A neutral has no grid

Every departure this collection has measured over two rounds has turned out to be a product of two things, and a product is zero when either factor is. This essay is about the factor belonging to the sample, and about how completely it empties.

The claim #

A sample whose reflectance is constant with wavelength has the same computed colour on every grid, and the equality is exact rather than approximate.

• Every tabulation choice vanishes on it: step, range, origin, slit width and fill-in rule, all at once, to 10⁻¹³ ΔE₀₀ and usually to zero.
• Every observer choice vanishes on it too, at any age, macular density or field size.
• The mechanism is the white point. A flat reflectance produces a stimulus that is a scalar multiple of the light itself, and dividing by the light’s own tristimulus values removes the scalar.
• And it is not a curiosity. It is the reason a collection whose samples are broad and smooth has been able to ignore three decisions for nineteen rounds, and it is the exact statement of when that stops being safe.

The arithmetic, in three lines #

A tristimulus value is X = k Σ R(λ) S(λ) x̄(λ) Δλ and its white point is the same sum with R set to one. If R(λ) = ρ for all λ, then every term of the first sum is ρ times the corresponding term of the second, so

X = ρ · X_white, Y = ρ · Y_white, Z = ρ · Z_white

whatever the grid, whatever the observer, whatever the slit. The ratios X/X_white, Y/Y_white, Z/Z_white are all exactly ρ, and CIELAB is a function of those three ratios alone. So the computed lightness is a function of ρ, the two chromatic coordinates are exactly zero, and nothing in the tabulation or the observer can reach any of it.

The identity survives the sum being wrong. It does not require the quadrature to be accurate, the range to be complete or the observer to be correct. It requires only that the same sum appears above and below, and it does, because nobody computes a white point on a different grid from the sample.

That is a stronger statement than it first sounds. The grid can be catastrophically wrong — the laser projector reduced to one line, the range truncated to a hundred nanometres — and a flat sample still comes out perfectly neutral at the right lightness.

What the collection’s own numbers do near it #

The identity is exact at ρ constant, and what matters practically is how fast the cost grows on the way out. Walking a red pigment from flat towards its own reflectance, with two observers differing in macular density watching, the excitation difference between them is exactly proportional to the distance travelled.

The middle column is straight to two parts in a hundred. The right-hand column is not, and the difference between them is what the colour-difference formula adds rather than anything about the eye: ΔE₀₀ has cube roots in it and a chroma weighting underneath, so a quantity that is exactly linear in the excitations is not linear in it.

The identity is about the eye and the curvature belongs to the unit, and that separation is only visible because both are computed.

How flat is flat enough is the practical form of the question. The identity is exact at a constant reflectance, and every real sample is only nearly one, so the practical question is how fast the cost grows on the way out. Mixing a notch-filter sample into a flat 0.5 by a fraction ε and asking what the five-nanometre grid costs under a fluorescent tube gives the sequence.

The first row is the identity. The second is a sample that is flat to one per cent — flatter than any paint, flatter than most papers — and it already costs six tenths of a unit under a light whose lines the grid cannot see.

So the identity is exact and its neighbourhood is not generous. The ratio of the cost at ε to ε times the cost at one runs from 2.7 at a hundredth down to 1.0 at unity, which means the approach is super-linear in the published unit: the first per cent of departure from flat buys almost three per cent of the full cost. That is the difference formula’s chroma weighting doing what it does near neutral, and it means the safe region around the identity is narrower than a linear reading of the endpoint would suggest.

The observer’s version of the same sequence is gentler. Two eyes differing in macular density, on the same mixture under daylight, read 2.2 × 10⁻³ ΔE₀₀ at ε = 0.001 and 1.714 at ε = 1, a ratio of 1.3 rather than 2.7. Two departures with the same identity at the same place and different curvature approaching it, which is a reminder that the identity is a statement about the mechanism and the shape of the approach belongs to the unit.

Why this is not the same as a grey card being useful #

A photographer’s grey card is neutral, and it would be easy to read the identity above as a restatement of why grey cards work. It is not, and the difference is worth being exact about.

A grey card is useful because it is a known reflectance, so a camera can be told what value to assign it. That works for a card of any colour, and cards of other colours are used for exactly that purpose. Its neutrality buys something different: it makes the card’s reading independent of the spectrum of the light in a way a coloured card’s reading is not — which is close to this essay’s identity but is the illuminant’s version of it rather than the observer’s.

The identity here is stronger and stranger. It says that two people with genuinely different eyes, using genuinely different tabulations, agree about the card exactly, and that they would still agree if one of them had made an arithmetical error in the sum. Nothing about the card’s usefulness follows from that. What follows is a diagnostic: a disagreement about a neutral is a disagreement about the white point, and never about the grid or the observer.

A neutral cannot test anything, because an identity is a place where nothing can be learned, and that has a consequence for anybody choosing a reference sample.

A perfectly neutral patch cannot falsify a claim about the grid, about the slit, about the observer’s age or about its field size. It cannot distinguish a well-tabulated calculation from a badly tabulated one. It agrees under every light and with every eye, which is precisely why it is useless as a test of any of them.

That is awkward, because a neutral is the reference sample most instruments carry and most workflows check against. A white tile and a black trap are what a spectrophotometer is calibrated with, and both of them are, to the extent they are spectrally flat, exactly the samples that cannot report a tabulation fault. An instrument can be perfectly calibrated on its tiles and wrong by a unit on the next saturated sample it reads, with nothing in the calibration record to suggest it.

The reverse also follows and is more useful. A disagreement about a neutral is a disagreement about the white point, and never about the grid or the observer. That narrows a diagnosis to one place in the pipeline in a single reading, which is a great deal more than most single readings do.

The other half of the condition #

A product has two factors and the sample is one of them. The other belongs to whatever is being varied, and it empties in its own way.

For the tabulation, the light’s factor is empty when the light has no structure the grid cannot hold — which is why a three-emitter LED costs nothing at five nanometres whatever sample it falls on. For the observer, the factor is empty when the two observers differ by a gain on each cone, because the white-point division is that gain’s inverse.

So there are two routes to zero and they are independent, and eight of the ten conditions in this round’s audit are identities in floating point rather than small numbers.

That figure is the identity seen from the arithmetic rather than from the sample. The cancellation that makes a coarse grid survivable in general is partial, light-dependent and worth between one and four; on a flat reflectance it is total, and it is total because the two sums are not merely correlated but proportional.

The neutral axis is where this collection lives #

The identity explains something about this collection that had never been stated and is slightly uncomfortable.

Its test surfaces are constructions: broad absorption bands on a pale base, of stated centre, width and depth. Its lights are daylight reconstructions and thermal radiators. Its samples are papers, plastics, eggshell paints. Almost everything it computes sits nearer the neutral axis than a saturated ink or a laser primary does, and every departure it has measured has therefore been measured on the samples where departures are smallest.

That is not a criticism of the choices. Those are the samples most measurement is done on, and a collection about ordinary colour should compute about ordinary colour. But the identity says exactly what the consequence is: the collection has been systematically near the empty end of one factor of every pairing it has published, and the numbers it quotes are lower bounds for a saturated world in a way its captions have not said. The one place it has gone the other way is the gamut work, where the surfaces at the boundary of what a reflectance can be are as far from neutral as a surface gets — and those essays are about the boundary rather than about a departure.

The distribution across the surface family makes the size of it visible. The macular departure runs from 0.26 ΔE₀₀ on the flattest member to 4.24 on the most saturated — a factor of sixteen, driven entirely by how far the sample sits from the light.

That last figure is the strongest form of the identity, because aliasing is the one tabulation fault with no smooth error term to cancel against. A light whose lines the grid cannot see produces sums that are wrong by amounts depending on where the grid began — and on a flat sample the sample’s sum and the white’s sum are wrong by the same factor, whatever that factor is, so the ratio survives an arbitrarily bad grid intact.

The figure above is the identity drawn, and it is the only chart in this collection whose content is that it has none. Every light, every step, every one of them exactly zero — and the same chart on a red pigment spans four decades. What separates the two is a property of the sample alone, which is the sharpest way to say that a departure belongs to a pairing rather than to either of its factors.

Where the identity fails without warning #

There is one class of sample that is neutral, is not flat, and does not satisfy the identity at all, and it is worth naming because it looks like the safe case.

A metameric grey — a reflectance with structure that happens to integrate to a neutral under one light through one observer — has R(λ) varying and X/X_white still equal to Y/Y_white. It computes as neutral, it measures as neutral, and it is not covered by anything above, because the derivation needed R constant rather than the answer neutral. Change the grid, change the observer or change the light and it moves.

That is the whole content of two spectra with one colour, arriving as the exception to an identity. A collection that used metameric greys as its neutral references would find every departure in this round reappearing in the place it had proved they could not, and the proof would still be correct.

The practical form of it is a warning about test charts. A neutral patch printed with three inks is a metameric grey and a neutral patch printed with black ink alone is much closer to flat, and the two behave differently under exactly the substitutions this round is about — which is a distinction the fourth ink was invented for and is used for something else entirely.

What was computed, and how #

The identity is asserted rather than argued. assertANeutralIsObserverInvariant computes the colour of a flat reflectance for each pair of observers in this round’s audit and requires the difference to be below 10⁻¹⁰ ΔE₀₀; the measured values are between 4 × 10⁻¹⁴ and 4 × 10⁻¹³, and one of them is exactly zero.

The distinction between an identity and a limit is made the way the previous round made it: a small number is not evidence of an identity, and the test is whether the number is at the floating-point floor rather than merely small. Eight of this round’s ten conditions are; the two that are not are reported as limits with their sizes, and they are 8.11 and 13.0 ΔE₀₀, which is not small at all.

The sequence in the table is computed by mixing the sample with a flat reflectance of 0.5 and is required to be linear in the excitation distance to two per cent. That assertion could fail: if the departure were not a pairing, the sequence would curve, and the check would say so.

Where the model stops #

The identity requires exact arithmetic in one place — that the same grid appears in the sample’s sum and the white’s — and any pipeline that violates it loses the identity entirely. Reading a sample at five nanometres against a white point taken from a published table at one nanometre is such a pipeline, and it is a common one.

It also requires the flat reflectance to be genuinely flat. A sample flat to a per cent is not covered; it is a sample a per cent of the way along the sequence in the table, which the sequence prices at about 0.02 ΔE₀₀ for the macular departure. That is small, and it is a limit rather than an identity, and the difference between those two statements is the whole discipline of this round.

And nothing here touches fluorescence. A brightened white is neither flat nor a reflectance, its object is a matrix rather than a curve, and the derivation above has no purchase on it.

The generalisation #

The habit is about how to look for an exact case inside an approximate one.

Most error analyses proceed by bounding — the error is at most this, on samples like these, under lights like those. An exact case is worth more than a tight bound, because it is a structural statement: it says which variable the error is proportional to, and therefore which direction makes it smaller. The way to find one is to look for a factorisation, and the way to confirm one is to require the residual to be at the floating-point floor rather than merely small.

The failure mode is to accept a small number as an exact one. Five of the ten conditions in the previous round’s audit looked identical to identities on a first reading and were limits; the sequences that distinguished them had to be constructed on purpose. A small number is not evidence of an identity; a floor is, and the difference is a claim about mechanism rather than magnitude.

Who found it, and when #

The algebra is as old as colorimetry and is usually stated as the reason a reflectance factor is dimensionless. Its use as a diagnostic — that a disagreement about a neutral cannot be an observer or a grid disagreement — is folklore in instrument calibration and is rarely written down.

The metameric-grey exception is Ostwald’s problem in modern dress and reached the printing industry through the grey-balance literature of the 1960s, where the distinction between a neutral made of three inks and a neutral made of one became a commercial matter rather than a theoretical one.

Where the ladder goes next #

Three tabulation choices have been measured and the condition under which all three vanish has been made exact. What is left of the grid is the question the whole section has been pointing at: which end of the tabulation a collection should actually buy, and the answer reverses depending on the lamp in the room.

After that this round leaves the index and opens the third factor of the integral. The observer is not a measurement either, and the arguments it does not admit to having are the next object.