The glow alone sets the light ink

A pale tint printed on brightened paper keeps a hue turn that a reader fully adapted to the sheet cannot remove. The brightener absorbs ultraviolet and glows blue, the ink covers some of it and shields the brightener beneath from the ultraviolet, and the tint loses glow that the bare paper around it keeps. A light ink can cancel the glow at one strength found that printing each pigment diluted, over more of the sheet, cancels that turn at one strength, and one light ink serves three papers within two degrees found the strength falling across three census stocks — a lightly brightened sheet, office paper and a premium sheet — along what looked like straight lines in the sheet’s glow.

An ink for a fading sheet is set for its middle age bleached the premium sheet’s brightener and found its strengths walking back along those lines a shade under them, and a yellowing base leaves the ink where the glow put it tested whether the shade came from the base paper, held white while real ageing would have yellowed it. It did not: yellowing the base moved no strength by more than 0.0023. The shade came from the line. The office sheet itself sits 0.016 under the straight line through the three census sheets and the light and premium sheets 0.008 above it — the line is a chord across a sagging curve, and the bleached sheets sag beneath it as the office sheet does.

That leaves a sharper question than the one the yellowing answered. The three census stocks differ in more than their brighteners: their base papers have different levels and different short-wave dips, so their whites differ and the light passing through an ink film differs. If the cancelling strength still lies on one smooth curve in the glow, the glow is the whole of what a printer needs to know about a sheet. The prediction was that the premium stock, bleached until its glow equals the office sheet’s 0.57 per cent and the light sheet’s 0.33, needs that sheet’s ink within 0.003 for every pigment. If it differed by more than the yellowing had moved the strengths, the strength would depend on how the glow is made, and the glow alone would not set the ink.

Matched in glow, matched in ink #

Bleached by 61 per cent, the premium sheet glows as the office sheet does to a millionth of the share; bleached by 83 per cent, as the light sheet does. On the first every pigment’s cancelling strength lies within 0.0005 of the office sheet’s, on the second within 0.0007 of the light sheet’s. The orange needs 0.5581 against 0.5578 and 0.6461 against 0.6456; the yellow and the violet differ most.

• The prediction holds, with a quarter of the room it allowed. No pigment’s strength on a matched sheet differs from its census sheet’s by a thousandth.
• The strength is a function of the glow. Two stocks bleached or loaded to the same glow need the same ink to three decimal places.
• The alternative does not arise. How a sheet’s glow is made — a strong brightener mostly bleached, or a weak one fresh — does not enter the ink.

The figure draws each pigment’s strength along the premium stock as it would be loaded from a tenth of its own brightener to all of it, and places the census sheets and the matched sheets on the same axes. Every census point lies on its pigment’s premium curve. The curves are smooth and every one bends the same way: steep where the glow is small, flatter where it is large. The green’s barely moves at all — its strength runs from 0.438 to 0.433 across the whole range — because the green’s light ink changes the tint’s hue very little whatever the paper.

What the thousandth is made of #

At the office sheet’s glow the matched premium sheet’s strengths differ from the office sheet’s by +0.0003 for the orange, +0.0001 for the red, +0.0002 for the magenta, +0.0003 for the violet, −0.0002 for the blue, −0.0001 for the cyan, +0.0002 for the green and +0.0005 for the yellow; at the light sheet’s glow by +0.0005, +0.0002, +0.0003, +0.0007, −0.0001, nought, +0.0002 and +0.0006. The warm and violet pigments need a shade more strength on the premium stock, the blue and cyan a shade less, and the differences roughly double from the office’s glow to the light’s.

The pattern has a plausible source that this calculation does not isolate. The premium stock’s ink film lets through a little less of the brightener’s excitation than the office sheet’s, so a tint on it loses a little more glow beneath the ink, and that would ask the pigments whose residue the glow pushes one way for a little more dilution and those it pushes the other way for a little less. The light sheet’s match is the more heavily bleached, and lies on a steeper part of the curve, where any such difference would be read more strongly. Testing that would mean varying the shielding alone at fixed glow, which the census does not do, and at a thousandth of strength the question is of no consequence to a printer; it matters only as the size of the one thing, besides the glow, that the stocks were seen to differ in and the ink might have felt.

At that size the second-order effects are measurable and irrelevant. A thousandth of strength moves a pale tint’s residue by a few hundredths of a degree on the steepest pigment; the residues the office ink leaves on the matched sheets, 1.214° and 1.810°, differ from the census sheets’ 1.212° and 1.804° by a fraction of a hundredth. A growing clump prints like the screen with as many dots measured the dot geometry’s effect on the residue at a few tenths of a degree, a hundred times larger than anything the paper’s base does here.

Two papers with one glow #

The matched sheets are genuinely different papers. The premium stock’s base reflects 0.90 across the long wavelengths against the office sheet’s 0.88 and the light sheet’s 0.87, so its total radiance factor stands two to four hundredths higher at every wavelength; its white’s Z is 125.9 against the office’s 125.6 and 118.4 against the light’s 118.0; and an ink film on it shields a different share of the brightener’s excitation, 0.1343 of its light passing against the office sheet’s 0.1354. Their emissions peak together at 430 nm and their glow shares are equal; nothing else about them is.

That is what makes the agreement informative. If the strength depended on the base’s level, the matched sheets would differ by the share of the residue the base sets, and the premium base reflects two per cent more light than the office’s. If it depended on the ink’s shielding, they would differ by the share of a per cent of light passing. They differ by less than a thousandth in strength, and the cancelling strength turns out to depend on the glow’s share of the sheet’s luminance and almost nothing else.

The reason is in how a tint’s residue arises. The residue is a hue turn for a reader adapted to the sheet, and adaptation divides out the sheet’s own colour: a whiter base raises the tint and the paper around it together, and the reader’s adaptation removes both. What is left is the difference between the glow under the tint and the glow around it, as a share of what the reader is adapted to — which is the glow’s share of the sheet’s luminance.

A curve that sags under every chord #

Read against the census line, the premium curve sags under it between the light and premium sheets, by at most 0.016 for the orange, and rises above it fast beyond the light sheet: at the dimmest loading, two thirds of the light sheet’s glow, it stands 0.056 above the line. Every interior loading of every moving pigment’s curve lies under the chord through its two neighbours. The curve is convex throughout, for the orange, red, violet, blue, cyan and yellow alike.

The census line was fitted to three points, all of them inside the range where a printer would expect to work, and between them it errs by under two hundredths. Outside them it errs fast. A sheet less brightened than the light sheet — an uncoated stock with a trace of brightener, or a premium sheet nearly fully bleached — needs an ink five or more hundredths stronger than the line predicts, and the error grows towards the dim end, where the strength climbs most steeply.

The strength falls fastest on a dim sheet #

The orange’s strength falls by 67 per unit of glow share between the premium stock’s two dimmest loadings and by 13 between its two brightest, five times less steeply; the red’s from 38 to 4.3 and the violet’s from 55 to 5.9, nine times; the blue’s, cyan’s and yellow’s by two to three times. Every pigment’s curve flattens as the glow grows.

A yellowing base leaves the ink where the glow put it read the halving of the orange’s slope between the light and office sheets and the office and premium as the source of the chord’s sag; the premium curve shows the halving as part of a longer flattening that runs from the dimmest loading to the brightest. A sheet with little glow needs a large change in ink strength for a small change in glow, because each step of dilution exposes brightener that hardly glows; a sheet with much glow needs a small change, because each step exposes a lot of glow.

That has a practical edge for stocks the census does not hold. The steepest part of the curve is where the census has no sheet at all, below the light sheet’s glow. A printer extrapolating from the census sheets towards a lightly brightened stock would underestimate the strength by an amount that grows quickly, and would do better to measure the stock’s glow and read the curve.

One ink serves a stretch of glow #

Printed with the office sheet’s light ink, the premium stock holds every residue from five to twenty per cent under one and a half degrees from a glow of 0.44 to 0.79 per cent; on the census sheets the office ink leaves 1.80° on the light sheet, 1.21° on the office sheet and 1.80° on the premium. The office ink’s own sheet sits near the middle of the stretch, and the stretch is set by the glow alone: a premium sheet at the office sheet’s glow is served exactly as the office sheet is, at 1.21°.

That is the practical form of the result. One light ink serves three papers within two degrees found one ink serving every census stock within two degrees; it serves within one and a half any stock whose glow lies between about 0.44 and 0.79 per cent of its luminance — from a little below the office sheet’s to nearly the premium’s — and a printer choosing which stocks to run with one ink needs only their glows.

The rim was never most of the glow found nine tenths of a tint’s residue coming from bare paper glowing beside ink that does not, and this calculation is that finding read from the ink’s side. The glow beside the ink is the glow’s share of the sheet; the ink’s strength is set to balance it; and nothing about the paper that does not change the glow’s share changes the balance.

What a printer should take from this #

Measure a stock’s glow, not its brand. The light-ink strength that cancels a pale tint’s turn depends on the share of the sheet’s luminance that is glow, within a thousandth, whatever the base and whatever brightener produced it.

Read the strength from the curve, not from a line through stocks. Between three stocks a straight line errs by under two hundredths; below the dimmest it errs by five or more, and the error grows.

An office-paper light ink serves a band of glows, from about 0.44 to 0.79 per cent, within one and a half degrees on every pigment.

How the sheets were built and read #

Each sheet is the census’s model: a base paper whose reflectance is a level less a short-wave dip of stated depth, a brightener of stated loading that absorbs ultraviolet and emits blue, combined under a wide D65 into a reflected and an emitted spectrum. The matched sheets are the premium stock — base level 0.90, dip 0.13, loading 2.0 — with its loading reduced by sixty steps of bisection until its glow share equals the office or light sheet’s. The curve is the premium stock at loadings of 0.2, 0.35, 0.5, 0.65, 0.8, 1.0, 1.2, 1.4, 1.6, 1.8 and 2.0. A pale tint is a uniform film of a pigment diluted to a strength, its coverage spread by Yule–Nielsen with an exponent of three and the brightener under it shielded by the ink vehicle’s ultraviolet absorbance; its residue is its hue in Oklab against the full-strength ink’s, for a reader adapted to the sheet by a Bradford transform; the cancelling strength is found by bisection on the residue’s sign at a ten-per-cent target.

What this does not reach #

Other brighteners. Every sheet here uses the census’s one brightener, with one excitation and one emission spectrum; a brightener emitting at 445 nm instead of 430 changes the glow’s colour, and whether the strength depends on the glow’s colour as well as its share is not tested.

Coated stocks. A coated sheet holds ink on its surface and shields the brightener differently; the census’s coated stock is not in this curve.

The film. Every tint is a uniform film; the dot edges that let glow back in are measured separately and treated as small.

Still open: whether the glow’s colour enters the ink #

Every sheet here glows with the same brightener, so its glow has one spectrum and differs between sheets only in amount. A different brightener, or a brightener partly quenched by photo-yellowing products, emits a glow of a different colour, and the cancelling strength balances glow of one hue against ink of another.

The calculation is the premium stock with its brightener’s emission moved fifteen nanometres longer and fifteen shorter, each at the loadings that give it the office and light sheets’ glow shares, every pigment’s cancelling strength read against the curve here. The prediction is that a longer-wave glow, nearer the ink’s own hues for the blue and violet pigments, moves their strengths by more than a hundredth and the orange’s and red’s by under a fifth of that, so that the curve in glow share holds for the warm pigments and splits for the cool ones. If no pigment moves by more than 0.003, the glow’s share sets the ink whatever its colour, and a single measurement of a sheet’s fluorescence suffices.

The glow’s colour sets the blue ink and spares the orange made that calculation: a 450 nm glow takes the blue ink to full strength and the violet down, moves the orange by a thousandth, and a 420 nm glow moves every ink.

A variable that was enough #

The habit here is about recognising when one number carries the effect.

Three census stocks differed in base, brightener and whiteness, and the cancelling strength had been fitted against one of their properties, the glow, by a line that left residuals. The residuals could have meant that the other properties mattered. Building a stock that matched another in glow and in nothing else answered it: the inks agreed to a thousandth, and the residuals were the line’s shape, not the papers’ differences.

The failure mode is to read a fitted relation’s residuals as the influence of the variables left out, before testing whether the relation’s form is wrong. The test is to construct a case that matches in the fitted variable and differs in the others, and read whether the outcome follows the fitted variable alone.