The reversals have a straight edge

A dark background moves every difference and no match established that CIECAM16’s background is one exponent, that no two samples change places in lightness or in hue when it moves, and that chroma is the exception: 21 of 276 pairs swapped between a near-black background and a near-white one. The explanation offered was structural — chroma is a product of a term carrying the exponent at half strength and a term carrying a plain multiplier, and a product of the two need not preserve order — and it ended by saying that 24 samples were enough to establish that reversals happen and not enough to say how common they are.

Both halves can be done better. The structural sentence is one algebraic step from an exact condition, and the condition turns the counting into arithmetic rather than a census.

A reversal is a straight line, and the line has a ceiling #

A pair of samples changes places in chroma when the background moves exactly when it is ordered one way in chroma and the other way in lightness and its chroma ratio is inside a bound set by its lightness ratio — a condition with two straight edges, which names every reversal and no others.

• The bound is half the change in the lightness exponent. Between backgrounds of 2 and 80 it is 0.2598, and 754 of 14,028 pairs are inside it.
• The condition disagrees with the model nowhere: across six spans of background it names 754, 556, 292, 154, 75 and 11 reversals and gets the same pairs the model does, with zero disagreements in 84,168 tests.
• The phenomenon has a ceiling nothing can exceed. The exponent’s base runs only from 1.48 to 2.48, so the widest the wedge can ever open is half the surround’s impact — 0.345 — and 13,050 of the 14,028 pairs keep their chroma order in every room the model accepts.
• There is no safe span. Backgrounds of 20 and 21 — one unit of luminance factor apart — still reverse 11 pairs.
• The reversals are rare as a property of a pair and ordinary as a property of a sample: 163 of the 168 samples take part in at least one.

Chroma’s whole dependence on the background is one number #

The model builds chroma as C = t^0.9 · (J/100)^0.5 · (1.64 − 0.29ⁿ)^0.73, where n is the background’s share of the white, t is a temporary quantity assembled from the sample’s adapted responses, and J is lightness. The background appears three times and each appearance behaves differently, which is why the structural sentence is true and unhelpful.

Take them in turn. The last factor is a function of n alone: it multiplies every sample by the same number and cannot reorder anything. Inside t, the background appears once, as the induction factor Ncb, and Ncb is likewise the same for every sample — so t is a background-free quantity times a common scale, and t^0.9 is too. Only the middle factor carries the sample. Lightness is 100 · r^(c z) with r = A / Aw free of the background, as a dark background moves every difference and no match established, so (J/100)^0.5 = r^(c z / 2).

Collecting them, a sample’s chroma at one background over its chroma at another is C₂ / C₁ = K · r^((p₂ − p₁) / 2), with p = c z and K the same for every sample. The background’s entire effect on chroma is one common scale and one power of the sample’s own lightness. Nothing about the sample’s hue, its level or its spectrum survives except through r.

The common factor is 0.82535 and the power is 0.2598, and the dots sit on the line to 1.2 × 10⁻¹⁵ of the factor. That is arithmetic noise, so the reduction is not a good approximation; it is the model rearranged. It is worth computing anyway for the same reason the exponent identity was worth computing, and for the reason the audit, read as appearances gives for putting a model’s own rearrangement to its own code: this is a claim about an implementation, several of the standard’s expressions carry Ncb and Nbb in places where a transcription error would be invisible, and the whole of what follows rests on this one step.

Two straight edges #

With the reduction in hand the reversal condition writes itself. Let x be the log of the ratio of two samples’ chroma at the first background and y the log of the ratio of their responses. Their chroma ratio at the second background is x + s·y, where s = (p₂ − p₁)/2 is half the change in the exponent. The pair changes places exactly when x and x + s·y have opposite signs.

That is a wedge in the plane, bounded by the line x = 0 and the line x = −s·y. It is the shaded region in the figure above, and it says three things at once.

A pair must be ordered oppositely in chroma and in lightness. If the more colourful sample is also the lighter one, x and y have the same sign, x + s·y is further from zero in the same direction, and no change of background can bring them together. That is not obvious from the model and it is the most useful half of the condition: every reversal is a trade between chroma and lightness, and pairs where the two agree are safe.

The pair’s chroma ratio must be small compared with its lightness ratio. Two samples whose chroma differs by a factor of two — x about 0.69 — need a response ratio of at least 2.7 the other way to reverse at this span, which is a lightness difference of more than sixty units. Nearly-equal chroma at plainly different lightness is the shape that reverses — which is also the shape which of two is worse found two difference formulae disagreeing about, for an unrelated reason and with the same geometry.

And the slope is the only thing the backgrounds contribute. The samples decide where the pairs sit in the plane; the two backgrounds decide only how far the wedge opens. So a count at one span predicts a count at another without recomputing the model at all, provided the cloud of pairs is known.

Across those six spans the condition and the model agree on every one of 84,168 pair tests. The counts run 754, 556, 292, 154, 75 and 11 against slopes of 0.2598, 0.1901, 0.1091, 0.0553, 0.0226 and 0.0038, and the relation between the two is nearly straight through the origin — which is what a narrow wedge sweeping a smooth cloud of points produces, and is a check on the cloud rather than on the condition.

The narrowest span is the one worth dwelling on. Backgrounds of 20 and 21 differ by one unit of luminance factor, which is a shade of grey nobody would distinguish on a mount board, and eleven pairs of samples change places in colourfulness across it. There is no span of backgrounds small enough to be safe — only spans small enough that few pairs happen to lie within the sliver. A specification that permits the background to vary at all permits chroma rankings to differ.

The ceiling, which is a guarantee #

The wedge’s slope is (p₂ − p₁)/2 and both exponents are c times a base z = 1.48 + √n. The background’s share of the white runs from nothing to one, so z runs from 1.48 to 2.48 and no more. The largest change in the exponent any two backgrounds can present is the surround’s impact itself, and the widest the wedge can ever open is half of it.

In an average surround that ceiling is a slope of 0.345, and 978 of the 14,028 pairs lie inside it — seven per cent. The other 13,050 keep their chroma order in every room the model accepts. That is a guarantee of the same kind as the lightness one, weaker in that it does not cover every pair, and it is the statement the earlier essay could not make: the reversals are not a nuisance that grows with the range of conditions asked for, they are bounded by the model’s own arithmetic.

The bound is different in other surrounds, and it is smaller. A dim surround’s impact is 0.59 and a dark one’s 0.525, so their widest wedges are 0.295 and 0.2625. A darker surround reorders less, which is the opposite of what a reader might guess from a dark room being the harder viewing condition, and follows from the exponent being smaller rather than from anything about dark rooms. The surround is three rows of a table found the three tabulated surrounds unevenly spaced in that impact, so the ceiling is unevenly spaced too.

Which correlate can be reordered at all, and why only one #

Setting the three correlates side by side makes the asymmetry a statement about form rather than about behaviour.

Hue: zero, and it could not be otherwise — the background is nowhere in the hue path. Lightness: zero — the background is one exponent on a background-free ratio, and a positive power leaves a list in order. Chroma: 754 — because the exponent enters at half strength and is multiplied by a scale, and a power times a scale is a power, which reorders whatever the power’s exponent moves.

The shape of the answer is what makes the wedge possible. A product of two arbitrary functions of the sample would give a reversal condition that has to be searched for, sample pair by sample pair, and would look like a scatter with no edge. A product of a power and a scale gives a condition that is linear in logarithms, and a condition linear in logarithms is a straight line. The reason the reversals have an edge is that the model’s chroma has almost no room in it.

Which samples, and how many places each moves #

A count of pairs says how often, not to whom. The second question has a different answer from the first and the difference is the useful part.

163 of the 168 samples take part in at least one reversal, and the most involved takes part in 25. Five per cent of pairs reverse and ninety-seven per cent of samples are in one, which sounds contradictory and is not: a sample has 167 partners, and needs only one of them to sit in the sliver.

The two readings answer two different questions and both get asked. Will these two samples keep their order? is a question about a pair, and the answer is almost certainly yes. Can this sample be relied on to keep its place in a ranking? is a question about a sample, and the answer is almost certainly no. A specification that ranks a set of colours by colourfulness — a shade card ordered by intensity, a set of ink strengths, a quality grade with a colourfulness floor in it — is asking the second question, and the second question is where the reversals live. A tolerance is a region makes the neighbouring point about acceptance: what a decision procedure is about decides which statistic describes it.

The five samples that take part in nothing are the ones with no near neighbour in chroma at a different lightness — the most colourful sample in the set, the least, and three isolated in between. Isolation is the protection, not extremity, and it is not a property a specification can arrange: a third of the appearance box is no surface measured how densely real surfaces crowd the model’s output, and they crowd most where a shade card sits.

The background as a dial #

The reduction has one more reading in it, and it is the one worth carrying away.

Chroma order at any background is the order of C · r^(s) at any other, with s running smoothly with the background. So the background is not shuffling a ranking; it is sliding it along a one-parameter family, and the family is a trade between chroma and lightness. At a dark background the ranking weights lightness slightly less; at a light background slightly more.

Read that way, the effect is not a defect in the model. It is a claim: that how colourful a surface looks relative to another depends on the wall, and that the dependence is a specific, small, monotone trade against lightness rather than an arbitrary reshuffling. Whether the claim is true of observers is a different matter, and nothing in the model’s derivation was aimed at it.

That is the same position the model has a hue shift it was never given found with the hue shift, from the other side. There the model reproduced a measured effect it was not fitted to, which is evidence for its structure. Here it predicts an effect nobody has measured, which is a prediction its structure makes and a place to look.

How the pairs were counted #

The samples are the widened surface family under D65 at full density — 168 surfaces spanning seven band centres and four reflectance levels — against the 24 the earlier census used, because the questions here are about how often rather than whether. Every reading holds the adapting luminance at 100 candelas a square metre and the surround at average, so only the background moves.

The background-free response is recovered from the model rather than reassembled from the standard’s constants: J/100 = r^p gives r = (J/100)^(1/p), and the common factor K is read off the samples as the mean of (C₂/C₁) / r^(s). If the reduction is right every sample returns the same K, and the spread over the samples is the test — which is why the spread is the number reported rather than a fit residual.

A reversal is counted when the sign of the chroma difference between two samples differs between the two backgrounds. The condition is evaluated independently, from the logarithms alone, and the two are compared pair by pair; a disagreement in either direction is a failure. The widest possible wedge uses the surround’s impact directly rather than any pair of backgrounds.

What this does not say #

The condition is exact for CIECAM16 and is a statement about that model’s algebra. Any appearance model whose chroma is a product of a power of lightness and a sample-free scale has the same wedge with its own slope; one that builds chroma differently does not, and the reversals would have to be searched for again.

The cloud of pairs is this sample set’s, in the sense the census under another observer makes of every count taken over a constructed family. The counts — 754, 978, 13,050 — describe surfaces with a single absorption band, and a set of real pigments, or a lattice through the object-colour solid, would put its pairs in different places and give different counts. The wedge’s slope is a property of the model; the counts are a property of the samples, and the two are separable here in a way they were not when the answer was a census.

Nothing here says which ordering is right. The model returns a different chroma ranking in a different room and no observer has been asked whether they do the same. An appearance is not always a stimulus is the standing reminder that the model’s outputs are a prediction rather than a measurement, and this is a prediction that has not been tested.

And the surround is held throughout. It enters the same exponent the background does, so it moves the wedge’s slope as well, and a comparison that changed both at once would need the slope computed from the two products rather than from the two backgrounds.

Still open: whether an observer’s ranking slides the same way #

The model says that the ranking of a set of surfaces by colourfulness slides smoothly with the background, that the slide is a trade against lightness, and that it is small — the fourteen most colourful samples here move by one or two places across the whole range of backgrounds.

The experiment is a ranking task rather than a matching one, and it is cheap. Give observers a set of surfaces whose chroma is close and whose lightness plainly differs, have them order the set by colourfulness against a near-black surround and again against a near-white one, and count the transpositions. The model predicts which pairs will transpose, by name, before anybody sits down — the wedge condition is computable from the surfaces’ own measurements.

Two outcomes are informative and only one is expected. If observers transpose the predicted pairs, an effect that has never been looked for has been found by rearranging a standard. If they transpose different pairs, or none, the model’s chroma is carrying a dependence on the background that nothing put there deliberately, and the place to look is the half-strength exponent — which arrived in the formula for C as a way of tying colourfulness to lightness rather than as a claim about surrounds.

A structural explanation is one step from a condition #

The habit is about what to do with a sentence that says why something can happen.

Chroma is a product of a term that carries the exponent and a term that does not, so it need not preserve order. That is a correct explanation and it has no predictive content: it says reversals are possible, which a census had already established, and it names no pair. Such sentences are the natural end of an argument about form, and they are usually one line of algebra short of something much better.

The move is to write the explanation as an equation and then divide. A ratio between two samples turns every common factor into nothing, and what survives is the condition — here, that one power of one background-free quantity decides everything. The step takes a minute and it converts a statement about what the model can do into a statement about what it does to each pair.

The failure mode is to stop at the mechanism. A mechanism explains a count; a condition replaces it, and the difference is that a condition can be evaluated on pairs nobody has computed, bounded over conditions nobody has tried, and handed to an experiment as a list of predictions.