Colour Science & Perception
what the eye settles, and what it cannot
Series

Cones — the series

73 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. A spectrum, weighted three ways, and the three numbers left over. The illuminant D65 above; below, the same spectrum multiplied by each matching function. The area under each product is one coordinate of XYZ. Everything else about the spectrum — its shape, its structure, all its remaining degrees of freedom — is discarded here.

    Three numbers

    A spectrum has as many degrees of freedom as anyone cares to give it. The eye reports three. Everything colour science can do, and every way it fails, follows from that one collapse.

    part 1 · eye
  2. Two different spectra that are the same colour. Two reflectance curves differing by 92 per cent RMS, and the two patches they produce under D65: identical to ΔE00 = 6.2e-14, which is arithmetic noise rather than a small number. Both patches are inside the sRGB gamut, so neither has been clipped into agreement.

    Two spectra, one colour

    Metamerism is usually described and almost never demonstrated. It does not have to be — the metameric black space is enormous, so a matching pair can be constructed to order, verified, and then made to come apart by changing the light.

    part 2 · eye
  3. The CIE 1931 colour-matching functions. The three functions that turn a spectrum into three numbers. They are all positive, which is why XYZ exists — the RGB functions they were derived from are not. ȳ is by construction the luminous efficiency function, which is why luminance comes out of Y.

    Why colour is exactly three-dimensional

    Matching every wavelength with three primaries requires, for some wavelengths, a negative amount of one of them. That physical awkwardness is why the colour-matching functions were transformed into XYZ, and why the horseshoe is curved.

    part 2 · eye
  4. The same two surfaces, weighed by each system. A long-wavelength and a short-wavelength surface under D65, with their relative luminance under the photopic curve and under the scotopic one. Under daylight vision the red surface is 1.33 times the blue; under rod vision it is 0.15 times, a reversal by a factor of 9.0. The swatches are the photopic appearance, which is the only one a display can produce: rod vision has no colour, and drawing a guess at it would be an invention.

    The eye that has no colour

    Rods outnumber cones twenty to one, work alone below a hundredth of a candela, and are absent from the centre of gaze. Between dusk and a lit room both systems run at once, and neither standard curve describes what is happening.

    part 2 · eye
  5. One palette under normal vision and two dichromacies. The same 7 colours simulated by the Brettel–Viénot–Mollon construction at severity 1.0. protanopia and deuteranopia collapse the red-green distinctions. This shows which discriminations survive, not what anybody sees.

    Three cones, two axes

    The retina does not send three receptor signals down the optic nerve. It sends a sum and two differences, and the reason falls out of the statistics of natural light rather than out of anything the eye intended.

    part 3 · eye
  6. Contrast sensitivity, three channels, normalised to each channel's peak. Spatial frequency in cycles per degree against relative sensitivity. The luminance channel runs to 50 cycles per degree, red–green to 12 and blue–yellow to 8, each by the same criterion of 5 per cent of that channel's own peak. Only the luminance channel dips at low frequency; the two chromatic ones are low-pass, so a chromatic edge of any size at all is seen at its full contrast.

    How fine a colour edge can be

    The eye resolves a lightness pattern to about fifty cycles per degree and a red–green one to twelve. Every colour difference here is quoted as though a patch had no size, and the same difference is plainly visible at one scale and gone at another.

    part 3 · eye
  7. Temporal sensitivity, and where each channel gives out. Modulation frequency in hertz against relative sensitivity. The luminance channel is band-pass, peaking at 8 hertz and running out at 60; an isoluminant modulation is low-pass and runs out at 15, which is 4.0 times sooner. Both cutoffs are at the same criterion of 5 per cent of that channel's own peak, so the ratio between them is a ratio between two measurements rather than between two conventions.

    The eye has a shutter

    An isoluminant flicker fuses at fifteen hertz and a luminance one at sixty, so a light whose colour changes forty times a second is a steady light of a colour it never emits. And the frequency at which flicker stops being visible is not a property of the eye — it moves twelve and a half hertz for every decade of light.

    part 3 · eye
  8. Longitudinal chromatic aberration of the eye, focused at 555 nm. Thibos's chromatic eye, plotted as defocus in dioptres relative to 555 nm. The curve crosses zero exactly once, at the wavelength the eye is accommodating on, and everything else is out of focus — by -1.22 D at 420 nm and 0.50 D at 680 nm. Through a 3 mm pupil that first figure is a blur circle of 12.6 minutes of arc, against a foveal acuity limit of about one. The eye is never in focus across the spectrum and never can be.

    Nothing is in focus at both ends

    The eye carries about two dioptres of chromatic aberration across the visible band, which is a strong reading prescription. Whatever it is focused on, most of the spectrum is landing somewhere other than the retina — and the cone class that gets the worst of it is the one the retina bothered least to sample.

    part 4 · eye
  9. The L cone's sensitivity at three axial pigment densities. Each curve is normalised to its own peak, so the only difference visible is shape. Raising the density from 0.1 to 0.9 widens the curve from 113.6 to 145.9 nm at half height while the peak stays within 5 nm of where it was. That is Beer–Lambert saturating: at the peak the pigment already absorbs nearly everything, so more of it can only catch more light in the wings.

    A cone absorbs its own light

    A photopigment's absorbance is a property of a molecule; a cone's sensitivity is that molecule stacked in a column deep enough to absorb most of what arrives. The stacking broadens the curve by thirty-five nanometres, and two observers differing in nothing else disagree about a match that is exact for one of them.

    part 4 · eye
  10. A screen at 45°, 8 c/°, and where its energy sits. Left, the pattern. Right, its power in the frequency plane with the zero frequency at the centre and the edges at the sampling limit of 23 cycles per degree, on a logarithmic scale over five decades. The closed curves are the visual system's own sensitivity at 5, 25, 60 per cent of its peak; they are not circles, because sensitivity is lower on the diagonals than on the cardinal axes by a factor of 2.0 at high frequency. Energy inside a curve is seen; energy outside it is not, whatever its size.

    A pattern has a direction

    Every spatial claim here is a claim about a frequency, and a frequency has no direction in it. Turning a printed screen forty-five degrees makes it exactly twice as quiet with nothing else changed — and the same rotation does nothing at all to a chromatic one.

    part 4 · eye
  11. Cone density, out from the centre of gaze. Quoted landmarks with logarithmic interpolation between them: 199,000 cones per square millimetre at the fovea, 9,500 at ten degrees — a factor of 21. The shaded bands are the two standard observers' fields. The 2° observer averages over a region whose density falls by 3.3× between its centre and its edge and the 10° observer by 12.4×, which is what "a 10° field" contains and is why the two sets of matching functions are different shapes rather than the same shape scaled.

    Colour stops at the edge of sight

    Cone density falls twenty-one-fold between the centre of gaze and ten degrees out, and the three channels give out at three different rates — so a colour difference in the periphery does not merely shrink, it turns. At the exact point of fixation there are no short-wavelength cones at all.

    part 4 · eye
  12. Three retinas of different composition, making identical colour matches. Seeded mosaics at L:M ratios of 1, 4, 16 to one — the range found between people, and a 16-fold difference in what the retina is made of. A colour match is the claim that two spectra produce equal excitation in all three cone classes, and changing how many of a class there are multiplies that class's excitation by a constant, which cannot disturb an equality. The relation between the two test spectra is identical to twelve decimal places across the whole sweep. Luminance, which is a weighted sum rather than an equality, moves by 1.33× over the same range. That asymmetry is why a standard observer exists and why V(λ) has a much larger between-observer variance than the colour-matching functions do.

    The mosaic is not the observer

    The ratio of long-wavelength to medium-wavelength cones varies between ordinary people by a factor of sixteen. Those people make the same colour matches, to twelve decimal places, and that single fact is the reason a standard observer can exist at all.

    part 5 · eye
  13. Photons caught by one cone in one integration time, under D65. Each curve is one cone class, counting isomerisations during a 0.1 s integration through a pupil that closes as the light rises. At 1000 cd/m² a long-wavelength cone catches about 11522 and at 0.001 it catches 0.12 — which is where the square root of the count stops being a small correction and starts being the signal. The rate works out at 30 isomerisations per second per troland, inside the published band of 5–50.

    Colour goes first in the dark

    A cone reports a count, and a count carries the square root of itself as noise. Counting the photons says where colour vision stops — a chromatic difference runs out four hundred times sooner than a lightness difference of the same size — and says just as clearly that in daylight the eye is nowhere near that limit.

    part 5 · eye
  14. The drift window, asked about each channel in turn. Every spatial frequency a channel can resolve, drifting at v degrees a second, arrives at f × v hertz; the bar is the range of v over which all of them stay above a quarter of that channel's temporal peak. The luminance band is closed at both ends — 0.018 to 0.71 degrees a second — because its temporal sensitivity has a dip at zero to fall into. The chromatic bands have no slow edge at all, because chromatic temporal sensitivity is low-pass: a stationary chromatic pattern sits at the top of its own sensitivity. The mark is the measured drift, and it is inside all three.

    The drift is a luminance mechanism

    The eye's own drift was shown to sit inside a band of speeds that keeps every spatial frequency modulating, and the band was quoted as though it were about vision. Asked about colour, it has no slow edge at all — a stationary chromatic pattern needs no eye movement whatever. And a stabilised chromatic pattern is the first thing to fade.

    part 5 · eye
  15. Coming back from a bleach, against the clock already measured. A 94 per cent bleach, and the pigment returning at its own time constant of 120 seconds. The lower curve is the site's slow neural adaptation constant, 60 seconds, started from the same place — it is finished while the chemistry is barely half done. Regeneration does not speed up because the light went away: the rate constant is the same one it always was, which is why the recovery is slow while the bleaching was fast.

    The slowest clock is chemical

    An earlier essay here joined the afterimage to the adaptation clock and named what was still missing — a third gain, upstream of both, in the pigment itself. It is twice as slow as anything measured before it, it leaves a coloured after-tint from a white field, and at steady state it cancels exactly, which is why nobody has ever needed to model it.

    part 5 · eye
  16. Where a pooled gain gives out, against where the eye does. The falling curve is how much of a pattern of each spatial frequency a local adaptation pool of 0.5° can see — and therefore how much of it a settled eye can cancel. It is half gone by 0.37 cycles per degree, which is a feature about 2.7° across. The three marks are the acuity limits of the luminance channel and the two chromatic ones. Every one of them is more than an order of magnitude finer than the pool, which is why a stabilised eye loses the fill of a picture and keeps its outline rather than losing the picture.

    What a still eye stops seeing

    A stabilised image is said to vanish, and nothing in a temporal filter predicts it — sensitivity at zero frequency is a quarter of the peak, not nothing. Give the adaptation gain a size and the answer falls out — fading is a high-pass filter that switches on over a minute, it takes the fill and leaves the outline, and a patch has to be about two degrees across before it goes at all.

    part 6 · eye
  17. What the same eye reports about one field, in the middle and at the edge. Each row is a uniform field, drawn at the most saturated version of itself this page can show — the percentage is how much of the full stimulus survived, the rest being the adapting light added to bring it inside the gamut. The left patch is what the centre of gaze reports and the right one what 10 degrees out reports, each adapted to the same light as that position sees it. The adapting white comes out identical to 5e-13, because an adapted eye cancels its own filter exactly. Nothing else does, and the largest difference is in the blue. tungsten light is not drawn: it cannot be shown at any useful saturation, and at full strength it differs by ΔE00 1.87.

    One person is two observers

    The macular pigment is a yellow screen over the fovea and nowhere else, so a cone at the centre of gaze and a cone ten degrees away have different colour-matching functions in the same eye. A match made in the middle comes apart at the edge by six units — and fitting one filter to the gap between the CIE's two standard observers gives a density of 0.40 against a measured 0.35.

    part 6 · eye
  18. The share of itself each change leaves behind, and the smallest is inside the eye. The residual as a fraction of the change rather than as a colour difference, which sorts the census differently. At the top is the macular pigment — the filter in front of the central few degrees of one's own retina — leaving 2.4 per cent of itself. It is a fixed transmittance multiplying the light and the white together, which is as close to a pure gain as anything here gets, and it is why nobody notices they have one.

    The filters inside the eye

    The macular pigment leaves 2.4 per cent of itself after adaptation — the smallest share of anything in this site's census of light changes, and less than half the next smallest. Fifty years of lens yellowing leaves 7.9 per cent, and the difference between the two says what a gain is actually good at.

    part 7 · eye
  19. A stabilised grating fades at a rate that depends on which way its bars run. What is left of a 0.3 cycle-per-degree grating, in multiples of its own threshold, as a function of its orientation, at four moments after the image was stabilised on the retina. The first curve is flat to floating point: at the instant the pattern arrives, orientation does not matter, because the filter that carries it has no orientation preference at this frequency. The anisotropy arrives with the fading, reaching ×2.09 after five minutes, and a pool with no axis predicts none of it.

    A fading pool has a shape

    Giving the local adaptation pool two axes instead of one costs a single parameter and produces a prediction the circular version cannot make — a stabilised grating fades at a rate that depends on which way its bars run. The obvious objection is the oblique effect, and the two act in bands that do not overlap.

    part 7 · eye
  20. A radiance factor, split into the part that was reflected and the part that was not. The two components of what leaves a heavily brightened sheet under M₁ — D50 including its ultraviolet. The lower band is the reflected component, which is what a reflectance curve means and is everything a reflectance-based model can hold. The band above it is light emitted at wavelengths it did not arrive at, and its total is decided by how much ultraviolet the source had rather than by anything about the sheet's colour. The line at one is the boundary a reflecting surface cannot cross; the sum reaches 1.21 at 430 nanometres.

    The eye weights where the light is not

    A brightened sheet returns a quarter more light than arrives at 430 nanometres, and it is three tenths of one per cent brighter for it. The luminous efficiency function is 0.017 there against 1.0 in the middle of the band, so the whole effect lands in the blue-yellow axis — brighter than white is a colour claim wearing a brightness word.

    part 7 · matching
  21. What reaches the retina, and why the observer's table stops at 360 nanometres. The transmittance of the eye's own optics across the short-wave band, at three ages, with the brightener's absorption shaded underneath. The upper curve is an eye whose lens has been removed — the cornea alone, opaque below about 295 nanometres and transparent above it. The photopigments absorb perfectly well in this band; what stops the light is a piece of optics in front of them, which is why the short-wave limit of colour vision moves with age and can be removed surgically. A twenty-year-old receives 21 times as much of the band a brightener works in as a seventy-year-old does.

    The eye stops at the lens

    Neither standard observer is tabulated below 360 nanometres, and the reason is not that the photopigments stop absorbing there. It is that the light never arrives — the cornea and the crystalline lens take it — so the short-wave limit of human colour vision is a piece of optics, it moves by a factor of twenty across a lifetime, and it can be surgically removed.

    part 8 · eye
  22. One observer's matching functions, in three of the bases the matches leave free. The three colour-matching functions after a change of basis 0.00 of the way from Hunt–Pointer–Estévez towards the set built from the dichromat confusion points. Every one of these triples predicts exactly the same matches as every other, because a match is an equality and a matrix applied to both sides of an equality changes nothing. What moves is where the peaks are and whether the curves go negative — these ones do not, and going negative is what the 1931 committee constructed XYZ to avoid.

    The matches do not name the cones

    Colour matching is the whole empirical basis of colorimetry, and it fixes the observer's three curves only up to a nonsingular 3×3 — nine numbers that no match, in any quantity, to any precision, can see. One particular choice of those nine is used throughout here, and it was made for a different purpose.

    part 9 · eye
  23. Where a dichromat's confusions converge. Every pair of colours a protanope cannot tell apart lies on one of these lines, and all the lines meet at a single point — at (0.7465, 0.2535) for this class. The point is the chromaticity of the missing cone's own response direction, which is why it need not lie inside the diagram or correspond to any light at all. Two of the three do not. The three points between them carry six numbers, and six is two thirds of what the matching data leave undetermined.

    A confusion point is a missing pigment

    The nine numbers colour matching leaves free are fixed by three points on a chromaticity diagram, each of them the place where everything one class of dichromat cannot tell apart converges. Two of the three lie outside the diagram entirely, which is not a defect — a direction in tristimulus space need not correspond to a light.

    part 9 · eye
  24. Every width at the wide end of its span, and at the narrow end. One line per published quantity, each spanning the value it takes when all four declared widths are read at the narrow end of their reported ranges to the value at the wide end, with a marker at the value as declared. The largest span is the deutan margin at a factor of 2.62; the smallest is 1.22. This is the reading the population model's own documentation promised for four phases and nothing ever took. It is not a confidence interval — the four ends are not quantiles and the widths are not independent draws — it is what a reader who distrusts all four at once sees.

    A width nobody varied

    Five numbers say how much people differ from one another, and every conclusion drawn here about a population rests on them. Each was written down with the range the literature reports beside it, so that a result could be re-read at the pessimistic end. Nothing ever was.

    part 9 · eye
  25. Four different bases, one adaptation model, one number. The middle row of the basis built from the confusion points multiplied by 0.21, 1, 3.7 and 11 in turn, with the resulting adaptation residual drawn as a bar in each case. The four bars are the same height to 9e-16 of a ΔE00, because the row's scale cancels exactly between the gain and the inverse. Three of the nine numbers a colour match leaves free are invisible to an adaptation model, which is why the six the dichromat data supply determine it outright with nothing left to fit.

    The three numbers a gain cannot see

    Colour matching leaves nine numbers free. Three dichromat confusion points fix six of them and three choices of unit fix the rest — and it turns out that a von Kries gain is exactly blind to those last three. So the dichromat data do not merely constrain an adaptation basis. They determine it, with nothing left over to fit.

    part 10 · eye
  26. How cone-like a basis is, against how well it adapts. Each basis placed by how far its own implied deuteranope confusion point falls from the measured one (horizontal) and by how much an adapted observer is left with in it (vertical). The construction from the confusion points sits at zero on the horizontal by definition and near the top on the vertical. Nothing near the left of the picture is near the bottom: the closer a basis is to the receptors, the more a von Kries gain leaves behind. The unconstrained winner sits at 1.63 on the horizontal, further from the measurement than any published transform except CAT02 and Bradford.

    The best axes are not receptors

    If the axes that make a von Kries gain work were nearly the cone fundamentals, the published adaptation transforms would be cone-like and their departures would be slack in a fit. They are not. Minimise the residual over all nine free numbers and the winner sits further from the measured dichromat confusion points than any of them.

    part 10 · eye
  27. Which width carries the answer, and which carries the doubt. Two columns of bars over the four things that differ between two pairs of eyes. On the left, the share of the population's disagreement each one accounts for — the attribution quoted here since the population was built, which puts the lens first at 81%. On the right, how much of the doubt each one puts on everything published here, which is its elasticity multiplied by how badly the width itself is known. The macular pigment comes first there, at 0.65 against the lens's 0.60 — a lead of 8%. The two lists agree exactly below the top.

    Which measurement is worth making

    Four things about an eye differ between people, and they have been ranked here by how much of the answers they carry since the population was built. Ranking them by how much doubt they carry gives a different order, and ranking them by which one takes a published claim closest to failing gives a third.

    part 10 · eye
  28. Nine eigenvalues, six of which exist. Nine points on a logarithmic vertical axis: the eigenvalues of the Hessian of the adaptation residual at its own optimum, largest to smallest. The first six run from 6.8×10² down to 7.6×10⁻¹, a condition number of 890. Then the axis drops: the seventh is 1.9×10⁻⁴, and the last three are separated from the sixth by a factor of 4.0×10³. Those three are not small curvatures. They are the finite-difference truncation error on directions along which the objective is exactly constant, and a shaded band marks them as the numbers the objective does not have.

    The rank is the invariance

    A von Kries gain cannot see the scale of a row of its basis. That is an identity, proved in a line, and it can be measured instead — as the rank of a second-derivative matrix. Both objectives this collection minimises over the observer's nine free numbers have a Hessian of rank exactly six, and the three directions they cannot see are the three scalings, to a hundredth of a degree.

    part 11 · eye
  29. A template that cannot place a point, fitted three ways. Three rows, one per set of stimuli the pigment template's cone matrix can be fitted over, each listing the three confusion points that matrix implies. The protanope's point wanders from (0.99, 0.20) to (0.76, 0.13) against a measured (0.75, 0.25), and the deuteranope's moves by 22.5 in chromaticity — further than the whole diagram is wide. A copunctal point is where two nearly parallel planes meet, so a template good to a few per cent, which is far more than enough to place a spectrum, is nowhere near enough to place this. It is why the population is built by moving the measured points rather than by deriving them.

    A template cannot place a point

    This collection's model of an eye is good to a few tenths of a per cent at predicting what a cone catches, which is far more than enough to place a spectrum. Asked where that eye's confusion points are, it puts the protanope's at (0.99, 0.20) against a measured (0.75, 0.25) and the deuteranope's anywhere from (1.1, −0.5) to (−18, 11) depending on which stimuli the fit was made over.

    part 11 · eye
  30. No one surface carries the answer, and the set is smaller than it looks. A falling bar chart of the 125 surfaces in the test set, ordered by how much each contributes to the published mean for daylight to tungsten. The tallest bar is 1.50 per cent of the total, so the mean is not a few awkward objects with a crowd behind them and a leave-one-out would move it by well under a per cent. The tail is the other half of the story: 5 surfaces contribute essentially nothing, because a flat grey is a surface an adaptation gain handles exactly. Counting the set by how evenly it contributes rather than by how many members it has gives 108.5 effective surfaces out of 125, which is what "a mean over a hundred and twenty-five surfaces" is really worth.

    The surfaces that answer nothing

    Five of the hundred and twenty-five test surfaces contribute exactly zero to every number the adaptation census reports — not approximately, exactly — and the reason is the one fact about von Kries adaptation that makes it worth having at all. Counting the set by how much it contributes gives about a hundred members rather than a hundred and twenty-five.

    part 11 · eye
  31. A confusion point is about the pigments that remain. Three groups of three bars. Each group is one dichromat's confusion point; each bar is how far that point moves in chromaticity when one of the three cone pigments has its absorption peak shifted by eight nanometres. In every group the bar for the pigment that dichromat is missing has length zero — exactly zero, to machine precision, not merely small. The protanope's point does not move when the L pigment moves, the deuteranope's does not move when the M pigment moves, and the tritanope's does not move when the S pigment moves. The reason is algebraic rather than physiological: a confusion point is the direction that excites only the missing cone, which is the null space of the other two receptors' rows, and rescaling a row does not move where the other two are zero. So the point at which a protanope's confusion lines meet is not a fact about the pigment a protanope lacks, which is why the claim about it restates in nanometres of the M pigment.

    A point about the pigments that remain

    The chromaticity at which a protanope's confusion lines meet does not move at all when the long-wave pigment moves — not slightly, exactly not at all. It moves a great deal when the medium-wave pigment does. A dichromat's confusion point is a fact about the two receptors they have rather than about the one they lack.

    part 12 · eye
  32. The same claim in nanometres of pigment, where no declared width can reach it. Five horizontal bars on a scale of nanometres, one per published chromatic-adaptation transform, each showing how far the medium-wave cone pigment's absorption peak would have to move for the receptors' own protan confusion point to land where that transform puts it. Zero is the measured peak. The bars run from -10.5 to 18.2 nanometres — in both directions, so two of the transforms want the pigment shorter and two want it longer. Drawn across them is the 25 nm separation between the L and M pigment peaks, which is the whole basis of red-green vision and is not a number this collection declared. The nearest transform asks for a displacement of 30 per cent of that separation, and the span across the table is 28.7 nanometres — larger than the separation itself. No population, cloud or standard deviation appears anywhere in the statement.

    The claim, in nanometres

    For four rounds the claim here has been that every published adaptation transform puts the protanope's confusion point outside any real population of eyes, stated in standard deviations of a population whose widths were declared rather than measured. Restated as a pigment displacement it needs no population at all — and the nearest transform asks the medium-wave cone to move thirty per cent of the way to the long-wave one.

    part 12 · eye
  33. How far this site's median observer sits from the 1931 standard, by template. Twenty-four natural reflectances under D65, each given a tristimulus value twice: once by the 1931 colour-matching functions and once by this site's median member, with each judged against its own white. The bar is the mean difference, which is the residual this collection bounds and calls inescapable. It is inescapable, and it is smallest for the simpler template: Lamb's 1995 nomogram gives 0.9006 against Govardovskii's 0.9522, and removing Govardovskii's secondary band brings it down again to 0.9354. Neither is an argument for changing template — a nomogram is fitted to measurements of individual receptors, not to colour matches, so agreement with the standard observer is not what either was trying to achieve. What it says is that the residual is a mismatch between two kinds of observer rather than a shortfall a better pigment model would close. The number after each bar is the template's tail ratio: how far the L cone's half-maximum reaches below its peak against how far it reaches above.

    The population rests on a template

    Two hundred observers here are built from one formula fitted to microspectrophotometry in 2000. The obvious alternative — the tabulated cone fundamentals — is not available, and the reason is the finding. A tabulated fundamental has no peak wavelength to move, so the moment it is used the population collapses to a single observer.

    part 12 · eye
  34. What is read at each distance from the edge of a lit region. Three materials under a half-plane of light, with the boundary at the centre of the horizontal axis and the lit side on the right. The vertical axis is the radiance leaving the surface as a share of what it leaves far inside the lit region. On the unlit side the sample is emitting light while receiving none, so the ratio the model calls a reflectance has a zero denominator there. The distance over which the curve runs from a tenth to nine tenths is 0.21 millimetres on coated paper and 5.1 on pale marble — which is the width of the neighbourhood a point's colour is decided by.

    Two blurs before the eye's own

    A translucent object arrives at the eye already blurred, by a kernel that is a property of the material rather than of the optics. On every material in this collection's table that blur is coarser at ordinary reading distance than the finest detail the eye can resolve — so the softness of marble or skin or wax is not a failure of vision, it is the object.

    part 12 · eye
  35. A template's asymmetry against what its observer costs. The horizontal axis is the tail ratio of the L cone's pigment absorbance — how far the curve reaches below its peak at half maximum against how far it reaches above — and the vertical is how far the observer built from that template sits from the 1931 standard. A real visual pigment has a long short-wavelength tail, so the three curves derived from a published nomogram sit above 1.1 and the two Gaussians sit below. The four are matched in width, so nothing here is about size. The ordering is the point: the two caricatures cost between two and four times what either nomogram does, and the axis they are separated on is the one feature the caricatures do not have.

    A template is mostly its tail

    Four pigment templates matched to the same width at the same peak, differing only in which side of the peak their half-maximum reaches further. Ordered by that one number, the observers they build are ordered by how far they sit from the standard one — and the two with the tail on the wrong side cost two and four times what either real nomogram costs.

    part 13 · eye
  36. The secondary band, dialled from nothing to twice what Govardovskii published. Govardovskii's template has a second, smaller absorption band below 400 nm, published at 0.26 of the α-band's peak. It is the one coefficient in the whole template whose contribution is somewhere else in the spectrum than the peak, and it is the part of the template a reader is least likely to have heard of. Sweeping it from nothing to twice the published value moves the median observer's distance from the 1931 standard from 0.9354 to 0.9790 ΔE₀₀, monotonically upwards. That is a small effect — about a fiftieth of the residual — and its being monotone is the interesting part: there is no interior optimum, so nothing here recommends the published value over any other, and a coefficient whose measured value is not the one that best fits an unrelated agreement is a coefficient to leave where the measurement put it.

    The band below four hundred

    The pigment template every observer here is built from carries a second, smaller absorption band in the ultraviolet, published at 0.26 of the main one. Dialling it from nothing to twice that moves the median observer monotonically away from the standard one, with no interior optimum — which is what a physical constant looks like when nothing downstream is pulling it anywhere.

    part 13 · eye
  37. The arguments a standard observer does not have. Seven choices inside a set of colour-matching functions, each with the shape it takes and what it is worth in ΔE₀₀ on a red pigment under a 6500 K radiator. Six are measurements: a field size, an age, a macular density, a cone optical density, three peak wavelengths and a rod contribution. The seventh is not — a change of basis is a change of curves and not a change of observer, and its entry is exactly zero because the space an experiment measures is what an observer is. Printing that zero beside the others is the clearest statement of what the other six are measurements of.

    The third factor is a construction

    Every figure in this collection names its observer, which was the whole point. None of them says what an observer is made of. Three curves are not a measurement of the eye; they are a projection of one, with a field size, an age, a macular density, three peak wavelengths and a luminance constraint inside them.

    part 13 · eye
  38. The conditions under which an observer's departure is exactly zero. A departure of the observer is the pairing of something belonging to the observer with something belonging to the stimulus, so emptying either factor empties the product. The axis is logarithmic in what is left when the condition is imposed. Six rows empty the stimulus's factor — a perfectly neutral sample is the same colour for every observer, at any age and any field size — and two empty the observer's, since a gain on each cone and a change of basis are both absorbed exactly. All eight are identities rather than small numbers. The last two are the same two conditions imposed in a published cone space rather than in the observer's own, and they are worth eight and thirteen units: the identity is about the eye, and the arithmetic everybody uses is in somebody else's coordinates.

    A neutral is everyone's colour

    Two eyes differing by fifty years of lens yellowing, by a factor of three in macular pigment and by six nanometres of long-wavelength peak agree about a grey card to four parts in ten thousand billion. The agreement is an identity rather than a coincidence, and it says exactly what an observer disagreement is a disagreement about.

    part 14 · eye
  39. The three cone absorptances at two settings of the cone optical density. Solid and dashed are the same construction at the two ends of two standard deviations of the reported spread. The curves are built from one pigment template through its ocular media, which is the same model its population of two hundred eyes is drawn from. The largest difference between the two sets is 18.8 per cent of the peak, and where it sits along the wavelength axis is what decides which stimuli the two observers disagree about — a departure concentrated in the blue is invisible on a sample with no blue in it.

    A gain is not an observer

    Multiply one eye's three cone sensitivities by 1.6, 0.7 and 2.4 and it is not a different eye. The white-point division is that multiplication's inverse, so the two agree exactly — and most of what a cone optical density change does is that multiplication, which is why the largest number in the table of individual variation is the one that matters least.

    part 14 · eye
  40. The arguments a standard observer does not have. Seven choices inside a set of colour-matching functions, each with the shape it takes and what it is worth in ΔE₀₀ on a red pigment under a 6500 K radiator. Six are measurements: a field size, an age, a macular density, a cone optical density, three peak wavelengths and a rod contribution. The seventh is not — a change of basis is a change of curves and not a change of observer, and its entry is exactly zero because the space an experiment measures is what an observer is. Printing that zero beside the others is the clearest statement of what the other six are measurements of.

    Three curves for one space

    Rotate a set of colour-matching functions by an arbitrary invertible matrix, undo the rotation at the end, and the computed colour is identical to eight parts in a thousand million million. An observer is a three-dimensional subspace, not a set of curves, and the literature keeps reopening a question that is a theorem.

    part 14 · eye
  41. Every departure under every light. Six departures across six lights, each cell the difference between two observers in ΔE₀₀, drawn as a bar whose length is the number. The rows are not multiples of one another: the lens is worst under tungsten and the pigment peaks are worst under a three-emitter LED, because a departure is a pairing and which light is being paired with decides it. The laser projector's row is empty, and that is not a fact about lasers — on this collection's five-nanometre grid a three-line spectrum is a one-line spectrum, and a single wavelength is a stimulus every observer agrees about exactly.

    The lens is worst under tungsten

    An ageing lens costs 4.38 ΔE₀₀ under a tungsten lamp and 2.20 under a fluorescent tube. The pigment peaks cost 2.84 under a three-emitter LED and 1.87 under the same tube. Six departures across six lights do not form a single ordering, which is what a pairing looks like and what a dominant term would not.

    part 14 · eye
  42. What choosing a space to divide the white out in is worth. Three pairs of routes to the same colour, over forty-two surfaces: dividing the white out in tristimulus values, in a published cone space, and in the observer's own cones. The first two agree to 0.59 ΔE₀₀ at the median. Either of them differs from the observer's own cones by more than fifteen. That is why the two exact conditions in this round are exact only in the eye's own coordinates: the identity belongs to the receptors, and every published arithmetic works in a basis somebody else chose.

    The identity is in the eye's own coordinates

    Two conditions in this round are exact — a gain on each cone is not a different observer, and three curves for one space are one observer. Imposed in a published cone space rather than the eye's own they leave 13.0 and 8.11 ΔE₀₀ standing. The identities belong to the physiology and every arithmetic in use works somewhere else.

    part 15 · eye
  43. Each departure over forty-two surfaces rather than one. The same six departures measured over a family of forty-two analytic reflectances — an absorption band of stated centre, width and depth — with the smallest, the median, the ninety-fifth percentile and the largest marked. Every one of them spans more than a factor of three, and the ranking between them is not stable across the family: what decides a departure's size is which sample it is asked about, because a departure is a pairing and the sample is one of the two factors. Quoting any single number for what an observer's age is worth is quoting a choice of example.

    The ranking is not stable

    On a red pigment under daylight the six observer departures run from 2.38 down to 1.20 ΔE₀₀. Over forty-two surfaces two of them change places, the top two separate, and every one spans between a factor of ten and a factor of thirty-five. A chart of six bars is a chart of one example.

    part 15 · eye
  44. The three cone absorptances at two settings of the field size. Solid and dashed are the same construction at the two ends of the CIE's own second observer. The curves are built from one pigment template through its ocular media, which is the same model its population of two hundred eyes is drawn from. The largest difference between the two sets is 16.0 per cent of the peak, and where it sits along the wavelength axis is what decides which stimuli the two observers disagree about — a departure concentrated in the blue is invisible on a sample with no blue in it.

    A field size is two changes

    The CIE publishes two standard observers and the difference between them is usually described as a field size. What actually differs is a macular pigment the light no longer passes through and a cone outer segment the light no longer travels the length of — two changes, in two places, with different signs and different sample dependence.

    part 15 · eye
  45. The three cone absorptances at two settings of the age of the lens. Solid and dashed are the same construction at the two ends of twenty years old against seventy. The curves are built from one pigment template through its ocular media, which is the same model its population of two hundred eyes is drawn from. The largest difference between the two sets is 25.5 per cent of the peak, and where it sits along the wavelength axis is what decides which stimuli the two observers disagree about — a departure concentrated in the blue is invisible on a sample with no blue in it.

    The observer has no age

    Five of the six arguments in this round are spreads — a population differs about them and the mean is a reasonable summary. The lens is not. Everybody's lens yellows in the same direction at about the same rate, so a standard observer with no age is not an average over a population; it is a snapshot of one moment in every reader's life.

    part 15 · eye
  46. The three cone absorptances at two settings of the rods. Solid and dashed are the same construction at the two ends of a tenth of the cone response, which is a dim room. The curves are built from one pigment template through its ocular media, which is the same model its population of two hundred eyes is drawn from. The largest difference between the two sets is 9.1 per cent of the peak, and where it sits along the wavelength axis is what decides which stimuli the two observers disagree about — a departure concentrated in the blue is invisible on a sample with no blue in it.

    The rods are a fourth curve

    Colorimetry has three numbers and a rod signal is a fourth. It cannot be absorbed by a gain, it cannot be removed by a white point, and adding a tenth of one to a three-curve observer costs 1.60 ΔE₀₀ — the narrowest distribution of the six departures, because an addition behaves quite differently from a filter.

    part 15 · eye
  47. The three cone absorptances at two settings of the macular pigment. Solid and dashed are the same construction at the two ends of two standard deviations of the reported spread. The curves are built from one pigment template through its ocular media, which is the same model its population of two hundred eyes is drawn from. The largest difference between the two sets is 31.5 per cent of the peak, and where it sits along the wavelength axis is what decides which stimuli the two observers disagree about — a departure concentrated in the blue is invisible on a sample with no blue in it.

    The macular is a band, not a filter

    The lens absorbs everything below 500 nanometres with a long tail; the macular pigment absorbs forty nanometres either side of 460 and nothing else. The two have similar sizes and completely different distributions, and the reason is that one is broad and one is narrow — which is what decides whether a sample is affected at all.

    part 15 · eye
  48. The three cone absorptances at two settings of the pigment peaks. Solid and dashed are the same construction at the two ends of two standard deviations, and the L/M polymorphism on top. The curves are built from one pigment template through its ocular media, which is the same model its population of two hundred eyes is drawn from. The largest difference between the two sets is 8.9 per cent of the peak, and where it sits along the wavelength axis is what decides which stimuli the two observers disagree about — a departure concentrated in the blue is invisible on a sample with no blue in it.

    The peaks move the flanks

    Shifting a cone's peak wavelength by three nanometres changes its sensitivity at its own maximum by almost nothing and on its flanks by several per cent, because a maximum is flat and a flank is steep. Every display primary sits on a flank, which is why a pigment polymorphism is worth 3.60 ΔE₀₀ under a laser projector.

    part 15 · eye
  49. The conditions under which an observer's departure is exactly zero. A departure of the observer is the pairing of something belonging to the observer with something belonging to the stimulus, so emptying either factor empties the product. The axis is logarithmic in what is left when the condition is imposed. Six rows empty the stimulus's factor — a perfectly neutral sample is the same colour for every observer, at any age and any field size — and two empty the observer's, since a gain on each cone and a change of basis are both absorbed exactly. All eight are identities rather than small numbers. The last two are the same two conditions imposed in a published cone space rather than in the observer's own, and they are worth eight and thirteen units: the identity is about the eye, and the arithmetic everybody uses is in somebody else's coordinates.

    Two observers and one metamer

    An observer departure is invisible on a single sample compared with nothing. It becomes a disagreement the moment two spectra are being asked to match, because a match is an identity between three integrals and a different observer takes different integrals. Everything in this round is a statement about pairs wearing a single sample's clothes.

    part 15 · eye
  50. What the rods cost a match, by where their signal enters and under which lamp. For five lights, the median colour difference over forty-two surfaces between the reference observer and the same observer with a rod signal a tenth of each cone's peak added — into all three cone channels, into the long- and middle-wavelength channels only, or into the short-wavelength channel only. Under daylight the first two are 1.03 and 0.90: whether the rods reach the S pathway hardly matters. Under a phosphor white LED they are 1.24 and 0.45, a factor of 2.75, and the S-only route alone costs 0.96.

    The rods' route is priced by the lamp

    A rod signal in a dim room disturbs a colour match, and how much depends on which of the cone pathways it reaches — a weight the physiology leaves uncertain, especially for the blue–yellow pathway. Under daylight the uncertainty is nearly free: a rod signal that skips the S pathway costs 0.90 at the median surface against 1.03 for one that enters all three. Under a phosphor white LED it is worth a factor of 2.75, 0.45 against 1.24. What decides it is one number per lamp: how large the rod signal is compared with each cone class's own catch of the light.

    part 16 · eye
  51. Which two lamps to stand a mesopic match between. Every pair of the five lights, by how far a match made under one and set under the other moves as the rod signal's weight into the S channel goes from nothing to equal — the median over forty-two surfaces, which is the signal an experiment has to resolve. The best pair is daylight against phosphor LED at 0.58 ΔE₀₀; the worst is tungsten against fluorescent tube at 0.09, a factor of 6. The count at the right is how many settings it takes to resolve the weight to a tenth at half a colour difference of scatter per setting.

    The reference lamp must not move

    To measure an uncertain weight, use the condition in which the answer depends on it most. That is right about half of an asymmetric colour match and exactly wrong about the other half: a match measures a difference of two displacements, and a reference field that also moves with the weight cancels the signal the test field carries. Daylight is the least sensitive of five lamps and belongs in every one of the three best pairs — 75 settings against a phosphor LED, 2,804 against the pair of lamps the principle as stated would have chosen.

    part 17 · eye
  52. Two ways to ask how much rod signal an older eye has. The rod signal's catch of each lamp divided by the S cones' own catch, against the observer's age, for five lamps — drawn twice. The upper curves take the rod signal at a fixed absolute size, and every one of them roughly doubles from twenty to seventy-five: an older lens cuts the blue before the S cones see it and the rods, peaking further into the green, lose much less. The lower curves take the rod signal at a tenth of each cone's own peak absorptance, which is the model's own definition, and they barely move at all. Nothing about the retina differs between the two; only the normalisation does.

    A rod signal has no natural size

    An older lens absorbs where the S cones are sensitive, so it should make the uncertain rod-to-S-cone weight cheaper. Measured, the rod signal's catch of a phosphor LED as a share of the S cones' own catch more than doubles from twenty to seventy-five — and the share the model actually uses falls by a fifth. The two differ by the S cone's peak absorptance, which the lens takes 60 per cent of, and which entered the model as a normalisation rather than as a claim.

    part 17 · eye
  53. How far a mesopic match moves as the S weight opens: two rooms against one field. For every pair of the five lamps, the median over forty-two surfaces of how far a match moves as the rod signal's weight into the S channel goes from nothing to equal: pale for the two-room match, each half adapted to its own lamp; dark for a bipartite field whose two halves share one adaptation. The field's signal is larger for every pair, by ×1.5 to ×7.7.

    One field keeps what two rooms divide out

    An asymmetric match can measure how strongly the rods feed the blue-yellow pathway, but set with the observer adapted to each lamp in turn it needs seventy-five settings on the best pair of lamps and hours of waiting between them. Putting the two lamps on the two halves of one field was proposed as the quick version, at the cost of a weaker signal. The signal is not weaker. Under one shared adaptation it is three times stronger for daylight against a white LED, and the best pair needs six settings. The adaptation that makes the slow version slow is also what was dividing the rods' contribution out of each half.

    part 18 · eye
  54. What a weight fitted under the wrong reading does across ages. If the retina's rod signal has a fixed absolute size and the weight is fitted with the model's scaled reading, the fitted weight is the true one times the ratio of the two readings' signals: ×0.80 at twenty, ×1 at thirty-two, ×1.62 at seventy-five. The other way about, the reciprocal. A cohort of observers sharing one weight would show it as a trend with age.

    Age can size the rod signal, if the lens is known

    A rod signal leaking into the colour pathways has no natural size in the model: read as a fraction of each cone's own peak it shrinks as the lens yellows, read as a fixed amount it grows relative to the cones, and the two readings differ by a factor of two over a lifetime. An asymmetric match on observers of different ages can tell them apart, because they predict opposite trends: one says a weight fitted at seventy-five will look sixty per cent larger than at thirty-two, the other says it will not move. A handful of observers per age would show it. But a lens mistaken by ten years looks like forty per cent of the weight, and a tenth of an optical density of macular pigment as much again, so the experiment is only as good as what it knows of each eye's lens and macula.

    part 18 · eye
  55. What each pair of lamps pays to fit the lens and the macula as well. For each pair of lamps, the standard error of the rod signal's S weight fitted from one session — with the observer's lens and macula known (top bar), with the lens fitted too (middle) and with both fitted (bottom). Daylight against the LED is the best pair with everything known, ±0.026, and pays most of its advantage for the macula; tungsten against the LED pays almost nothing for either.

    The matches carry their own lens

    A lens mistaken by ten years moves a colour match as far as half the rod signal being measured, and a tenth of an optical density of macular pigment as far again, so an experiment on the rod signal looked as if it needed a densitometer for every observer. It does not. Across a family of surfaces the three move the matches in different patterns, and one session of forty-two settings fits all three together — the lens to within a year and a quarter, better than a densitometer was asked for. The price is the weight's precision, and a second pair of lamps nearly removes it.

    part 19 · eye
  56. Where a lens edge 10 nanometres off goes in a one-session fit. A sixty-year-old observer whose lens edge sits 10 nanometres longer (upper bars) or shorter (lower bars) than the model's, fitted by the model from one daylight-against-LED session: the shift in each fitted parameter, in that parameter's standard errors. The macula takes the most, 2.4 of its standard errors; the weight moves by 0.59 of its own, 0.035 in absolute terms, against a prediction of under a tenth for an edge ten nanometres off.

    A lens of the wrong shape lands in the macula

    One session of colour matches can fit an observer's rod signal, lens and macular pigment together — but it fits a lens of the model's shape, one exponential in wavelength scaled by age, and a real lens need not be that shape. Give a sixty-year-old a lens with the right density at 400 nm and its edge ten nanometres off, and the fit puts the error mostly in the macula, moves the rod signal's weight by six tenths of its standard error — six times the predicted tenth — and leaves a misfit the session cannot see. A second lamp pair makes it worse: 3.5 standard errors. Fitting the edge as a fourth number removes the bias, costs the weight three to fourteen per cent, and finds the edge.

    part 20 · eye
  57. Where one session puts a band in the lens, by the band's wavelength. A band of a tenth of an optical density in a sixty-year-old's lens, centred from 410 to 460 nm, fitted from one daylight-against-LED session by the model with weight, lens age, macula and the lens edge free: each parameter's shift in its own standard errors. Near 410 to 430 nm the band is read as lens; from 440 it is read as macula; the edge takes little of it anywhere; and the rod weight moves at every centre, by 0.37 to 1.39 of its standard errors.

    A band in the lens reads as age or as macula

    A lens whose absorption edge is steeper or shallower than the model's biases a matching session's fit, and fitting the edge's position repairs it. An ageing lens also grows bands: yellow pigments that add a shoulder of absorption just past 400 nanometres. The edge parameter takes none of one. A tenth of an optical density centred at 410 to 430 nm is read as four to six years of extra lens age; centred at 440 to 460, as three to ten hundredths of macular pigment. Either way the rod signal's fitted weight moves, by up to 1.4 of its standard errors in one session and 1.8 in two, and the session's misfit stays within its noise. The lens model has to carry the band, or the experiment has to state what it does not know.

    part 21 · eye
  58. More rods make a lens band's pull on the rod weight worse, unless the band is fitted. A sixty-year-old's lens with a band of a tenth of a density at 450 nm, fitted by the model with the rod signal a tenth to a half of each cone's peak: the shift in the fitted rod weight, in its own standard errors. With weight, lens age and macula fitted, one session's shift grows from +1.73 at a tenth to +2.53 at a third, and two sessions' more. With the band's amplitude fitted as a fourth number, one session stays under half a standard error at every share.

    A dimmer session hides a lens band in the rods

    A band of extra absorption in an older lens is read by a colour-matching session as lens age or as macular pigment, and it pulls the fitted rod weight with it. The rods see the lens and not the macula, so a dimmer session, where they carry a third of the signal instead of a tenth, was predicted to tell the band from the macula. It does the opposite. With the band at 450 nm left out of the fit, the rod weight's shift grows from 1.7 of its standard errors to 2.5, the misfit a session could notice shrinks by a third, and the macula and the band trade more tightly. The rods see the band — and the rod weight, which the fit is free to move, is exactly what absorbs it. What separates the band from the macula is fitting its amplitude as a fourth number, which works in daylight already.

    part 22 · eye
  59. How well a matching session places a lens band's centre. A sixty-year-old's lens with a band of a tenth of a density centred from 410 to 460 nm, fitted with the rod weight, lens age, macula, the band's amplitude and its centre all free: the centre's standard error, from one daylight-against-LED session and from designs adding a second session. One session places the centre to 11.5 nm at worst below 430 and within 5.7 above it; adding the LED against the three-emitter lamp brings every centre within 5.5.

    A second lamp pair finds a lens band's centre

    A band of extra absorption in an older lens is found by a colour-matching fit only if the fit has a parameter for it, and a band parameter at a fixed wavelength finds only a band near that wavelength. Freeing the band's centre as a fifth number asks whether the matches can place it. From one daylight-against-LED session they place a band above 430 nm to within six nanometres and one below it only to eight to twelve, where moving the band and ageing the lens change the matches together — and there they barely see the band at all. A second session, the LED against a three-emitter lamp, places every centre within five and a half nanometres. The fluorescent tube's line at 436 nm helps where it sits and nowhere else.

    part 23 · eye
  60. From which starts a fit finds a lens band, two sessions. For a band of a tenth of a density at 415, 430 and 445 nm in a sixty-year-old's lens, the number of twenty simulated experiments of two sessions each in which a fit of all five numbers, started with the band's centre at each of 400 to 470 nm, finds the centre within twice its standard error. The ticks at the left edge are the same count for a fit started at the true observer: 18, 18, 18 for the three bands.

    A lost lens band goes to age or the macula

    Two colour-matching sessions determine the centre of a band of extra absorption in an older lens to about five nanometres, if a fit can find it. Fitted in full from eight starts across 400 to 470 nm on twenty simulated experiments, two sessions find a band at 415, 430 or 445 nm from every start in fifteen to eighteen experiments of twenty, against eighteen for a fit started at the truth. The prediction was that a low band would be lost from high starts to the macula's flank. The start barely matters with two sessions, and a lost band does go somewhere — but a band at 415 or 430 goes to lens age, three to five years of it, and only a band at 445 goes to the macula. Every loss is a worse fit than the session's best, so eight starts find the band as often as the truth does. With one session the start decides.

    part 24 · eye
  61. The lens band a fit reports for a macula that peaks off 460 nm. For an observer at sixty with no band in the lens and a macula peaking 6 and 3 nm short of the model's 460, at 460, and 3 and 6 nm long, the band the five-number fit reports from the exact matches, as its amplitude over its standard error, with one and two sessions. A macula 3 nm short is read as a band of 0.061 at 438 nm from two sessions, 3.1 standard errors; a macula peaking long has no band to become, and at 6 nm long the fit runs to the edge of the range a band may take.

    A macula three nanometres short is a lens band

    Two colour-matching sessions can fit a band of extra absorption in an older lens, its size and its centre, and the test proposed for real older observers was to count those with a band more than twice its standard error. That test assumes a significant band can only be a band. The observer model's macula peaks at 460 nm; a real one is two carotenoids whose mix moves its peak by a few nanometres. Simulated with its macula peaking three nanometres short and no band at all, an observer at sixty is fitted with a lens band of 0.06 at 438 nm, three standard errors, and noisy sessions find it in nineteen of twenty experiments. Six nanometres short, the phantom is larger than the band the earlier essays searched for. The confusion runs both ways: a real band fitted with the macula's peak free reads as a macula two to three nanometres short. Neither fit alone can tell them apart. Comparing the two can, with two sessions — never preferring the macula for a real band, and preferring it in sixteen to nineteen experiments of twenty for a short macula.

    part 25 · eye
  62. How often a six-number fit finds both departures. For nine observers at sixty, each with a lens band of a tenth of a density at 415, 430 or 445 nm and a macula moved by −3, 0 or +3 nm, the number of twenty noisy experiments in which a fit of all six numbers finds the band beyond twice its standard error, and finds the macula moved beyond twice its standard error — with two sessions and with one. With two, both are found in most experiments wherever both are present; with one, neither is.

    A second session keeps a band and the macula apart

    A lens band and a macula peaking a few nanometres short look alike to a colour-matching fit that has only one of them, and an older eye may have both. Fitted with all six numbers — rod weight, lens age, macular density, the band's amplitude and centre, and the macula's peak — nine simulated observers at sixty, each with a band at 415, 430 or 445 nm and a macula moved three nanometres either way or not at all, test whether the matches can carry both. The prediction was that the band at 445 and a short macula would trade almost completely, a correlation beyond 0.95 and the band's standard error tripled. That is what one session does: correlations of 0.91 to 0.95 at 430 and 445 nm, and a band found in at most twelve experiments of twenty. Two sessions do not. Their correlations stay between 0.42 and 0.83, the band is found in twelve to twenty experiments for every observer, and a moved macula in twelve to nineteen — while an unmoved one is found moved in at most three. The price of the extra number is paid by the band nearest the macula, whose standard error rises by three quarters; the rod signal pays almost nothing.

    part 26 · eye
  63. A line on the macula's peak frees the band; one on its flank does not. For the band at 445 nm, its standard error in the six-number fit over its value when the macula's peak is not fitted, with the third session's line at 465 to 510 nm. Two sessions leave it at ×1.78 and a repeated second session at ×1.41. The cyan line at 487 nm leaves it at ×1.51, worse than the repeat; at 465 nm, near the macula's 460 nm peak, it comes to ×1.16, inside the prediction's quarter.

    A line on the macula's flank buys less than a repeated session

    Fitted with six numbers, two sessions of colour matches keep a lens band and a moved macula apart, but the band at 445 nm pays for it: correlated with the macula's shift at 0.82, its standard error 1.8 times its value when the macula is not fitted. A moved macula changes its long flank past 470 nm, where no lens band reaches, so a third session — the LED against a lamp with a narrow line at 487 nm — was predicted to bring the correlation under 0.6 and the band's error within a quarter of its band-only value. The correlation falls to 0.80 and the error to 1.5 times. A third session that simply repeats the second does better on both errors. The flank line pins how much macular pigment there is, not where it peaks, and it ties the band's height to its centre. Moved to 465 nm, onto the macula's peak, the same session brings the band within about a fifth of its band-only error — but no position of the line brings the correlation below 0.78. The band at 445 nm and the macula's peak are told apart by their short flanks, and a third session adds matches rather than a new kind of evidence.

    part 27 · eye
  64. A known density frees two sessions, not the line on the peak. For an observer at sixty with a lens band at 445 nm and an unmoved macula, the correlation between the band's amplitude and the macula's shift when the macular density is fitted with the other five numbers, and when it is known exactly and fixed, under two sessions, the second session repeated, and the cyan session with its line at 465 to 510 nm. Fixing the density takes two sessions from 0.83 to 0.62 and the line at 465 nm, on the macula's peak, only from 0.86 to 0.74: the line that freed the band's error best is left with the strongest trade.

    Knowing the macula's density frees two sessions, not the line on its peak

    Fitted with six numbers, an older observer's colour matches cannot fully split a lens band at 445 nm from a macula whose peak has moved: their correlation stays above 0.78 under every third session tried. The proposal was to measure the macular pigment's density from outside the matches, leaving five numbers, and the prediction was that a line on the macula's peak would then split the pair below 0.6 while two plain sessions stayed near 0.8. It runs the other way. With the density known exactly, two sessions fall from 0.83 to 0.62 and the line at 465 nm only from 0.86 to 0.74. The density was the road the band and the peak traded along — correlated with each at about −0.75 in two sessions' fit — and the designs that had already narrowed that road gain least from closing it. A realistic outside measurement closes almost none of it: the matches already know the density to 0.027, so a flicker measurement good to 0.05 moves the correlation by three hundredths. And far from the macula, at 415 nm, a known density binds the band to the peak more tightly than before.

    part 28 · eye
  65. A density a tenth wrong moves the band by more than its error. For an observer at sixty with a lens band at 445 nm and an unmoved macula, fitted from two sessions with the macular density fixed at 0.8, 0.9, 1.1 and 1.2 times its true 0.35: the bias in the band's amplitude and in the macula's fitted peak, in units of the fixed fit's standard errors. A tenth too high lowers the band by 1.4 standard errors and moves the peak 1.25 nm short; the prediction was a third of a standard error and under half a nanometre.

    A density a tenth wrong costs the band more than fitting it

    An older observer's colour matches carry a lens band and a macula that can trade against each other, and fixing the macular pigment's density from an outside measurement breaks much of that trade when the density is right. A flicker measurement and a colour match average the pigment over different fields, so the fixed density can be wrong by a tenth or more, and the prediction was that a tenth too high takes about a third of a standard error off a band at 445 nm and moves the macula's peak by under half a nanometre. It takes 1.4 standard errors off the band and moves the peak 1.25 nm short. At 415 nm the same error pushes the band the other way, up by 2.4 standard errors, and runs its fitted centre 5.6 nm shorter and the lens's age three years younger. Bias and error together, a density fixed a tenth wrong gives the band at 445 nm a larger error than leaving the density free; the break-even is about eight per cent. Given as a measurement with its own standard error of 0.03 instead of fixed, the same wrong density biases the band by under half its error.

    part 29 · eye
  66. Near the macula a wrong density moves every experiment together. For an observer at sixty with a lens band at 445 nm and an unmoved macula, two sessions: each of twenty simulated experiments' fitted band amplitude with the macular density held a tenth low, at its truth and a tenth high. A tenth high moves the band by -0.031 on average with a spread of 0.0024 between experiments; a tenth low by 0.030, spread 0.0022; the band's own standard error is 0.023.

    A wrong density shifts every experiment alike near the macula

    Read from the expected matches, a macular density fixed a tenth too high lowers an older observer's lens band at 445 nm by 1.4 of its standard errors and raises a band at 415 nm by 2.4. Real experiments carry noise, and the prediction was that their fits scatter about those biases with the fixed fit's own error, so that a tenth's mistake puts the band beyond twice its error in about a third of experiments at 445 nm and two thirds at 415. Twenty simulated experiments per case, each fitted with the density free and held at 0.9, 1 and 1.1 of its truth, give five of twenty at 445 nm and sixteen at 415 under two sessions — eight and eighteen with a line on the macula's peak. Near the macula the bias is a rigid shift: every experiment moves by the first-order amount to within a tenth of a standard error. Far from it the bias is lopsided — up by 0.058 for a density too high, down by 0.032 for one too low — and a density too low erases the band altogether in one experiment of twenty.

    part 30 · eye
  67. One session, two valleys: the macula where it is, or far short. For the observer with a band at 415 nm and an unmoved macula, one session: each of twenty experiments' census fit in band amplitude and macula peak, and, for the 7 the census missed, the better fit, joined to the census's. Every better fit puts the macula seven to nineteen nanometres short: a second valley in which a macula moved far short does part of the band's work.

    One session has a second valley

    Every essay on older observers since the second session has read its fitted lens bands and macula shifts from one census: nine simulated observers, three designs, twenty noisy experiments each, every fit a damped descent from two starts. Two of forty experiments refitted in the last essay found a better fit than the census had. The prediction was that the census misses one fit in twenty, almost all where the band is small, and that the misses change no count it reported by more than a few hundredths. Refitted from three more starts, 66 of its 540 experiments have a better fit — 45 of 180 under one session, 9 and 12 under two and three. Sixty of the 66 better fits move the macula's peak more than two nanometres further short, most shrinking the band, and 31 lie lower by under one unit of χ²: a second valley, nearly level with the first. Two sessions' counts stand to within one or two experiments. One session's do not: with the best fits an unmoved macula is reported moved in eight experiments of twenty, where the census said one.

    part 31 · eye
  68. Bounded, one session stops finding a macula that never moved. For the three bands with the macula unmoved, how many of twenty one-session experiments report it moved beyond twice its standard error: from the census's two starts, from the best of five starts, and from the best of five with the shift held within ±6 nm. The bound brings the false shifts at 415 and 430 nm from 8 and 6 to 1 and 2.

    A bound on the macula trades false moves for hidden ones

    Fitted thoroughly, one session of an older observer's colour matches finds a second valley in a quarter of its experiments — the macula's peak moved ten or more nanometres from the model's, where no real macula sits — and reports a moved macula where there is none in up to eight experiments of twenty. The prediction was that holding the macula's shift within six nanometres either way, the range real pigment mixtures span, restores every observer's one-session counts to within two of the census's and leaves about one fit in four pressed against the bound. The false moves vanish: an unmoved macula is reported moved in one, two and one experiments of twenty for the three bands, against eight, six and two unbounded. Seven of nine observers keep their count of moved maculas within two of the census's. But only one fit in six ends on the bound, and beside the band at 445 nm the bound hides real moves: a macula truly three nanometres short is found in three experiments of twenty, where the census found seven. Two sessions barely notice the bound.

    part 32 · eye
  69. The prior trades the macula's moves for the lens band. For each observer, how many more experiments of twenty find the lens band under the prior than under the census's fits, against how many more find the macula moved. The observers moved short or unmoved gain up to seven bands and lose one to five moves — false ones where the macula never moved, real ones where it did; those moved long gain no band and lose one or two moves, the 445 nm observer losing three bands as well.

    A soft prior on the macula frees the lens band

    One session of an older observer's colour matches, fitted thoroughly, can drift into a valley where the macula's peak sits ten or more nanometres from any real eye's, and a hard bound at ±6 nm removes those false moves at the cost of hiding some real ones — beside a lens band at 445 nm, a macula truly three nanometres short is found in three experiments of twenty where the census found seven. The prediction was that a Gaussian prior four nanometres wide, which lets a fit go further when the matches insist, keeps the false moves away and restores those real ones. It keeps the false moves away — none, one and one for the three bands. It does not restore the real moves: it finds them as seldom as the bound or more seldom, two of twenty beside 445 nm, because it pulls every fitted shift towards nought, to a mean of 1.2 to 1.9 nanometres for moves of three. What it buys instead is the lens band. With the macula held near the standard, the band beside it is found in ten to twelve experiments of twenty, where the census found three to seven.

    part 33 · eye
  70. Each widening of the prior buys moves with bands. Over the six observers whose macula truly moved three nanometres, the real moves one session finds beyond twice their error, against the lens bands found for the four observers whose band the four-nanometre prior freed — under priors four, six and eight nanometres wide, the census's own fits and the hard bound. 4 nm: 14 moves, 44 bands; 6 nm: 22 moves, 34 bands; 8 nm: 25 moves, 25 bands; census 27 and 22; bound 19 and 20.

    A wider prior gives the moves back by spending the band

    One session of an older observer's colour matches, fitted under a Gaussian prior four nanometres wide on the macula's peak shift, finds the lens band beside the macula in ten to twelve experiments of twenty and real three-nanometre moves of the macula as seldom as a hard bound does. The prediction was that a prior six nanometres wide keeps the false moves at two or fewer, finds the real moves within two of the bound's counts and keeps two thirds of the band's gain, and that at eight the far valley reopens at 415 nm. Three parts hold. The fourth fails: six nanometres keeps 55 per cent of the band's gain. Read across the widths the prior traces one trade — 14 real moves and 44 bands at four nanometres, 22 and 34 at six, 25 and 25 at eight, against the unconstrained fit's 27 and 22 — and the hard bound, at 19 and 20, is worse than six nanometres on both.

    part 34 · eye
  71. Even two sessions fall into the far valley, and the prior lifts them out. For the observer with its band at 415 nm and its macula truly three nanometres short, every two-session experiment's fitted shift at the best of five starts and under a prior six nanometres wide. Three best fits lie beyond 7.5 nm short — −11.84 drawn to −1.90, −7.73 drawn to −3.05, −11.00 drawn to −0.94 — and the rest move by a tenth or two; the mean goes from −3.89 to −2.57.

    A second session leaves the prior one job

    Under one session of an older observer's colour matches, a prior on the macular pigment's peak shift trades the lens band against real moves of the macula, six nanometres wide at the knee of the trade. Two sessions already find the band in twelve to twenty experiments of twenty and real three-nanometre moves in twelve to eighteen, and still report an unmoved macula moved in two to three. The prediction was that the six-nanometre prior, applied to two sessions, removes those false moves and costs nothing else. It removes none of them — two, two and none remain, about the rate at which a test at twice the standard error calls a move by chance. It costs nothing at five observers of six. Beside 415 nm, with the macula moved short, it finds four fewer moves: three of the best fits there were in the far valley, eight to twelve nanometres short, and counted as moves because their sign was right. With two sessions the far valley is the prior's only remaining job.

    part 35 · eye
  72. A third session finds the band beside the lens's edge. For each observer, how many of twenty experiments find the lens band beyond twice its error and within 20 nm of where it was put: two sessions under the six-nanometre prior, and three sessions at their best of five starts. Beside 415 nm with the macula unmoved, two sessions find it in 12 and three in 18; beside 430 nm with the macula short, 20 and 16.

    A third session finds the edge band and counts bands misplaced

    Two sessions of colour matches find an older observer's lens band in eighteen to twenty experiments of twenty at every place but one: a band at 415 nm, beside the lens's own short-wave edge, found in twelve. The prediction was that a third session — a phosphor LED set against a lamp with a cyan line — finds it in at least seventeen, and that the experiments losing it under two sessions fit the lens three or more years old. Three sessions find it in eighteen, and every loss under two did fit the lens older, but by under three years in five of eight. The count hid something else. It credits a band wherever a fit puts it, and up to twelve of 180 fits put it sixty to eighty nanometres from its place, on the macula's long flank. With three sessions the six-nanometre prior no longer closes that valley.

    part 36 · eye
  73. Held near the lens, the band is found where it was put. For each observer, how many of twenty three-session experiments find the lens band beyond twice its error and within 20 nm of its place, with its centre free to range from 390 to 500 nm and held to 400–460. Beside 415 nm the bounded fits find it in 19, 20, 19 with the macula short, unmoved and long, against 16, 18, 15 free.

    A band held near the lens closes the valley the prior could not

    Three sessions of colour matches find an older observer's lens band beside the lens's own edge as predicted, but twelve of 180 fits place the band sixty to eighty nanometres away on the macula's long flank, each with the macula fitted far short — a valley the six-nanometre prior on the macula's shift no longer closes. The prediction was that holding the band's centre to 400–460 nm, where a lens band can be, empties that valley, finds the 415 nm band with the macula short in at least eighteen experiments, and raises the moves found beside 430 nm from nine to twelve. The first two hold: no fit is left in the valley, and the band is found in nineteen. The third does not: 430 nm stays at ten. And the valley turns out to have been nearly level with the truth — holding the band costs ten of the twelve fits under four units of the fit's chi-squared, and every one's macula comes home.

    part 37 · eye

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