The glow's pull on the white sets every ink

A brightened sheet glows violet-blue under daylight, and a light tint printed on it turns in hue because the glow escapes through its open paper and not through its ink. A light ink can cancel the glow at one strength found the remedy — dilute the ink until the turn cancels — and a run of essays since has asked what sets that cancelling strength. The glow alone sets the light ink answered for every sheet glowing with one brightener: a single curve in the glow’s share of the sheet’s luminance, to 0.0007.

The glow’s colour sets the blue ink and spares the orange moved the brightener’s emission from 435 nm to 420 and 450 with the same photons absorbed and emitted. At 450 nm the blue ink’s strength rose from 0.81 to 0.98 because a glow near the blue’s own hue barely turns a blue tint, the violet’s fell by 0.06, the orange’s moved by a thousandth though the glow’s share of luminance nearly doubled; at 420 nm every ink moved, the orange by 0.04. Read at a moved sheet’s own share, the curve in glow missed by 0.09 to 0.21.

So the share of luminance is the wrong number once a glow’s colour can change, and the obvious repair adds the colour. The prediction was that a line in the glow’s share and its chromaticity coordinate y, fitted for each ink across the census sheets, the premium stock’s eleven loadings and the moved sheets, reads every ink within 0.01 on every sheet left out, the blue following y and the orange following y only below 435 nm. If any ink missed by more than 0.02, its turn would depend on the glow’s spectrum beyond its colour, and a sheet’s ink would need its glow measured as a spectrum.

Eight peaks, two loadings, thirty sheets #

Three colours of glow are too few to fit a line in colour and test it on a sheet left out, so the census was widened first. The premium stock’s brightener was given emission peaks every five nanometres from 420 to 455, each at the two loadings that give the 435 nm brightener the office and light sheets’ glow — fourteen new sheets, each absorbing and emitting the same photons as its 435 nm partner.

Across the eight peaks at the office loading the orange needs 0.596, 0.579, 0.566, 0.558, 0.554, 0.555, 0.559 and 0.569 — least at 440 nm, more on either side. The violet falls steadily from 0.703 to 0.552. The blue rises from 0.777 to 0.984 at 450 nm, and at 455 no light blue ink cancels its turn at all. The yellow, like the orange, bottoms out, at 435 nm; the cyan rises gently after a shallow dip.

None of those shapes is monotone in the way a single variable of the glow would make it. The orange’s U says its strength depends on something about the glow that rises and then falls as the peak moves; the blue’s runaway says something crosses a threshold near 450 nm. Both are shapes the share of luminance, which rises steadily with the peak, cannot give.

The share and its colour miss #

Over the thirty sheets, each left out, a line in the glow’s share, its square and its chromaticity y misses the orange by up to 0.073, the violet by 0.062, the yellow by 0.050 and the red by 0.040; the share alone, read quadratically, misses the blue by 0.249, the yellow by 0.149 and the orange by 0.121. The prediction fails by a factor of seven, beyond the alternative’s own threshold of 0.02.

The chromaticity was a reasonable choice and the wrong one, for a reason the reader’s adaptation makes plain. Chromaticity describes the glow’s colour as light on its own. What turns a tint is not the glow’s colour as light but what the glow does to the sheet as seen by someone adapted to it — and adaptation to a brightened sheet, as adapting to a brightened sheet leaves the glow found, takes the sheet’s white as its reference. The glow is part of that white. A reader adapted to it sees the paper’s own reflection, without the glow, as displaced from white by exactly what the glow added, in a direction and by an amount set by the adaptation.

The glow’s pull on the white #

That displacement can be computed for every sheet. Take the sheet’s reflected light alone, without its glow, adapt it by Bradford to the glowing sheet’s white, and express it in Oklab: its a and b, negated, are how far the glow moves the paper to the white the reader sees. Call it the glow’s pull.

The eleven loadings of the 435 nm brightener lie along one track through the census sheets, the pull growing in chroma with the loading at a hue that drifts by under one and a half degrees. Moving the emission swings the pull round: at the office loading its hue goes from −64° at 420 nm to −74° at 455, and its chroma rises from 0.023 to 0.030 at 445 nm and falls again. The two loadings draw two such arcs, the lighter one at about three fifths of the office’s chroma.

That is the picture the ink strengths needed. Loading moves the pull in one direction only, outward along a fixed hue, which is why every census sheet’s ink lay on one curve: one number, the pull’s chroma, is a monotone function of the share when the hue is fixed. Colour moves the pull along an arc, with a chroma that peaks, and that is a two-number change the share cannot represent.

The pull reads every ink #

A line in the pull’s a and b alone reads every ink within 0.046; with their squares and product, six terms, within 0.014 — the orange within 0.012, the red and violet within 0.014, the yellow within 0.006, the blue within 0.003 and the cyan within 0.002. Across thirty sheets that differ in loading, in brightener colour and in base paper, every sheet left out, the pull’s two numbers set the ink to about a hundredth.

The misses that remain are not concentrated on the moved sheets. The census sheets, the loadings and the peaks scatter alike about nought, and the largest misses fall at the ends of the grid, where a quadratic in two variables is extrapolating: the violet’s on the 455 nm sheets, the red’s and the orange’s on the lightest loading, a fifth of the premium’s brightener. So the residue is the line’s form, not a third variable it is missing: the prediction’s alternative, that the ink would need the glow’s whole spectrum, does not arise at this precision.

The pull reads the aged sheets it never saw #

The thirty sheets were chosen to vary the glow’s amount and colour. The run of essays has also aged sheets in two other ways, and those sheets were held out of the fit entirely. The brightener is being used up found daylight bleaching a sheet’s brightener; an ink for a fading sheet is set for its middle age bleached the premium stock by a quarter, a half and three quarters; a yellowing base leaves the ink where the glow put it yellowed its base in step. A yellowed base changes the paper’s own reflection, and so changes the pull, by a route none of the thirty sheets took.

Fitted on the thirty sheets and read on the six aged ones, the pull line reads every ink within 0.0031 — the orange within 0.0031, the red within 0.0026, the violet within 0.0018 and the blue within 0.0005. That is four times closer than its worst miss on the sheets it was fitted on, which is what a reading in the right variable does on sheets that lie inside its range: the aged sheets’ pulls fall among the loadings’, and their inks fall where the pull puts them, whether the pull moved because the brightener faded or because the base yellowed.

It also explains the earlier finding that a yellowed base moved no ink. Yellowing the base lowers the paper’s short-wave reflection, which the adapted reader discounts along with the rest of the paper’s colour; what is left of the glow’s pull barely changes, and the ink with it. At each bleach, yellowing the base moves the pull by under 0.0003 in a and b, about a hundredth of the pull itself. One light ink serves three papers within two degrees found three census papers needing different inks; in the pull’s terms they needed different inks because their glows pulled by different amounts, and the pull line reads all three within its hundredth. The census’s word for this was that the ink stays where the glow put it. In the pull’s terms, yellowing changes the paper and leaves the pull, and the pull is what the ink answers to.

The warm inks need least where the pull is strongest #

The pull’s chroma peaks at 445 nm on both loadings, at 0.0299 and 0.0182. The orange needs least strength at 440 nm on both, the yellow at 435. The warm inks’ U is the pull’s hump turned over, shifted five to ten nanometres short.

The reading is direct. A stronger pull is a larger glow, as the adapted reader sees it, and on every census sheet a larger glow needed a weaker ink — the cancelling strength falls as the glow grows. Moving the emission toward 445 nm strengthens the pull without adding a photon, so the warm inks need less strength, and past 445 the pull weakens again and their strength recovers. The few nanometres between the two minima are the pull’s hue at work: as the pull swings toward −70° it moves a little across the warm inks’ hue directions, and the strength responds to that as well as to the chroma, which is what the quadratic’s cross-term carries.

Past 450 nm the blue tint turns the other way #

A full-strength blue tint at ten-per-cent coverage turns +1.22° with the glow at 420 nm, +1.08° at 435, +0.51° at 445, +0.08° at 450 and −0.43° at 455, at the office loading; at the light loading the turn crosses nought between 450 and 455 as well. A light ink cancels a turn by making it smaller, and it cannot make a turn change sign; once the full-strength turn is reversed, the diluted ink’s turn is reversed too, and there is no strength at which it is nought.

The blue ink’s hue in the adapted Oklab lies near −91°, and the pull’s hue swings toward it as the emission lengthens. One reading — consistent with the numbers here, not derived from them — is that a tint’s turn follows the component of the glow’s pull across the ink’s hue direction, and that this component changes sign as the pull’s hue swings past a certain angle relative to the ink’s. For the blue that happens between 450 and 455 nm. A blue ink on a sheet whose brightener emits at 455 needs not a lighter ink but an ink that turns the other way.

What a print engineer should take from this #

Characterise a sheet’s glow by its pull on the adapted white, not by its fluorescence. Two numbers — the Oklab a and b by which the glow moves the paper’s reflection to white, for a reader adapted to the sheet — set every light ink’s strength within about a hundredth across loadings, brightener colours and papers. The glow’s share of luminance sets it only for sheets that share a brightener.

The pull’s chroma is the warm inks’ variable. The orange, yellow and red follow how strongly the glow pulls, and a brightener that pulls hardest per photon needs the weakest warm inks.

A blue ink has a brightener beyond which no light ink helps. When the glow’s pull swings past the blue ink’s own direction the tint’s turn reverses, and the cancelling strength ceases to exist. On this model that happens for a brightener emitting between 450 and 455 nm.

How the sheets and inks were computed #

The premium stock is the census’s paper base with a brightener absorbing in a Gaussian band at 358 nm and emitting in a 25 nm Gaussian band at its peak, quantum yield 0.85; the eleven loadings run from a fifth to twice the premium’s own, and the emission grid from 420 to 455 nm every five, each at the loading that gives the 435 nm brightener the office or the light sheet’s glow share. Each sheet is lit by D65 extended to 300 nm and read through the 1931 observer. The pull is the sheet’s reflected light alone, adapted by Bradford from the sheet’s white to D65, in Oklab, subtracted from the adapted white. A tint at ten-per-cent coverage is a Yule–Nielsen halftone with the glow through the open paper in full and through the ink in proportion to its ultraviolet transmission; the cancelling strength is the dilution at which a matched-lightness light tint’s turn is nought. Each line is fitted by least squares on twenty-nine sheets and read on the thirtieth.

What this does not reach #

One brightener shape. The emission moves as a fixed 25 nm Gaussian; a real brightener whose width or shape differs could pull differently at the same peak.

A quadratic in two variables. Six terms over thirty sheets read every ink within 0.014; whether the residue is the form or a third quantity is argued from where the misses fall, not tested with a third variable.

The sign reversal is the model’s. That the blue tint’s turn reverses near 452 nm depends on this blue pigment’s hue and this adaptation; another blue or another adaptation would put it elsewhere.

Still open: whether a reader adapted elsewhere moves the pull #

Every pull here is seen by a reader adapted to the glowing sheet itself, which is how a sheet is read in isolation. A sheet read beside an unbrightened proof, or under a lamp with little ultraviolet, is seen by a reader adapted to something else, and the glow’s pull on that reader’s white is a different pair of numbers.

The calculation is every sheet’s pull and every ink’s cancelling strength with the reader adapted to the unbrightened base paper’s white instead of the sheet’s own, the pull line refitted and read on each sheet left out. The prediction is that the pull still reads every ink within 0.015, its chroma larger by about a third because the reader no longer discounts the glow, and that the blue’s reversal moves to a shorter peak, near 445 nm, because a larger pull crosses the blue’s direction sooner. If the pull fitted under one adaptation fails by more than 0.03 under the other, the inks depend on the adaptation directly and not only through the pull, and a light ink must be set for the viewing condition as well as the sheet.

The number the census held still #

The habit here is about choosing the variable a law is written in.

The census’s law was exact in the glow’s share of luminance because, with one brightener, the share stood in for the glow’s pull. Adding the glow’s chromaticity was the natural repair and it failed by a factor of seven, because chromaticity describes the glow as light and the ink responds to the glow as seen by an adapted reader. Writing the glow in the reader’s own terms — the displacement of the paper from the white the reader takes as white — made two numbers enough.

The failure mode is to repair a law by adding the next variable that comes to hand, when the law’s variable was a stand-in for one the census never computed. The test is to ask what the quantity being predicted physically depends on, express the inputs in those terms, and see whether the law’s form survives the change of variable.