the only way it can fail is by reaching the other line with its flank. That flank is its steep side, turned towards the line it spares, and the wide side runs off the other way — beyond 578 nanometres when the band absorbs the yellow, below 546.5 when it absorbs the green — into wavelengths where the clear mercury lamp has no line at all. Nothing the wide side does can be seen. The best placement slides the band until its steep flank just clears the passed line while its centre still covers the absorbed one, and that balance involves the steep side alone.

So the crossing of one per cent is a property of the steep side too. It falls at a steep half-width of 15.6 nanometres: a skew of 0.555 at thirty-five, 0.611 at forty, and at thirty already a symmetric band. The prediction put the thirty-five-nanometre crossing at 0.57, a steep side of fifteen; the true crossing is half a nanometre wider.

The one-per-cent rule, broken and replaced #

The forty-nanometre essay read its count off the single band’s error with a clean division: ten or more directions lost while that error was above one per cent, five or fewer once it was below. At thirty-five nanometres the rule fails at once. At 0.575 one band errs by 0.77 per cent, comfortably under the line, and the pigment still loses eight directions.

The rule was never about one per cent. On the forty-nanometre series no searched skew had its single band between 0.52 per cent and 1.16, so a division drawn anywhere in that gap would have separated the counts as cleanly. The thirty-five-nanometre series puts two pigments inside it — 0.575 at 0.77 per cent and 0.6 at 0.52 — and they fall on opposite sides: eight lost and four. With every width’s pigments placed by steep side, the count drops through five between a single-band error of 0.77 per cent and 0.52, a steep side of fourteen and seven eighths and fourteen, and that is the knee.

Why there and not at the census’s tolerance is the question the earlier essay half answered. A direction is lost when the pigment falls short of the ideal by more than one per cent, but the pigment’s shortfall in a direction is not the single band’s error: a direction that asks for only part of the green line, or a little of the yellow, can tolerate a band whose flank costs it more than the census’s line, while a direction that asks for the full cut feels every fraction. The directions’ own demands spread the single band’s error over a range of shortfalls, and the knee sits where the error has passed most of them.

The same failures, not just the same count #

A staircase in the count could still hide two pigments failing differently. They do not. The exactly matched pair, fourteen nanometres each, loses directions 54, 61, 120 and 131 — the same four — with shortfalls of 1.31, 1.34, 1.26 and 1.64 per cent at one width and 1.30, 1.34, 1.26 and 1.64 at the other. The pair a quarter of a nanometre apart, 15.75 and sixteen, loses the same nine directions, the wider steep side short by between a tenth and three tenths of a point more in each — the direction the staircase says it should.

The searches were independent. Each started from the census’s structured starts at its own width and skew, and each ran its own two thousand steps of Nelder–Mead; nothing passed from one width to the other. Two independent searches of differently shaped pigments landing on the same shortfalls in the same directions is the strongest evidence the census can give that a shortfall is the shape’s and not the search’s — and, by the same reasoning, a shortfall that differs between twins would be the first place to look for a search that went wrong.

One direction the search missed #

That test found one. The first census of the thirty-five-nanometre series lost a direction at 0.65 that it had reached at 0.6 and 0.625: direction 30, short by 1.81 per cent. A stronger skew can in principle lose what a milder one reached, since the shapes are not nested, but a band’s centre, width, depth and steep side mean the same thing at every skew, so the milder skews’ best answers are fair starting points. Searched again from them, direction 30 at 0.65 is reached within 0.18 per cent. The loss was the search’s.

The same direction carries the same signature on the forty-nanometre series — reached at 0.55, lost at 0.6 and 0.65 — and searched again it is reached within 0.52 and 0.33 per cent. Every other loss at the stronger mild skews of both widths was searched again the same way, and none moved by more than four tenths of a point or crossed the line; the losses at 0.55 have no milder mild skew to start from and stand as searched. The forty-nanometre counts at 0.6 and 0.65 are nine and four, not the ten and five the earlier essay reported, and that essay now carries the correction. Its conclusion stands — the gain comes late, and three to two does not halve the losses — but its knee moves by one direction at each of those two skews, which is exactly the amount that put its forty-nanometre counts a direction above the thirty-five-nanometre twins.

A harder search pulls the band pigments away from the slope limit had already shown that searches from many starts could find what the structured starts missed. What the twins add is a cheap way to know where to look: wherever two pigments that should be equivalent disagree, or a stronger shape loses what a milder one held.

The flank the lamp can see #

Drawn with their steep sides placed together, the two twins are one curve from the green line to the band’s centre and part only beyond it. At 546.5 nanometres both reflect the same fraction; at the yellow line both absorb the same. Past 590 nanometres the forty-nanometre band absorbs further into the red, and the thirty-five recovers sooner — but the clear mercury lamp emits nothing between its yellow line and the end of the visible, so the difference is invisible to every direction the census asks about. A green-passing pigment’s wide side lies in that dark stretch; a yellow-passing pigment’s wide side reaches back from the green towards 436 nanometres, a hundred and ten nanometres away, and fades long before.

This is the line lamp’s peculiarity, and it is what makes the result exact rather than approximate. The gap has to be dark, not the line narrow found that what spares a slow pigment under a line lamp is darkness between the lines, not the lines’ own narrowness; the steep side is the same principle seen from the pigment’s end — the part of a band that falls in the dark costs nothing. Under a lamp with light on both sides of the cut — a fluorescent tube’s phosphor bands, or a continuum — the wide side would run into light it should reflect or absorb, and the full width would matter again. The lamp outweighs the pigment model found the lamp mattering more than which pigment model is used; here the lamp decides which half of the pigment model matters at all.

What a dye chemist should take from this #

Under a line lamp, the half-width of a band’s steeper side is the whole of its shape. Two dyes whose bands differ in width and asymmetry but share a steep side reach the same colours under clear mercury. A chemist trying to reach the green–yellow splits can narrow the band or skew it or both, and should judge the result by one number: fourteen nanometres at half depth leaves four of the twenty-five directions short; thirteen leaves one; twelve and a quarter leaves none.

Depth is part of the steep side. Every band here is twelve deep at its centre, and a sharp edge is bought with depth found that a deeper band draws a reflectance edge steeper than its own width. A shallower dye band would have to be narrower on its steep side to do what these do, and the staircase’s numbers belong to this depth.

Check the single band, not the count. A single band’s best split of the two lines, swept in fractions of a second, depends on the steep side exactly as the census does, and places a proposed shape on the staircase without searching anything.

The full width still matters elsewhere. Absorbing the blue line at 436 nanometres, or the broad bands of a tube, the wide side is in play again, and nothing here says a narrow steep side on a wide band is as good as a narrow band overall.

How the pigments were searched #

The pigment is the band census’s — a reflectance e^(−a), a the sum of a floor and three absorption bands no deeper than twelve, searched by Nelder–Mead from structured starts for two thousand steps — with each band thirty-five nanometres wide at half depth and split at its centre: on one side a Gaussian whose half-width at half height is the stated share of the full width, on the other the rest, a fourth number per band choosing which side is steep. The shares are 0.55, 0.575, 0.6, 0.625 and 0.65, beside the symmetric and two-to-one bands of the earlier census at thirty and thirty-five nanometres and the forty-nanometre series. The lamp is clear mercury, its lines one and a half nanometres wide on a one-nanometre grid, and a direction is lost when the pigment’s best reach falls more than one per cent short of the ideal’s. Every direction lost at a mild skew was searched again for four thousand steps from its own best fit and from the best fits at every milder mild skew of the same width, and the better reach kept. The single band is swept through every offset in fiftieths of a nanometre and its larger error on the two lines kept; the steep side is the full width times one less the wide side’s share.

What this does not reach #

Gaussian flanks. Every steep side here is a Gaussian, so its half-width fixes its whole profile. A band whose steep side has a heavier foot or a sharper one at the same half-width might fall elsewhere on the staircase, and the census cannot say whether half-width or something further down the flank is what it measures.

One lamp. The wide side drops out because the clear mercury lamp is dark on its far side. Under any lamp with light there, the full width returns.

Structured starts. The re-search found one missed direction in the twenty-nine losses it re-searched; it searched only from answers the census already had, and a wider search might find more.

Still open: whether the half-width or the foot is what counts #

A Gaussian flank is fixed by its half-width, so the staircase cannot tell whether the census responds to the flank’s half-width or to how much absorbance is left in its foot at the passed line, more than three half-widths from the band’s centre. Real dye bands are often closer to Lorentzian in their wings than Gaussian, with far heavier feet at the same half-width.

The calculation is the band census at a steep side of fourteen nanometres with the steep side drawn as a Lorentzian and as a sharper-footed Gaussian of power four, the wide side unchanged, searched and re-searched as here, beside the single band’s error for each profile. The prediction is that the count follows the single band’s error and not the half-width: a Lorentzian steep side, whose foot at the passed line carries far more absorbance, loses most of the twenty-five directions at fourteen nanometres, and a sharper foot loses none. If instead the three profiles lose about the same at the same half-width, the census responds to the flank’s middle, and the staircase is a property of half-width that any dye’s band will obey. At one half-width a flank can lose nothing or nearly everything makes that calculation.

Two numbers that were one #

The habit here is about the parameters a model is written in.

The census’s bands were written with a width and a skew because that is how a chemist would describe a lopsided band, and the essays followed the parameters: one varied the width, one the skew, and the prediction asked how they trade. But the lamp only ever saw one combination of them. A model’s parameters are a coordinate system chosen for writing it down, and the physics may run along a direction the coordinates do not name.

The failure mode is to map a response in the coordinates a model was written in and report an exchange rate between them, when the response depends on one combination alone. The exchange rate exists and is real and is a statement about the coordinates, not about the pigment. The test is to find the pair of different settings that ought to be equivalent and see whether they are — here, two widths whose steep sides match — and, when they agree to a hundredth of a point, to rewrite the map in the quantity they share.