A surface that is not a multiplication

Almost everything on this site is one multiplication. Light arrives with a spectral power distribution, a surface has a reflectance, and what leaves is the product of the two, band by band. A bounce off a wall is that product. An ink on a sheet is that product twice. A change of illumination is a matrix because the product is linear in the surface’s three coefficients, and adaptation is a diagonal because the product is diagonal.

There is a class of surface for which none of that begins, and most white office paper is in it.

Which pair of lamps is chosen decides how far apart the two curves are and not whether there are two of them.

A reference that is not a real light at all is the clean case, since whatever the sheet does under it is a property of the sheet rather than of somebody’s lamp.

Two more pairs, one of which includes a reference nobody has built, say the same thing without a real lamp anywhere in the answer.

The claim #

A fluorescent surface is a full operator on the spectrum, so it does not have a reflectance and none of this site’s arithmetic applies to it.

• Light in at one wavelength comes out at another. A brightener absorbs between 380 and 420 nm and emits around 438, so the operator has entries off its diagonal and the product E · ρ is not a description of anything.
• The same sheet returns different apparent reflectances under different lamps — the worst band differs by 0.186 between D65 and tungsten — so the quantity a spectrophotometer prints is a property of the pair, not of the sample.
• The factor goes above one, to 1.06 under equal-energy light, which no diagonal operator on a spectrum can do.
• Every argument above fails to start, not by a little: there is no matrix T, no basis, no residual, and adaptation has nothing to work on because the change is not a change of context at all.
• And the industry’s response has been to standardise the lamp rather than the sample, which is the only available move and is not the same as measuring the sheet.

What a reflectance is supposed to be #

The word reflectance carries an assumption strong enough that it is almost never stated: that the number is a property of the material. Two per cent at 450 nm means two per cent at 450 nm, whatever else is going on, and that is what makes it possible to measure a sample once and compute what it will look like under any light.

The assumption is exactly the statement that the surface acts as a diagonal operator on the spectrum. Each band in, the same band out, scaled by a number that belongs to the sample.

A fluorescent sample violates the assumption at its root, and not marginally. It moves energy from one part of the spectrum to another, so the operator relating the incident spectrum to the returned one has entries connecting different wavelengths — the Donaldson matrix, if it is written out in full.

The measurement that shows it #

The test is simple enough to state as a syllogism. If a sample has a reflectance, the ratio of what it returns to what arrives is the same function under any lamp. Compute that ratio under two lamps. If the two disagree, the sample has no reflectance.

Under D65 and under tungsten, this site’s brightened sheet disagrees with itself by 0.186 in the worst band — an enormous number for a quantity that is supposed to be a material constant. Under equal-energy light the ratio peaks at 1.06, which is a surface returning more light in a band than reached it in that band.

That last figure is the practical form. Measure the sheet under a lamp with little ultraviolet, compute what it will look like under daylight, and the prediction is out by more than any tolerance in the trade — because the computation assumed a multiplication and the sheet is doing something else.

Where each of these arguments stops #

It is worth walking through them, because each fails at a different point and for the same reason.

The matrix. The census’s central result is that a change of light is a 3×3 on tristimulus values, and it follows from the response being linear in the surface’s three coefficients with the light as a diagonal factor. With a fluorescent surface the light is not a factor at all — it appears inside the operator — so there is nothing to factor out and no matrix to write.

The basis. The eigenvector argument diagonalises a change of light. There is no change matrix to diagonalise.

Adaptation. A gain is applied to the whole field and removes what is common to it. Fluorescence is not common to the field: it is a property of some objects in the scene and not others, so even in principle it is outside what a gain addresses.

The substrate rule. Media-relative colorimetry divides by the sheet’s tristimulus values, and the sheet’s tristimulus values depend on the lamp in a way that is not shared by the inks printed on it. Dividing by a number that moves for a different reason than the numerator does is not a normalisation.

What it looks like, and why it looks like that #

A brightened sheet looks whiter than white, and the phrase is not marketing. It is a description of where the sheet lands.

Whiteness in the colorimetric sense has a ceiling: a surface cannot return more light than reaches it, so a perfect diffuser is the whitest object there can be, and every real paper sits below it. A brightened sheet is above it in the blue. It is returning more at 440 nm than arrived at 440 nm, having taken the difference from the ultraviolet, and the result is a sheet whose chromaticity sits on the blue side of the illuminant with a luminance factor near one.

That is exactly the direction the eye reads as clean. A slightly yellow paper reads as aged; a slightly blue one reads as new, and the industry has been exploiting the asymmetry for eighty years. The essay that measured it put the peak apparent reflectance above 1.05.

The consequence for this essay is that the effect is not a subtlety at the edge of measurement. It is the largest single reason a sheet of paper looks the way it does, it is present in most of the white paper anybody handles, and the arithmetic that the whole colour-management stack is built on does not describe it.

An instrument that agrees with itself #

There is one more consequence and it is the one that reaches a laboratory first.

Two spectrophotometers of the same class, calibrated against the same standard, measuring the same ordinary sample, agree to a small fraction of a unit. Measuring a brightened sheet they can disagree by several units, and the reason is not that either is out of calibration: it is that their lamps have different ultraviolet content, so they are measuring two different quantities and both are measuring them correctly.

That is a failure mode with no equivalent anywhere else in inter-instrument agreement. A bandpass difference, a sampling difference, a geometry difference — each of those is a difference in how the instrument looks at a fixed sample. This is a difference in what the sample is, produced by the instrument, and no amount of care with the optics removes it.

What the industry does instead #

Since the sample cannot be characterised on its own, the response has been to characterise the pair. A measurement of a brightened sheet is quoted with the lamp it was made under, and the standards define lamps whose ultraviolet content is specified: D50 simulators for graphic arts are graded on how well they match in the ultraviolet as well as the visible, and instruments are sold with switchable ultraviolet filters so that a sample can be measured with the excitation present and absent.

That is the only available move and it should be recognised for what it is. It does not measure the sheet. It measures the sheet under a stated lamp, and it converts a material property that does not exist into a procedure that can at least be repeated — which is what a standard illuminant is for everywhere else too.

The residual problem is that the reader’s lamp is not the standard’s. A brightened sheet under a window, under an office tube with a phosphor that emits nothing below 400 nm, and under a warm domestic lamp is three different papers, and no measurement made in a booth predicts all three.

The band the site cannot see #

There is a limit here that belongs to this site rather than to the subject, and it should be stated where it bites hardest.

Every spectrum here runs from 380 to 780 nm at 5 nm intervals, and a brightener’s excitation band runs from about 340 to 420. So more than half of what excites the sheet is outside the grid, and every fluorescence number on this site is a declared floor rather than an estimate.

Widening the range is not a small change. It touches the observer functions, every illuminant, the metamer construction and every figure downstream, and it has been deferred deliberately twice rather than drifted into. What can be said with the present grid is that the effect is already large enough to break the arithmetic, which is the claim this essay makes.

The two-mode measurement, and what it cannot say #

The trade’s compromise deserves examining rather than only naming, because it is a good compromise that answers a narrower question than it appears to.

A modern instrument measures a sample twice: once with its ultraviolet output present, and once with a filter cutting it. The first reading is the sample as it behaves under a lamp with ultraviolet in it; the second is the sample with its brightener unexcited, which is as close as the instrument can get to a non-fluorescent version of the same sheet. Quoting both is honest and is standard practice.

What the pair gives is a bracket, not an interpolation. The two readings are the sample under two specific excitation conditions, and a third lamp with some intermediate ultraviolet content does not produce a reading somewhere between them by any rule the pair contains. The amount emitted depends on how much energy the lamp puts into the excitation band, which is an integral over a part of the spectrum the instrument’s two modes sample at two points.

That is the difference between a bracket and a model. A model of the sample — the full Donaldson matrix — would let any lamp’s reading be computed. The two-mode measurement lets exactly two lamps’ readings be quoted, and every other lamp in the world is somewhere unstated between them.

The gap matters most where it is least visible. An office lit by a phosphor-converted LED has almost no ultraviolet at all, so a brightened sheet there behaves close to the filtered reading; the same sheet by a window behaves close to the unfiltered one. Both readings are correct and neither is the reading for the room the work will be judged in, which is usually a mixture of the two.

Who found it, and when #

Optical brighteners were introduced commercially in the 1940s and were in mass-market laundry products and papers within a decade. The measurement problem was recognised immediately, because the paper industry’s own whiteness numbers stopped agreeing between instruments.

Donaldson’s matrix — the full description of what comes out at each wavelength for light in at each wavelength — is from 1954, and a bispectral instrument capable of measuring it exists but is rare and slow. The practical trade has run on the two-mode compromise ever since: measure with the ultraviolet in and with it filtered out, and quote both.

The framing that a fluorescent surface is a non-diagonal operator is standard in the fluorescence literature and does not usually travel to the colour-management side, where the arithmetic that assumes a diagonal is embedded in every profile.

What was computed, and how #

The brightener is a stated absorption band between 380 and 420 nm, a Gaussian emission at 438 nm of 45 nm width, an efficiency of 0.85 and a base reflectance of 0.82. Energy absorbed in the excitation band is redistributed into the emission band, so the model conserves what it should conserve and puts nothing in by hand.

The apparent reflectance is the returned spectrum divided by the incident one, which is scale-invariant — doubling the lamp doubles what is absorbed, what is emitted and what is returned, so the ratio does not move. It is the shape of the lamp that moves it, which is what makes the comparison between D65 and tungsten meaningful.

The gate requires the apparent reflectance to differ between two lamps by more than 0.02 in the worst band, and requires it to exceed one under equal-energy light. Both are properties no diagonal operator has, so an implementation that quietly stopped fluorescing would stop the build rather than reporting an ordinary paper.

What the lamp can actually move, on this grid #

The two headline numbers — 1.06 under equal energy and a worst-band disagreement of 0.186 between D65 and tungsten — are both consequences of one lever, and the lever is computable from the illuminants alone.

The emitted power is proportional to what the lamp puts into the excitation band, and the apparent reflectance at the emission wavelength is that divided by what the lamp puts in there. So the whole lamp dependence is the ratio Σ(380–420) / E(440):

Between D65 and tungsten the lever is 1.36, and that is all of it. Everything else about the brightener — its efficiency, its emission width, its base reflectance — divides out of the comparison, because both readings are of the same sheet.

That number is worth having on its own. On this site’s grid a change of lamp can move a brightened sheet’s apparent reflectance by 36 per cent of whatever the effect is, from the most ultraviolet-rich daylight to a tungsten radiator. The rhetoric of a sheet being three different papers under three lamps is about a lever that, on the visible grid, is a third.

The two numbers do not fit the same brightener #

Applying the lever to the stated 1.06 gives a much smaller disagreement than the one reported.

An apparent reflectance of 1.06 on a base of 0.82 is an emission excess of 0.24 under equal energy. Scaled by the ratios above, that is 0.171 under D65 and 0.125 under illuminant A — a difference of 0.046, not 0.186.

Running it the other way, a D65-to-A difference of 0.186 needs an excess of 0.70 under D65, which is an excess of 0.98 under equal energy and an apparent reflectance there of about 1.80.

So the two numbers describe brighteners about four times apart in strength, and the essay’s own statement of the equal-energy figure is the weaker of the two. Which is the one to keep is not settled here; what is settled is that they cannot both come from a sheet whose excitation band lies between 380 and 420 nanometres, because in that case the lamp’s whole influence is the 1.36 above.

And the resolution is probably the grid. The essay says so in its own terms: a real brightener’s excitation runs from about 340 nanometres, so more than half of what excites the sheet is outside the grid — and the part outside is exactly the part where D65 and tungsten differ most. Off-grid the lever is not 1.36 but something nearer ten, which is the ratio of their ultraviolet content. The 0.186 is what a sheet excited over its real band would do; the 1.06 is what the model on this grid produces.

That makes the essay’s declared floor caveat quantitative rather than rhetorical. The truncation costs about a factor of four on the size of the effect and about a factor of seven on the lamp’s lever, and the two headline numbers are one on each side of it.

What the emission width has to be #

One parameter of the model is stated ambiguously and the arithmetic settles it.

A Gaussian emission at 438 nanometres of 45 nm width is either a full width at half maximum or a standard deviation, and the two differ by a factor of 2.35 in area. Read as a full width at half maximum, the emission’s area is 9.6 five-nanometre bands, so a peak excess of 0.24 is a total emission of 2.30 band-units and — at an efficiency of 0.85 — an absorption of 2.70 spread over the nine bands of the excitation region: a mean absorption of 30 per cent.

Read as a standard deviation the same peak requires 71 per cent absorption across 380 to 420, which would make the sheet visibly dark in the violet and is not a description of white paper.

So the width is a full width at half maximum, and the model’s implied absorption is 30 per cent of whatever arrives between 380 and 420. That is a plausible loading for a brightened stock and it is a number the model never states, recoverable only by working back through the emission it produces.

A radiance factor above one is a statement about how much light leaves the sample, so no choice of observer can bring it back below the line.

Where it stops #

The model is a single excitation band and a single emission band, which is a caricature of a real brightener and is stated as one. Real brighteners have structured excitation spectra and the emission has a tail; the Donaldson matrix of a real sheet is not rank one.

Nothing here addresses fluorescent colorants — the pigments used in safety clothing and warning signage, which absorb in the visible and emit further into the visible. Those are the same phenomenon with the excitation inside this site’s grid, and they would be computable here; they are simply not built.

And the numbers are floors twice over: once for the truncated ultraviolet, and once because the efficiency used is a plausible middle value rather than a measurement of any particular product.

Where the ladder goes next #

The obvious extension is the one the grid blocks. A site that ran from 300 nm would be able to compute a brightened sheet properly and would be able to say how much of the effect its present numbers are missing — and would have to redraw every figure that names an observer, which is why the decision has been deferred rather than taken.

The other direction is the one this essay opens by closing. If the whole phase’s arithmetic depends on a surface being a diagonal operator, then the interesting question is which other things on this site are quietly not diagonal. A gloss pedestal is one — it adds rather than multiplies, and no adaptation removes it either. Between them, the two mark the edges of what a multiplication can describe.