The claim, in nanometres

A robustness audit found one sentence in this collection that would stop being true if a number this site had chosen were a third larger. The right repair is not a wider margin. It is a sentence that does not contain that number.

The claim #

Every published adaptation transform implies a protanope’s confusion point that corresponds to a medium-wave pigment displaced by a large fraction of the distance between the two long-wave pigments, and none of them agrees with any of the others.

• The nearest is CAT02, at −7.5 nanometres — thirty per cent of the 25 nm L–M separation.
• The furthest is a plain XYZ scaling, at +18.2 nanometres, seventy-three per cent of it.
• They disagree in both directions. Bradford and CAT02 want the pigment shorter; Hunt–Pointer–Estévez and CAT16 want it longer.
• The span across the table is 28.7 nanometres — larger than the L–M separation itself.
• No population, cloud, standard deviation or declared width appears anywhere in the statement.

What was wrong with the old form #

The observation is four rounds old and is not in doubt. Two of the three confusion points implied by every published transform sit outside any plausible population of eyes, and the third does not — which is why the claim has to name the points it rests on rather than being stated as a slogan.

The form of the statement was a distance divided by a cloud radius. The cloud is a population of two hundred modelled observers drawn from five reported spreads: the lens as an age range, macular pigment density, cone outer-segment density, and the three pigment peaks. Each of those five is a declared modelling input, quoted as a round number chosen to be defensible rather than precise, and this collection says so wherever they appear.

The round before this one measured what that costs. The headroom on the protan claim — the factor by which one declared width would have to be larger for the claim to stop holding — is 1.34. The claim’s margin looked comfortable and its exposure was not, and a table of margins cannot be read as a table of risks.

A statement whose truth turns on a modelling input the author picked is not a measurement of anything. It is a measurement of a decision.

The unit the evidence has #

The evidence is not a spread. It is a set of pigment absorption peaks, and microspectrophotometry reports those in nanometres. So the question to ask is the one a spectroscopist would ask:

How far would a pigment have to move for its confusion point to land where the published transform puts it?

Answered in nanometres, against a scale the eye fixes rather than one this site declared: the 25 nanometres between the L and M pigment peaks, which is the basis of red–green vision and is not a quantity anybody here chose.

And the pigment is the medium-wave one, not the long-wave one, because a dichromat’s confusion point is determined by the two receptors that remain. Moving the L pigment does not move the protan point at all — exactly — so the only axis on which the question can be asked is M.

The answer #

The nearest of the five asks the medium-wave pigment to move nearly a third of the way to the long-wave one. Cone pigment peaks are measured by microspectrophotometry and by genetics, they agree between methods to a couple of nanometres, and the common L-pigment polymorphism is worth about three. Seven and a half nanometres is not a plausible measurement error and eighteen is not in the same conversation as one.

And they disagree with each other by 28.7 nanometres, from Bradford’s −10.5 to XYZ’s +18.2. That is larger than the separation between the two pigments the whole comparison is about. Five transforms cannot all be describing the same eye, and the amount by which they fail to is expressible without reference to anybody’s population.

What the leftover says #

None of the five is on the locus a shifted pigment traces, and the residual is reported rather than hidden.

A pigment peak is one parameter and a chromaticity is two, so the locus is a curve in a plane and a point off the curve stays off it. The residuals are small — a few thousandths of a chromaticity unit — but they are not zero, and what they say is that the published points are not the confusion points of any observer whose only difference is a shifted M pigment. They are the confusion points of a fitted matrix, which is a different kind of object, and the nanometre figure is the closest a real receptor gets to it.

Allowing the S pigment to move as well closes most of the gap — Bradford’s residual falls to 0.0004 with an additional −2.5 nm on S — which is the expected behaviour of a two-parameter fit to a two-dimensional target and is not evidence that a real observer is there.

The other two points, in the same unit #

The protan point is the exposed one and is the reason for the exercise. Doing the other two costs nothing extra and is what turns a repair into a check, because the nanometre form has to reproduce the sigma form’s ordering if it is measuring the same thing.

It does. The deuteranope’s point is determined by the L and S pigments, so its restatement is in nanometres of the long-wave pigment; the tritanope’s likewise.

Read the three columns against the sigma table and the agreement is exact in ordering. The old form said every published transform is far outside the population on protan and deutan and inside it on tritan; the new form says the protan and deutan columns need pigment displacements of four to twenty nanometres and the tritan column needs, for four of the five, under six.

That the two forms agree is worth more than either. They share no denominator — one divides by a modelled population’s spread and the other by a measured pigment separation — so their agreeing about which points are far and which are near is evidence that the observation is about the transforms rather than about either scale.

And the residuals separate the three points too. The protan column’s closest approaches leave 0.0001 to 0.0050; the deutan column’s leave 0.0036 to 0.0688, more than ten times worse. So the deuteranope’s implied points are not merely far from the receptors’ own — they are far from the locus any single-pigment shift can reach, which is a stronger statement about the transforms not describing eyes and one the sigma form has no way to make.

What the restatement buys #

It cannot be broken by a declared width. Nothing in the sentence depends on how wide anybody thinks a population is. Scaling all five population widths by a factor of ten leaves the nanometre table bit-identical, because the table is computed on the reference observer.

It is falsifiable by a measurement somebody could make, which is more than the old form’s denominator allowed. If microspectrophotometry established that the M pigment’s peak is really at 549 rather than 541, the table would change and the claim would have to be restated — which is what it means for a claim to be about the world. The old form could be falsified only by somebody disagreeing about a modelling choice.

And it is comparable across the table. Two transforms whose points differ can be compared in nanometres and the difference means something; two transforms whose sigma-distances differ can only be compared inside the same population model.

What it does not buy is a stronger claim. It is the same observation in a different unit, and the observation was already secure — the round before this one found it robust and found the sentence stating it exposed. The whole of the repair is that the sentence now says what the observation says.

Why the old form was written that way #

Not carelessness, and the reason is worth having because it recurs.

A distance in chromaticity is uninterpretable on its own. A protan point off by 0.13 could be somebody else’s eyes, in which case the published transform is describing a real observer, or it could be a hundred times any real variation, in which case it is not describing eyes at all. Dividing by a population’s spread answers exactly that question, and it was the right instrument for the question the earlier round was asking.

The trap is that the instrument’s denominator is a modelling input, so it converts an interpretability problem into a robustness problem. The old sentence was interpretable and fragile; the new one is interpretable and not, and the difference is that its denominator is measured.

The general form is worth stating, and it is the same lesson a set’s own declaration teaches: when a quantity needs a scale, prefer a scale the world fixes to a scale the model provides, even when the model’s scale is the more natural one to compute. The L–M separation is not the obvious denominator for a chromaticity distance — it is not even in the same units — but it is the one nobody here chose.

The other confusion point traces its own locus, and putting the five on that one instead is what tests whether the nanometre axis was the right axis.

What is still exposed #

Two things, and neither is the width.

The transfer construction. The points are computed by moving a member’s basis away from the published one and reading the confusion points off the result, because a pigment template cannot place a point and is out by 0.24 in chromaticity when it tries. That construction is a modelling choice and the nanometre figures depend on it. It is defensible and it is documented and it is not measured.

And the reference peaks themselves. 566 and 541 nanometres are quoted, not derived, and their difference is the scale the whole restatement is against. They are measured quantities with a real literature and an uncertainty of a nanometre or two, which is the honest exposure of the new sentence: it fails if a measurement is wrong, rather than if a modelling choice is unpopular.

That is the exchange the restatement makes. It does not remove the claim’s dependence on anything; it moves the dependence onto quantities that other people measure.

What a specification could say #

The practical form, because the restatement is worth something to somebody writing a document rather than only to somebody auditing this one.

A colour-management specification that names an adaptation transform is choosing between the five, and the usual grounds are institutional — CAT02 because ICC profiles use it, Bradford because the profile-connection space does, CAT16 because it is current. None of those grounds says anything about eyes, and the nanometre table is what a document could say instead: this transform’s implied protan confusion point corresponds to a medium-wave pigment displaced by N nanometres, against a 25 nm L–M separation.

That is a sentence anybody can check with the matrix and no model, and it makes the choice legible. It does not decide the choice — a transform is chosen to make adaptation behave, not to describe receptors, and getting the receptors wrong may be exactly the right trade — but it names the price in a unit that will not move when somebody’s population model does.

The unit generalises past adaptation. Any claim about a receptor-derived quantity can be stated as how far would the pigment have to move, and any such claim so stated is immune to disagreements about how variable people are. The construction is always the same: find which pigment the quantity actually depends on, sweep it, and read off the displacement that reaches the published value.

The table does not rank the transforms #

The closing section proposes the nanometre figure as something a colour-management specification could state, so that a choice between the five becomes legible. It would be, and the three-point table says the statement has to name a deficiency, because the five transforms do not have one ordering.

Ranking them by how near each comes to a real receptor, on each of the three points separately, with 1 nearest:

The rank correlations between columns are +0.60, +0.70 and +0.50 — positive, and nowhere near one. CAT02 is the nearest transform on the protanope’s point and the third furthest on the deuteranope’s; Hunt–Pointer–Estévez is the reverse, best on deutan and third on protan. A document that said this transform’s implied confusion point corresponds to a pigment displaced by N nanometres would therefore have to say which point, and two documents choosing differently would recommend different transforms from the same table.

Only the bottom row is unambiguous, and it is the one nobody needed a table for: a plain XYZ scaling is worst on all three, by a wide margin on each, because it is not a cone basis at all.

That is a limitation of the instrument and not of the observation. The observation — that no published transform is near a receptor on the two long-wave points — survives every column. What does not survive is the idea that the nanometre figure is a single score. It is three scores, and the transforms trade against each other across them, which is what a fit to a different objective would be expected to produce.

The third confusion point can be traced the same way, and it is the one the published transforms are furthest apart about.

The short-wave axis is where they agree #

The three columns differ in more than their ordering, and the difference is the sharpest thing in the table.

Setting aside the XYZ row, the mean displacement needed is 9.4 nanometres on protan, 8.8 on deutan and 2.25 on tritan. The tritan column is four times smaller than either of the others, and its two smallest entries — 0.6 and 0.7 nanometres — are inside the uncertainty on the reference peaks themselves.

So the four real transforms are, on the short-wave point, indistinguishable from receptors, and on the two long-wave points they are four to twenty nanometres away. The disagreement with the eye is concentrated almost entirely in the L and M axes, and the S axis is nearly free of it.

That has a mechanism, and it is not flattering to the transforms. An adaptation transform is fitted against corresponding-colour datasets, and the changes of light those datasets contain — daylight to tungsten, one phase of daylight to another — move a scene mostly along the warm–cool direction, which is carried by the ratio of the two long-wave cones. That is where the data have leverage, and it is exactly where the fitted basis departs from the receptors. The short-wave axis is barely constrained by the fitting, so it stays close to where a cone basis would put it by default.

The transforms are furthest from the receptors precisely where they were fitted hardest, which is the expected shape when a basis is optimised for something other than describing eyes, and it is a stronger version of the essay’s own point about a transform being chosen to make adaptation behave.

The deutan point is worth seeing in both units at once, because it is the point where the old statement and the restated one disagree most about how large the discrepancy is.

What the repair still declares #

One honest residue, because the essay’s case is that the new sentence depends on nothing the author chose.

For each of the three points there are two remaining pigments, and the restatement sweeps one of them. The protan case is argued explicitly — the L pigment does not move the protan point at all, so M is the only axis with content — and that argument is about the missing pigment rather than about the two that remain, so it does not carry over. A protanope retains M and S; the essay sweeps M and notes separately that letting S move as well closes most of the residual. The deutan and tritan columns are quoted in L with no argument given for why not S and M respectively.

The choice of sweep axis is a modelling decision, and it is the one thing the repair swapped in rather than out. It is a much smaller decision than a declared population width — a nanometre figure in the other axis would be a different number about the same disagreement, not a different verdict — and the span across the table, 28.7 nanometres or 115 per cent of the L–M separation, is a difference between rows and cancels the axis choice entirely. That last number is the one to quote if a single figure is wanted, because it is the only one in the essay that no modelling decision touches.

Where the model stops #

The restatement is about the protan point, which is the exposed one. The deutan point’s claim has more headroom and would restate the same way, in nanometres of the L pigment, and has not been done here. The tritan claim is the one that says the published transforms are inside the population, and it is a claim in the opposite direction that a nanometre form would state as no plausible pigment shift is needed, which is true and is a weaker sentence.

And nothing in the restatement says the published transforms are wrong for their purpose. They are fitted to make chromatic adaptation behave, not to describe receptors, and this collection has measured what that imposition costs at seventy per cent on one objective. The nanometre table is a measurement of how far from receptors that fitting takes them, in a unit that says so without needing a population to be agreed on first.

Who found it, and when #

The confusion points are old measurements — Pitt’s, Judd’s, and the values quoted here are the ones colour science has carried since — and the fact that adaptation transforms imply confusion points is the fundamental-primary construction, which is König’s.

What is new is the unit. The observation had been in this collection for four rounds in the sigma form; the round before this one measured its headroom at 1.34 and named the repair in one sentence — a restatement in a quantity that does not scale with the population’s spread is an afternoon’s work and would close the round’s own most exposed row. It was, and it did, and the part that was not anticipated is that the axis would be the pigment nobody was talking about.

Where the ladder goes next #

The five transforms disagree with one another by more than the separation between the two pigments they disagree about, which is a fact about the table rather than about any row in it.