The grid hid the observer

Two audits in one round, run on the same six lights and the same forty-two surfaces, so that they could be crossed. The crossing produced the round’s sharpest single number and it is a zero.

The claim #

A tabulation coarse enough to miss a light’s emission lines does not add to an observer departure; it removes it, and the removal is total.

• On the site’s grid the laser projector’s six departures are all exactly zero. On a quarter-nanometre grid they are 1.28, 2.74, 3.78, 1.04, 3.60 and 1.57 ΔE₀₀ — the largest peak departure anywhere in the round.
• The mechanism is not an approximation. On a five-nanometre grid the projector is a one-line spectrum, and a single wavelength is a stimulus every observer agrees about exactly.
• Five of the six lights are unaffected, agreeing between the two grids to within 0.04 ΔE₀₀, which is what a well-sampled calculation looks like.
• And the concealment is silent. Nothing about the coarse table’s appearance distinguishes the row that is a physical result from the row that is an artefact of arithmetic.

Two audits, one set of stimuli #

The first half of this round audited the index the colour integral is summed over and the second half audits the observer it is summed against. Both were computed on the same six analytic lights and the same forty-two analytic surfaces, and that was a deliberate choice made before either had produced a result.

The reason is that departures are only comparable when the things they depart from are the same. The previous round put four departures of the sample on one footing and found that doing so was most of the work; putting two whole audits on one footing is the same discipline one level up.

What it buys is the ability to ask whether two departures compound. Two errors in one calculation might add, might cancel, or might be independent, and none of those is predictable from either alone. The previous round asked that question of its own four and found two that partly cancel by more than a hundred per cent of the smaller.

Here the answer is neither. One departure does not add to the other and does not cancel against it. It conceals it, which is a third relation and the one nobody looks for.

What the coarse grid does to the projector #

The three-laser projector emits at 465, 532 and 638 nanometres with lines a fifth of a nanometre wide. The site’s grid runs 380, 385, 390 and so on, so 465 is a grid point and 532 and 638 are not.

Evaluated at the grid points, the spectrum is a single line at 465 nanometres and zeros everywhere else. That is not a poor approximation to three lines; it is a different stimulus, and it happens to be one with a very strong property.

A single-wavelength stimulus is the same colour for every observer. The sample’s cone responses are R(465)·lᵢ(465) and the white’s are lᵢ(465), so the relative excitations are (R, R, R) for any three sensitivities whatever. That is the neutral identity arriving by a different route: a monochromatic stimulus is a scalar multiple of its own white, exactly as a flat reflectance is.

So the zero is real arithmetic about a real property. It is a fact about a one-line spectrum, and the coarse grid has turned a three-line spectrum into one.

This is the interesting kind of error. Errors that make a number too large are irritating and errors that make it too small are dangerous, and this one is worse than either because it makes the number exactly right for a different question.

The coarse answer is not wrong about anything it computes. It is the correct observer departure for a monochromatic source at 465 nanometres, computed to the floating-point floor, with every identity in this round holding exactly as it should. No assertion fails. No convergence check complains, because there is nothing to converge to — the answer is stable under any refinement that does not cross the linewidth.

And the row it produces looks like a result. A zero in a table of numbers reads as these observers agree here, which is a strong and interesting claim about laser projection, and it is the opposite of what the fine calculation says.

The only way to catch it is to compute the same thing at a resolution the grid does not have, which is the one operation the coarse calculation cannot perform on itself. That is the same obstacle the tabulation audit had to build its way around and the same solution: an analytic observer and an analytic light, evaluable anywhere.

What the fine answer says instead #

On a quarter-nanometre grid the projector becomes the light it is, and the six departures on a red pigment read:

The peaks entry is the largest of any light-and-departure pair in the whole round, and that is not a surprise once stated: three narrow lines sample the cone sensitivities at three points, and moving a pigment’s peak by three nanometres moves what each line contributes by a great deal more than it moves a broad integral.

This is the mechanism behind a phenomenon the field already knows about. Narrow primaries make observer metamerism worse, and the reason is exactly this: an observer difference is a difference between curves, and a narrowband source samples the curves rather than integrating over them. What is new is the number, and that the collection’s own arithmetic had been unable to produce it.

A chart of six zeros is an unusual thing to publish and it is here because the alternative is worse. Removing the row would leave the round with five lights and no record of the failure; drawing it without comment would leave a reader with a claim about laser projection that is the reverse of true. Drawing it beside its fine-grid twin is the only arrangement that reports what happened.

It also makes a point about assertions that this collection keeps rediscovering. Every gate in the figure family passed on that chart. The identities held, the conditions held, the pairing held, and the arithmetic was exact throughout — because all of them are statements about the stimulus the calculation had, and the calculation had a perfectly well-behaved monochromatic one. An assertion checks the arithmetic against the object it was given, and no assertion in this collection asks whether the object is the one that was meant.

The other five rows #

Five of the six lights agree between the two grids to two decimal places or better: the tungsten lamp at 2.373 against 2.374, daylight at 1.766 against 1.760, the white LED at 1.833 against 1.833, the three-emitter LED at 2.075 against 2.075, and the fluorescent tube at 1.635 against 1.667.

That agreement is worth as much as the disagreement, for two reasons.

It says the observer audit’s other results are safe. Every ladder, distribution and comparison in this round is computed on the site’s grid, and five of six lights are indistinguishable from their fine-grid answers — so nothing else in the round is contaminated.

And it draws the boundary of the effect sharply. The fluorescent tube’s lines are a nanometre wide and its two answers differ by 0.03 ΔE₀₀, which is nothing; the projector’s are a fifth of a nanometre and its answers differ by everything. The concealment is not gradual. A grid either catches enough of a light’s structure to represent it or it does not, and between those two states there is very little.

What this collection owes as a result #

Three consequences, all of them recorded rather than repaired.

Any absolute colour quoted here for a laser primary is a point-sampled colour at whichever of its lines happens to land on a grid point. That was already true from the tabulation audit; what is new is that it reaches the observer results too, and there it presents as a zero rather than as a wrong number.

The narrowband-display work in this collection is computed on the coarse grid. Its lamps are constructed with emitters of ten to thirty nanometres’ width, which is comfortably resolved, so those results stand. Anything narrower than about two nanometres does not, and the collection has no figure at that width except the projector constructed for this round.

And the observer audit’s laser row should not be quoted. It is in the table because removing it would conceal the finding, and it is drawn beside its fine-grid twin for the same reason. That is the same treatment the shortfall queue gives to anything this collection knows to be wrong and has not repaired.

The general relation nobody looks for #

Two errors in one calculation can add, cancel or be independent, and the literature on error budgets covers all three. This is a fourth relation and it does not appear in that literature at all.

Concealment is when one error changes the object so that a second error has nothing to act on. It is not cancellation, because nothing is subtracted; the second error’s cause is removed rather than its effect. And it is not independence, because the two are strongly coupled — the concealed quantity is a function of the concealing one.

The signature is a zero where a small number was expected, and the diagnosis is to remove the first error and see whether the second appears. That is a cheap test and it is not standard practice, because a zero is normally read as good news.

There is a version of this in the previous round’s findings that was not recognised as one at the time. Its measurement of what a translucent sample’s aperture costs came out exactly zero on an opaque sample, and it was reported as a condition rather than as a concealment — correctly, because there the opacity was the physics rather than an artefact. The two situations are algebraically identical and differ only in whether the object was changed by the world or by the arithmetic.

That figure is included as a reassurance rather than as a result, and the reassurance is the useful part of an audit that finds a fault. Knowing precisely which of a body of work is affected is worth as much as knowing that something is, and here the boundary is sharp: any figure whose light has a feature narrower than about two nanometres is suspect and every other figure is not.

By that criterion the round’s own laser row is the only affected figure in this collection. Its narrowband display work uses emitters of ten nanometres and upwards, its discharge lamps have lines of a nanometre and are affected at the level of 0.03 ΔE₀₀, and everything else is smooth. One figure, identified, drawn twice, and recorded — which is the outcome an audit should produce and is a good deal better than a general warning.

What was computed, and how #

The fine grid is a quarter of a nanometre from 380 to 780, which is 1,601 points and is comfortably finer than the tabulation audit’s own tenth-nanometre reference is coarse. It is not the same grid, and the difference is deliberate: an observer built on a quarter-nanometre grid costs about twenty times a five-nanometre one and the audit runs six observers per light per departure.

Both grids build their observers from the same analytic template, and the normalising peak of each pigment is read off the site’s own grid whatever the caller asks for — so a change of grid samples the pigment rather than rescaling it. Without that, a fine-grid observer would be a slightly different observer and the comparison would measure two things at once.

The assertion the figure carries is that the laser row is below 10⁻⁹ on the coarse grid and above 1 on the fine one, which is a claim that the concealment is total rather than partial. A partial concealment would be a much less interesting result and would fail the same gate.

One further consequence belongs to anybody who has ever written a numerical audit and been reassured by its agreement with itself. The coarse calculation here agrees with a coarse calculation at a different step, agrees with its own assertions, agrees with every identity in the round, and is internally consistent to the last bit. It is consistent with everything except the world.

That is the shape of failure a self-consistent system has, and there is no internal test for it. A calculation cannot check whether the object it was handed is the object that was meant, and the only remedy is an external comparison — a finer grid, a different method, a measurement — which is precisely the thing an internally consistent system makes look unnecessary.

Where the model stops #

The projector is a construction with three Gaussian lines of a fifth of a nanometre. Real laser projectors use diode and diode-pumped sources with linewidths from picometres to a nanometre depending on the technology, and a real speckle-reduced projector is deliberately broadened. So the extreme case here is more extreme than most hardware.

The grid offset is the site’s own, beginning at 380. A different offset would catch a different subset of the three lines, and the spread across offsets is thirty-five colour differences — so the coarse row is not even stable, it is one arbitrary member of a family of arbitrary answers, one of which happens to be a clean zero.

And the fine-grid numbers carry the model residual in full, because they are absolute rather than comparative. The 3.78 for the macular departure is what this construction gives; a different template would give a different number, though not a different sign or order of magnitude.

The generalisation #

The habit is about what to do with an unexpectedly clean result.

A zero, an exact agreement or a suspiciously round number is usually one of three things: a real identity, a bug, or a degeneracy in the setup. All three look identical in output, and the discipline that separates them is to ask what would have to be true for the clean result to be real, and then to check that thing directly.

Here the clean result was real of the object being computed and the object was not the one intended. That is the hardest of the three to catch, because both the arithmetic and the identity are correct and only the correspondence between the calculation and the world has failed.

The test that catches it is to vary something the answer should not depend on. A colour should not depend on the grid it was summed over, so summing it over a different grid is a test that costs nothing and is almost never run — for the same reason nobody re-runs a calculation at a different resolution when it already agrees with itself.

Who found it, and when #

Observer metamerism’s dependence on primary bandwidth is well established and the display industry has measured it directly; the CIE published a technical report on the assessment of it in 2016, prompted by exactly the shift to narrow primaries this essay is about.

What is not in that literature is the interaction with tabulation, and the reason is that the literature computes from measured spectra, which arrive band-integrated through an instrument’s slit. A slit is what makes a coarse table legal, and a discipline whose data always come through one has no occasion to discover what happens when they do not.

Where the ladder goes next #

The identity underneath the zero is worth an essay of its own, because it has a consequence that runs the opposite way from the concealment. A single wavelength is everyone’s colour — and a display made of three of them is where observers disagree most.