The row a fourth dimension improves

Almost everything gets worse when the surfaces gain a dimension, which is not surprising. One row gets better, and which one it is changes when the dimension changes shape.

The claim #

Neither the row a fourth reflectance dimension hurts most nor the row it helps is a property of the census. Both are decided by how the surface’s new structure and the source’s own structure line up.

• Under a third cosine, a three-primary display is hurt most (+90%) and a triphosphor tube is improved (−6%) — the only row in the table that improves.
• Under a narrow band at 550 nm, the triphosphor tube is hurt most of all (+45%) and nothing is improved.
• Under a fourth cosine, the macular pigment is hurt most (+96%).
• Under a dye edge, a three-primary display again (+138%), and a blackbody is very slightly improved.
• So the same lamp appears at both extremes of the table depending on the shape, and the mechanism that explains one explains the other.

Why almost everything gets worse #

Take it in the direction that is obvious first, because the exception only reads as an exception against it.

A change of light is exactly a 3×3 matrix on the three-dimensional family, and what the census measures is how much of that matrix an adapted observer’s diagonal gain fails to undo. Add a fourth reflectance dimension and two things happen at once: the matrix stops being exact, and the surfaces acquire structure the observer’s three broad channels cannot follow.

Both make the residual larger, and neither is subtle. A surface whose extra structure differs between two lights arrives at the eye as two different triples that no fixed transformation relates, so the mismatch has nowhere to go. Thirteen of the fourteen rows rise, by between three and ninety per cent at twenty per cent amplitude.

The one that falls #

The triphosphor tube’s residual goes from 2.323 to 2.177 ΔE*₀₀, a fall of six per cent, when the surfaces acquire a fourth cosine dimension.

It is the row that looks most fragile in the table. A triphosphor fluorescent tube is three narrow emission peaks with very little between them, it is the second harshest change of light in the census, and it is the standard example this collection uses for what three numbers cannot see. A perturbation that makes every other row worse and makes this one better is not what anyone would predict.

The mechanism is the same one that makes the tube harsh in the first place, run backwards.

Integrating a surface against a three-line source is close to sampling that surface at three wavelengths. The tube’s residual is large because the three samples it takes are a poor summary of the surface — two surfaces that agree at those three wavelengths and differ elsewhere arrive identical under the tube and different under daylight, which is exactly a metameric failure.

Now add a fourth dimension shaped like a cosine of three half-cycles across the visible. Its period is around 130 nanometres, and the tube’s three lines are roughly 60 to 80 nanometres apart. The new modulation therefore alternates in sign between adjacent lines, so its contributions to the three samples partly cancel when they are combined into a tristimulus — while under daylight, which weights the whole band, they do not cancel in the same way. On this particular pairing, the extra structure moves the two lights’ readings closer together rather than further apart.

That is a fact about a beat between two periodicities and not a fact about the lamp.

And the one that is hurt most #

Change the fourth dimension to a narrow absorption band 25 nanometres wide at 550 nm and the tube goes from the table’s only beneficiary to its biggest loser: +45 per cent, ahead of the three-primary display’s +41 and the macular pigment’s +34.

Same mechanism, different phase. A 25 nm band at 550 falls into the gap between the tube’s green and red emission lines. The tube therefore cannot see it at all — none of its three samples lands on it — while daylight, which is present at every wavelength, sees it in full. Two surfaces differing only in that band are identical under the tube and different under daylight, and there is no transformation between the two lights that can be right for both.

That is metameric failure in its purest form, and it is what the tube’s residual is largest for — the same mechanism a notch no pigment can cut is about, seen from the source’s side. A cosine spread across the whole band is partly seen by all three lines and partly cancels; a narrow feature in a gap is not seen at all.

Four shapes, three different rows at the top, and only two of the four improve anything at all.

The beat argument works, and not for the stated reason #

The mechanism offered for the negative row is a beat between two periodicities, and it is checkable from the two numbers the essay gives. Neither survives, and the mechanism does.

A cosine of three half-cycles across 380 to 780 nanometres has a period of 267 nanometres, not the 130 the essay quotes. What is 133 is its half-period — the distance from a maximum to a minimum — and the half-period is the quantity the argument needs, since alternating in sign between adjacent lines means advancing half a cycle between them.

That matters because of the second number. The lines are said to be roughly 60 to 80 nanometres apart, and a spacing of 70 nanometres advances the phase by 0.53π, which is 94 degrees. Ninety-four degrees is not a sign flip; a feature would have to be near 133 nanometres from its neighbour to alternate reliably. On the two figures as stated, the alternation the argument requires does not follow.

It does happen, and it happens because of where the lines sit rather than how far apart they are. Evaluating the third cosine at the mercury and phosphor positions a triphosphor tube emits at — near 440, 545 and 610 nanometres — gives +0.16, −0.73 and +0.65: alternating, decisively, with the two outer lines near the peaks and the middle one near the trough. The spacings that produce it are 105 and 65 nanometres, which advance the phase by 142 and 88 degrees. The first is nearly a sign flip and the second is not; the alternation comes from the two of them together landing the three lines on alternate lobes, which is a fact about the absolute positions.

So the essay’s conclusion is right and its explanation is a coincidence dressed as a rule. A spacing argument would predict alternation for any lamp with lines about 133 nanometres apart; a position argument predicts it only for lamps whose lines fall where these do, and the second is what the numbers support. That is a weaker and more honest statement, and it explains why the effect vanishes when the fourth dimension’s shape changes rather than when the lamp’s does.

The same slippage runs through the macular claim #

The essay explains the fourth cosine hurting the macular pigment most by saying that cosine’s period matches the pigment’s own absorption scale. A fourth cosine’s period is 200 nanometres and the macular band’s full width at half maximum is about 70. Those do not match.

The half-period is 100 nanometres, which is within a factor of one and a half of the band’s width, and that is the comparison the mechanism wants: a modulation whose lobe is about as wide as the absorption it is beating against. Reading period as half-period makes both explanations work and neither is written that way.

It is worth fixing rather than letting stand, because the essay’s closing section offers the pairing argument as something that predicts — and a prediction made with the wrong factor of two would put the expensive shape an octave from where it is.

The shape that produces the exception also produces the widest table #

One more reading of the four-shape table, which supports the essay’s thesis more directly than anything in it.

Under the third cosine the census’s rows move by factors from 0.94 to 1.90 — a span of 2.02. Under the narrow band they move from 1.03 to 1.45 — a span of 1.41. The shape that produces the only negative row also produces a table 44 per cent wider than the shape that produces none.

That is what a pairing predicts and a property does not. A general degradation would move every row by a similar factor whatever the perturbation looked like, so the span would be roughly constant across shapes. A pairing moves each row by an amount set by how that shape lines up with that light, so a shape that interacts strongly with some lights and weakly with others produces a wide table — and the same shape is the one able to produce a negative entry, because strong interaction can have either sign.

The span across the census is therefore a second, cheaper signature of the same mechanism, and unlike the negative row it does not depend on one lamp: it is a property of the whole table under each shape, and it separates the four shapes without needing any of them to produce an exception.

What the general rule is #

Not narrowband sources are worse, which is what the table looks like it says and is not what it says.

The quantity that matters is the overlap between the source’s spectral structure and the surface’s, and it can have either sign.

• Where the surface’s new structure sits in a gap in the source, the source cannot see it and a second source can: maximum disagreement, maximum residual.
• Where it sits on the source’s own features and alternates with them, part of it cancels in the tristimulus integral, and the two sources can disagree less than they did before.
• Where the source is smooth, neither happens; the extra structure is averaged with a slowly varying weight and its contribution falls away with frequency, which is why the smooth-source curve has a maximum and the narrowband one does not.

So the useful statement is a pairing rather than a property: a source and a surface family have a spectral relationship, and the residual is a measurement of that relationship rather than of either one. A three-primary display is hurt by a dye edge at 600 nm because its red primary sits near there; the macular pigment is hurt by a fourth cosine because that cosine’s period matches the pigment’s own absorption scale; a blackbody is barely touched by anything because it has no structure to interact with.

Why this was nearly published wrong #

Worth recording, because the mistake is this round’s own and it was caught by machinery rather than by reading.

The figure that draws the census against amplitude carried an assertion requiring at least one row to fall — written when the only shape tried was the third cosine, where one does. Drawing the same figure with a band-shaped fourth dimension made the assertion fail, and the caption it was attached to said the row that falls is the same row, which was false.

The assertion was the only thing that knew. The numbers were plausible under both shapes, the picture looked the same, and the sentence was the kind of tidy generalisation that reads as a finding. What made it catchable was that the claim had been written into code rather than only into prose — and that the code was run at a second set of parameters rather than at the one it was written for.

The replacement assertion states the disagreement rather than either observation: which row suffers most differs between shapes, and whether any row is helped differs between shapes. That is a claim about four measurements rather than one, and it cannot be broken by trying a fifth shape — a fifth shape either agrees with the others, which is not what the claim says will happen, or it does not, which is what it says.

What it means for a real collection #

Three practical consequences, and the first is the one that generalises past this site.

A colour-rendering claim about a lamp is a claim about a set of surfaces — the same statement the adaptation census turned out to need. The colour-rendering indices are computed on standard test-colour samples for exactly this reason, and this essay is a demonstration of why the choice of samples is not a detail: the same lamp can be the best or the worst thing in a table depending on whether the samples have structure in its gaps. That is not a criticism of any index; it is what an index is.

A lamp with gaps is not uniformly bad — it is unpredictable. The triphosphor tube’s residual moved by −6 per cent and +45 per cent under two equally plausible fourth dimensions. A smooth source’s residual moved by a few per cent under both. The right thing to say about a narrowband source is not that it renders worse but that what it renders badly depends on the surface in a way a smooth source’s does not, which is the property that makes it difficult to specify for.

And adding a dimension to a test set is not a refinement. It is a different measurement, and it can move a conclusion in either direction. The census’s extremes hold under every shape tried, and its middle was never ordered anyway.

Where the model stops #

Four shapes is four shapes. Nothing here bounds how large the effect can be, and a shape chosen adversarially against a particular lamp would do worse than any of these — a comb matching the lamp’s own line spacing would be the obvious construction and has not been tried.

Nor does the essay establish which shapes real surfaces have. It establishes that the answer depends on the shape, sharply, with either sign, and that a variance figure cannot carry the information needed to predict it. Anyone with a measured collection and a specified lamp can compute the number; anyone without one can only state the sensitivity, which is what this does.

Why the sign is worth the essay #

A six per cent improvement on one row of a fourteen-row table is not a large effect, and the essay is about it because of what a negative sign rules out rather than what it establishes.

It rules out reading the fourth dimension as a general degradation. The obvious summary of what a fourth dimension costs is the surfaces are more complicated than the model allows, so everything gets worse — which is true of thirteen rows and would have been stated as the finding. One row with the opposite sign turns that into a statement about a mechanism, because a general degradation cannot have exceptions and a pairing can.

And it rules out a monotone correction. If every row rose by a similar factor, a published residual could be corrected for the surfaces’ true dimensionality by multiplying. It cannot: the factors run from 0.94 to 1.90 under one shape and from 1.03 to 1.45 under another, with different rows at the extremes. A correction that has to be computed per row and per shape is not a correction; it is a measurement.

That is why the negative row is asserted as part of a comparison between shapes rather than reported on its own. On its own it is a curiosity about one lamp. Beside the shapes that do not produce it, it is the evidence that the whole table is a pairing rather than a property.

Who found it, and when #

The underlying object is the metameric black — the part of a spectrum an observer’s three channels cannot see — and the fact that a metameric black under one light is not one under another is the whole content of metamerism failing. That is Wyszecki’s, from 1953, and the machinery this site uses to construct such spectra has been here since its second phase.

What is new is running it backwards: instead of asking what can be added to a surface without changing its appearance under this light, asking what happens to the disagreement between two lights when something is added that neither was designed for. The answer has a sign, the sign is not always positive, and the case where it is negative was found by an assertion refusing a caption.

The general statement, and what it costs to make #

The essay’s finding is a pairing, and pairings are harder to state usefully than properties. It is worth doing the work of stating it.

A residual after adaptation measures how differently two lights sample a surface’s spectrum. Write the surface’s structure as a function of wavelength and the two lights as weightings; the residual grows with how much the two weightings disagree about that structure. A fourth reflectance dimension adds structure at a stated scale and position; whether it increases the disagreement depends on where it falls relative to each light’s own features.

Three regimes follow and all three are visible in the measurements:

Both lights smooth. The extra structure is averaged with slowly varying weights under both, so it partly cancels twice and the residual rises modestly. The daylight-to-blackbody row rises by three per cent at twenty per cent amplitude.

One light structured, the surface’s feature in its gaps. The structured light cannot see the feature at all and the smooth one sees it fully: maximum disagreement. The triphosphor tube under a 550 nm band, +45 per cent.

One light structured, the surface’s feature beating against its features. Partial cancellation in the structured light’s own samples, and the disagreement can fall. The triphosphor tube under a third cosine, −6 per cent.

The cost of stating it this way is that it takes a paragraph rather than a sentence, and the benefit is that it predicts. A dye edge at 600 nm should hurt a three-primary display most, because that display’s red primary is near 600 — and it does, by 138 per cent, the largest single figure anywhere in these tables.

Where the ladder goes next #

The set’s dimension has now been varied. Its shape — how bright and how saturated its members are — is three declared numbers that have never been moved, and they turn out to be the most elastic input in the collection.