Two matrices hold to almost tungsten, and the blue turns long before

An appearance model is two operations in a row. It adapts — scales each of three cone-like signals by a gain that makes the room’s white look white — and then it compresses, passing each signal through a curve that rises steeply near nought and flattens towards the top, the way a receptor’s response does. CIECAM02 did the two in different spaces: adaptation in CAT02, which was sharpened to fit corresponding-colour data, and compression in Hunt–Pointer–Estévez, which is closer to the cones — the distinction the cones an appearance model uses measured, finding CAT16’s axes committed to a confusion point far from any receptor’s. CIECAM16 merged them into one space, and the blue’s hue turn was lost with a matrix found what the merger cost: the Abney turn at blue, the drift in hue a saturated blue shows as it is mixed with white. Compressing in Hunt–Pointer–Estévez puts it back.

Two matrices fail under a warm white found what the merger bought. Converting adapted signals from one space to another can send a colour to a negative signal, which a compression curve built as a power law cannot take. Adapted to the display white, the two-matrix front end fails only for spectral lights past 560 nm, and by less than a hundredth of their luminance. Adapted to illuminant A, it fails for every spectral light from 380 to 474 nm and at the blue corner of every display, sRGB’s included. Somewhere between the two whites the failure reaches the colours people reproduce.

The closing section of that essay turned the question into a walk. Take the adapting white down the radiator locus and the daylight locus from 6500 K, and find the temperature at which sRGB’s boundary first fails, with the display blue’s hue turn at each. The prediction was that sRGB’s blue fails near 3500 K, a warm-white LED’s temperature, and that the blue’s two-matrix turn stays within five degrees of its daylight value down to about 4000 K before rising steeply. A two-space model would then be sound for daylight and neutral rooms and unsound for warm ones. If instead sRGB held nearly to illuminant A, the two-matrix design’s working range would be wider than the merger assumed.

The boundary holds to almost A, and the turn does not wait for it #

sRGB’s boundary holds down to 2872 K and first fails at 2871, fifteen kelvin above illuminant A — not near 3500 K. Display P3, which shares sRGB’s blue primary, fails at exactly the same temperature. BT.2020’s bluer primary fails first, at 3626 K. Between 5700 and 4700 K no colour of any kind goes negative, not even a spectral light. But the display blue’s two-matrix hue turn does not stay near its daylight value: from 14.5 degrees at 6500 K it is above 19 by 4500 K, 22 by 4000 K and 43 at 2880 K, the last temperature before sRGB fails. CIECAM16’s own turn moves from 3.9 to 5.9 degrees over the whole walk.

• The failure is late. A two-matrix front end is arithmetically sound for every colour sRGB or P3 can show under any radiator warmer-looking than a tungsten lamp is not.
• The failure is the blue primary’s. sRGB and P3 fail together because they share a blue; BT.2020 fails seven hundred kelvin sooner because its blue is further out.
• The turn is early. The restored hue turn grows steadily as the white warms, well before any signal goes negative.
• They are one thing seen twice. The turn follows how close the blue’s long-wave signal comes to nought, and the failure is that signal reaching it.

Read from left to right the figure is a walk from a cool daylight-coloured white to a candle-coloured one. At the cool end, P3’s and BT.2020’s red corners fail by a few thousandths — the tiny red failures the display white already showed, with sRGB’s red corner just inside. In the middle there is a band where nothing fails at all. At the warm end each display fails once its blue corner is reached, BT.2020 in the middle of the range of warm-white LEDs and sRGB only at the doorstep of tungsten.

Walking the white along two loci #

The walk takes two families of white. Black-body radiators from 6500 K to 2700 K are the whites of incandescent lamps, and illuminant A is one of them, at 2856 K. CIE daylights from 6500 K to 4000 K, where the daylight series is defined, are the whites of skylight and of lamps built to imitate it. The two loci run close together and part slightly: a daylight at a given correlated colour temperature is a little greener than a radiator at the same one.

At every white the census is the earlier essay’s. The white sets CIECAM16’s adaptation gains in CAT16 space, at the degree of adaptation CIECAM16’s own rule gives for a bright, average room; a colour’s adapted CAT16 signals are carried through XYZ into Hunt–Pointer–Estévez; and each of the three resulting signals is checked for sign. Because adaptation is a set of gains, whether a colour fails depends only on its chromaticity, so each display’s boundary is its triangle’s edges sampled at six hundred points, and the locus is the spectrum at every nanometre. At 6500 K the daylight walk reproduces the earlier census’s display white, turn for turn to a fifth of a degree, and the radiator at 6500 K differs from it by under half a degree — which fixes the walk to the earlier result at its starting point.

sRGB holds until fifteen kelvin short of A #

sRGB’s safety margin rises before it falls. Under a cool white the weakest point on sRGB’s boundary is its red corner, whose short-wave signal sits at eight hundredths of the colour’s luminance. Warming the white raises the short-wave gain — adaptation boosts the signal the white is poor in — and the red corner’s margin grows, to a fifth of its luminance at 3500 K. Meanwhile warming lowers the long-wave gain, and the blue corner’s long-wave signal, which was comfortably positive under daylight, is falling towards it. At 3250 K the blue corner becomes the weakest point, and from there the margin falls fast: 0.077 at 3000 K, 0.018 at 2900, and nought between 2872 and 2871 K.

So the prediction’s 3500 K is where sRGB is safest, not where it fails. A warm-white LED at 3000 or 3500 K is well inside the region where a two-matrix front end handles every colour sRGB and P3 can show — with the caveat a lamp is not a blackbody makes about every lamp sold by a temperature, that an LED’s white need not lie on the radiator locus at all. What it does not handle is BT.2020’s boundary: its blue primary sits further out towards the violet, where the long-wave signal is smaller to begin with, and it crosses nought at 3626 K and reaches −0.60 of its luminance by 2700 K. The earlier essay found BT.2020 failing by 0.45 under illuminant A; the walk says where that began.

Display P3 fails at exactly sRGB’s temperature, to the kelvin, and the reason is in its definition: P3 widened sRGB’s red and green primaries and kept its blue. Every display failure the walk finds on the warm side is a blue-corner failure, and a display’s blue primary is what sets its warm limit. A display with a deeper blue than BT.2020’s — a laser projector’s 440 nm line, say — would fail warmer still.

A window where nothing fails #

The spectrum locus is the hardest test, because every real colour lies inside it. Under a 6500 K radiator the two-matrix front end fails for spectral lights from 557 nm to the red end; under a 2700 K one it fails from the violet end to 476 nm. The two failures shrink towards each other as the white moves towards the middle, and between 5700 and 4700 K — a thousand kelvin, twenty-one steps of fifty — neither is present: every spectral light, and so every colour at all, keeps three positive signals.

That window is not an accident of this model. Both matrices are normalised so that the equal-energy white gives three equal signals, so for that white CIECAM16’s adaptation gains are all exactly one; with nothing to undo, the two matrices compose to Hunt–Pointer–Estévez applied to the colour’s own tristimulus values, whose signals are non-negative for every real colour. The failures are caused by the gains departing from one, and the equal-energy white’s chromaticity lies close to a radiator’s at about 5450 K, inside the window. The display white of the earlier census sits just outside it on the cool side, which is why its failures were real but tiny. A two-matrix front end is exact for every colour under a neutral white a little warmer than daylight, the white of many photographic viewing booths and of north-light studios, and strains in both directions away from it.

The blue’s turn does not wait for the failure #

The turn is what the two-matrix design is for. White turns a hue, and the models part at blue found the Abney effect’s largest disagreement between models at the display’s blue, and the matrix essay found CIECAM16’s one-space front end turning that blue by under four degrees where Oklab and the two-matrix front end turn it by about fourteen. Under daylight the two-matrix turn is 14.5 degrees at 6500 K, and the prediction was that it stays within five degrees of that down to about 4000 K.

It does not. Under radiators the turn is 16.3 at 5500 K, 19.4 at 4500 — past five degrees above its start — 22.0 at 4000 and 26.2 at 3500. The daylights stay within a few tenths of the radiators, drawing up to six tenths above them by 4000 K. Below 3500 K the rise steepens: 29.6 at 3250, 35.6 at 3000, 40.8 at 2900. At 2880 K, the last step at which every signal is still positive, the blue turns 43 degrees, three times its daylight value, towards purple. CIECAM16’s own turn, meanwhile, moves from 3.9 to 5.9 degrees over the whole walk and would not be noticed.

The last hundred and fifty kelvin #

Closer to the failure the turn accelerates. From 3000 to 2880 K it climbs from 35.6 to 43.2 degrees, faster with every step; at 2870 K, with the blue’s long-wave signal just below nought, it is 48.2; by 2850 K, 50.8. The turn does not jump at the failure. It runs up to it. A compression that carries a negative signal’s sign through the power law — as CIECAM02’s did — produces a hue for the failed colour, and the hue it produces is continuous with the ones before it. Nothing in the output flags the moment the input stopped being a cone signal.

That is the practical danger of the failure the earlier essay found: not a crash but a smooth drift into a region where the numbers mean nothing, arrived at through a region where they mean something increasingly strained. A model deployed with the two-matrix front end would report the blue’s hue under a 2900 K lamp as forty degrees away from where it is under daylight, and nothing would say whether that is the Abney effect or the arithmetic.

What the turn follows #

Across every radiator and daylight where the blue’s signals stay positive, its two-matrix turn ranks with how close its least signal comes to nought, at a rank correlation of 0.995. The least signal is the long-wave one on the blue’s way to grey: 0.68 of its luminance at 6500 K, 0.42 at 4000, 0.08 at 3000 and 0.018 at 2900. The compression rises most steeply near nought — a compression goes below the floor is about what that steepness does to small differences — so as that signal shrinks, the same mixture with white moves it through a steeper and steeper part of the curve, and its share of the hue calculation grows. The turn and the failure are the same quantity read at two thresholds: the failure is where the least signal reaches nought, and the turn is how close it has come.

That changes what the restored turn under daylight is evidence of, in the way the model has a hue shift it was never given found the compression producing a Bezold–Brücke shift nobody fitted. The blue resists through the model’s denominator and the matrix essay traced the daylight turn to the compression acting on a small long-wave signal. The walk shows that signal shrinking steadily as the white warms, so the two-matrix turn is a function of the adapting white as much as of the blue. White drains the blue last in every model found the blue the hardest hue for white to reach in every space, and the hue scale has four corners that the model’s hue quadrature treats blue as a corner of its own; a turn of forty degrees at blue crosses most of a quadrant. Whether a real blue’s Abney turn grows under a warm white is a question for observers, and nothing here answers it; what the census says is that the two-matrix model predicts it tripling, and CIECAM16 predicts it barely moving.

What a model builder should take from this #

The two-matrix front end’s arithmetic is sound much further than the merger assumed. Every colour sRGB or P3 can display keeps three positive signals under any radiator down to 2872 K, which includes every warm-white LED and most tungsten-halogen lamps. BT.2020 content is the exception: its blue corner fails from 3626 K.

Its hue behaviour is not sound as far as its arithmetic. The blue’s turn triples before any signal fails, driven by the same shrinking signal. A model that restored the blue’s daylight turn by compressing in Hunt–Pointer–Estévez would be predicting a much larger turn under warm light, and that prediction needs data before it is trusted.

A guard on the least signal would catch both. The failure is a signal crossing nought; the strained region is that signal approaching it. A front end that reported how close any signal came — a tenth of the luminance, say — would flag the warm-light blues before their hue became meaningless and long before the power law failed.

How the walk was taken #

The adapting whites are Planck’s law at each temperature on the site’s 5 nm grid and the CIE daylight series from its S-basis, each integrated against the 1931 observer and scaled to a luminance of a hundred. The front end is the earlier census’s without change: CIECAM16’s degree of adaptation for an adapting luminance of 100 cd/m², a background of twenty and an average surround; gains applied in CAT16 space; the adapted signals carried to XYZ and into Hunt–Pointer–Estévez. A boundary fails if any of its six hundred sampled points has a negative signal; a display’s first failing temperature is bisected to a kelvin between 4500 K, where it holds, and 2700 K, where it fails. The spectral window is the stretch of radiator temperatures, in steps of fifty kelvin, at which no wavelength from 380 to 700 nm fails. The blue is the display’s HSV blue at full saturation, mixed with the adapting white at its own luminance down to a fifth of its purity; its turn is the change in hue angle through each front end’s compression and opponent stage, and its least signal is the smallest two-matrix signal, over the colour’s luminance, at the five purities.

What this leaves out #

The adapting condition is fixed. A bright room, a background of twenty, an average surround. A dimmer room lowers CIECAM16’s degree of adaptation, the gains move less far from one, and every failure temperature would move warmer; the window and the turn’s dependence on the least signal would not change in kind.

Only two loci are walked. A white LED off the radiator locus — pinker or greener than a radiator of the same temperature — sets different gains, and a green-tinted warm white could fail sooner.

The Abney turn under a warm white has no data here. The census compares two models’ predictions; it does not say which, if either, an observer adapted to 3000 K would report.

Still open: whether a warm room turns an observer’s blue #

The census reduces the two-matrix design’s claim to one testable prediction. Under a 6500 K white the two-matrix and Oklab turns agree at about fourteen degrees for the display blue, and CIECAM16’s is under four. Under a 3000 K white the two-matrix front end predicts thirty-six degrees and CIECAM16 still under six.

The experiment is a hue match: observers adapted to a 6500 K and to a 3000 K surround each set a mixture of the display blue with the adapting white to the hue of the unmixed blue, at a fifth of its purity. The prediction worth recording is that the turn grows under the warm surround, but by less than the two-matrix model says — perhaps to eighteen or twenty degrees — because a real long-wave cone signal does not approach nought the way the model’s Hunt–Pointer–Estévez signal does when it is computed from CAT16-adapted values. If observers’ turns stay near their daylight value, the two-matrix model’s warm-light turn is an artefact of converting between spaces; if they triple, the strained region is where the model is most right.

A threshold hides the approach to it #

The habit is about what a failure census counts.

The earlier essay asked where signals go negative, and the answer here is: later than predicted, almost at tungsten. Read as a safety limit, that says the two-matrix front end is safe for nearly every white. But the quantity that crosses the threshold was moving the whole way, and something that depends on it — the very hue behaviour the design exists to restore — moved with it long before the threshold was reached.

The failure mode is to report where a quantity crosses a line and not how it approaches. The test is to plot what the model is for against the quantity that fails, over the whole range, and see whether it waits for the threshold. Here it did not wait at all.