A notch a pigment cannot cut

A worst case that sits on the wall of its own box is a number about the box, and the repair is a bound with a mechanism in it.

The claim #

The family of painted rooms does have a worst case. It is at a band six nanometres wide, which is narrower than any pigment can make and narrower than the floor of the box the previous search used — which is why nobody saw it, and why the previous conclusion was wrong in principle and right to half a per cent in practice.

• The previous round searched a box and hit its wall in three of four parameters, and reported the honest thing: this family has no worst case, and a number quoted for one is a number about a constraint nobody wrote down.
• The residual does not rise for ever as the band narrows. A band narrow enough returns too little light to move the white much, so the profile turns over — at 6.0 nanometres, measured by profiling the width rather than searching over it.
• The peak is 28.54 ΔE00 against 28.38 at the box’s own floor of ten. Half a per cent. The old statement’s practical content survives entirely.
• Bounded by physics instead — an absorption band no narrower than forty nanometres, which is what a vibronic progression on a molecular transition gives — the worst case is 21.9, and the answer is again sitting on the bound.
• So the repair does not remove the dependence on a bound. It gives the bound a mechanism, which is the difference between a number about a box and a number about a pigment.

What the previous search found #

The worst case is where the box stops established the shape of the problem. A painted wall is a reflectance with a centre wavelength, a width, a depth and a base; a room applies it once, twice or three times; and searching that family rather than listing four members of it reaches 21.3 ΔE00 against the census’s 3.375.

The search ends on the wall of its box in three or four of its four parameters. The residual rises monotonically towards a narrower band, at a shorter wavelength, on a darker base — so the answer is the constraint, and the constraint was a set of ranges typed into a source file.

The record left the repair written down: a bound in terms of a physically meaningful quantity would turn a sensitivity into a number, and it needs a claim about pigments this collection does not currently make. This essay makes three of them.

The bound with a mechanism #

A coloured surface is coloured because an electronic transition absorbs. In a solid at room temperature that transition is coupled to the vibrations of the molecule or lattice around it, so what would be a line becomes a vibronic progression — a series of sub-bands spaced by a vibrational quantum — and the progression is smeared further because every absorbing molecule sits in a slightly different local environment.

Both broadenings are of order the vibrational quantum. For the bonds that make organic pigments that puts a band at some tens of nanometres wide and never at ones, which is why a paint’s reflectance curve is a gentle hill rather than a spike.

The exception proves it. The lanthanides absorb through f–f transitions on electrons shielded by filled outer shells, barely coupled to their surroundings, and neodymium and holmium glasses are sold as wavelength calibration standards precisely because their bands are narrow. They are also weakly absorbing, expensive, and not paint.

So three floors are declared with the physics named beside them: ten nanometres for a lanthanide glass, forty for an organic pigment, seventy for an inorganic one. They are modelling inputs and not quoted measurements, which this site’s fourth invariant requires to be said out loud — and every result below is reported as a function of the floor so that a reader who prefers different numbers can read a different answer off the same curve.

The turnover #

The curve at the top of this essay is a profile rather than a sweep of bounds, and the difference is what makes it trustworthy.

Searching four parameters under a floor and reading off the width it lands on asks a simplex to find a maximum that is usually on a boundary. Fixing the width and optimising the other three is three parameters with an interior answer, done sixteen times. The first version did it the other way and reported an answer at the narrowest floor that was lower than at three times that floor — which is not a shape any function has. It was the simplex failing on a boundary and reporting a local optimum, and it would have been perfectly publishable.

The profile rises from 26.94 at two nanometres to a peak of 28.54 at six, and falls away to 21.91 at forty and 9.07 at a hundred and sixty. Fitted through the floors from ten to a hundred and twenty, the log–log slope is −0.31: halving the narrowest band a pigment is allowed to have buys about a quarter more worst case.

The mechanism at the narrow end is simple and is worth stating because it is what the box hid. A Gaussian band of width w on a dark base returns light in proportion to w. As the band narrows the light it lets through falls linearly while its selectivity rises more slowly, and below about six nanometres the first effect wins. The maximum is where the two cross.

How flat the turnover is #

A maximum found by profiling a parameter deserves to be reported with the shape around it, because the shape decides how much the bound matters.

The residual against the band’s width: 26.94 at 2 nm, 27.64 at 3, 28.27 at 4, 28.52 at 5, 28.54 at 6, 28.49 at 8, 28.38 at 10, 28.07 at 14, 27.28 at 20, 24.72 at 30, 21.91 at 40, 18.38 at 55, 16.19 at 90 and 9.07 at 160.

The peak is a plateau rather than a point. Every width between 4 and 14 nanometres is within one per cent of the maximum, and every width between 3 and 20 is within five. A bound placed anywhere in that range returns the same answer — which is why the old box’s floor of ten gave 28.38 where the true peak gives 28.54.

And the fall is almost entirely on the wide side. From 6 nanometres down to 2 the residual loses 5.6 per cent; from 6 up to 160 it loses 68. So the turnover is real and shallow, and nearly everything that varies across this profile varies among bands wide enough for a pigment to make.

That changes what the physical bound is doing. Forty nanometres is not near the peak — it is a third of the way down the far side, at 21.91 against 28.54 — so the mechanism-based bound is a genuine constraint rather than a formality, and it costs 23 per cent of the unconstrained worst case where the box’s floor cost half of one.

What the correction is worth #

Very little, numerically, and that is the honest headline.

The peak at six nanometres is 0.56 per cent above the value at the box’s own floor of ten. The previous round’s statement — this family has no worst case — was a statement about the box it was given, the box’s floor was on the wrong side of the turn, and the practical content of the finding is entirely unchanged.

What changes is what kind of statement it is. No worst case is a claim about a family and is false. A worst case at a band narrower than any pigment can cut, worth half a per cent more than the box’s floor is a claim about a family and a bound, and it is true — and it tells a reader something the first version could not: that the answer is insensitive to the narrow end and entirely sensitive to the broad one.

That asymmetry is the useful part. Nobody needs to argue about whether a pigment can have a six-nanometre band. The argument that matters is whether forty is the right floor or seventy, and there the answer moves by a factor of 1.3.

The bound that does not bite #

The third of the three nested bounds is the one a reader would have named first, and it changes nothing.

A wall’s excitation purity — how far its chromaticity reaches from the white towards the spectral locus — is what “how saturated is this paint” means quantitatively, and a ceiling on it is the obvious way to say a paint somebody sells. Declared at 0.6, it never binds: the worst wall under the physical bound reaches 0.52, comfortably inside, and the two bounds return the same answer to a part in ten thousand.

The worst wall is dark, not colourful. At the arithmetic bound its purity is 0.18; at the physical bound 0.52. What breaks an adapted observer is a wall that takes most of the light away and returns a narrow band of what is left, and such a wall is dim rather than vivid — a deep bottle-green rather than a signal orange.

That is worth reporting as a finding rather than as a null result, because the intuition it contradicts is strong and would have led an experiment somewhere useless. A saturated paint is the one that looks like it should cause trouble, and it does not.

The parameter that is not bounded by paint #

Three of the four parameters are now held by something: the width by physics, the depth and base by the requirement that a reflectance not exceed one. The fourth is not.

The band’s centre wavelength runs to the short-wavelength end of every range the search is given — 400 nanometres when the box says 400, and 380 when it says 380, which is the edge of this collection’s own wavelength grid.

Nothing about a pigment stops it. What stops it is the observer: below about 400 nanometres the eye’s sensitivity is falling away so steeply that a band there stops changing the white, and where that happens is a fact about the short-wavelength cone and the lens in front of it rather than about paint.

So a worst case quoted for a room is partly a statement about an eye, and the curve is the evidence rather than an assertion. It is also a U rather than a slope: the long-wavelength end rises back to 14.9, so the worst wall is at one extreme of the visible band or the other and never in the middle.

What was computed, and how #

The objective is the residual an adapted observer is left with after the change of light, in the CAT16 basis, over this collection’s own family of reflectances — the same quantity the census reports, so the numbers here and the numbers there are comparable without a conversion.

At each fixed width, three parameters are optimised by a simplex with restarts, under three constraints: the centre inside the box, the depth and base inside theirs, and the sum of depth and base no greater than one, which is what a reflectance is.

That last one is worth a sentence because the published box does not enforce it. A base of 0.85 with a depth of 0.9 is inside the previous round’s paint box and describes a wall reflecting 1.75 of the light that falls on it. Adding the constraint alone leaves the answer where it was — to 1.9 × 10⁻¹³ — because the search wants a dark wall and a dark wall satisfies it comfortably. Checking that is what distinguishes a bound that changes an answer from one that tidies a box.

The bounce count is handled separately from the four continuous parameters, because it is an integer and a simplex would interpolate it into something meaningless — a wall applied 2.4 times. Each count is searched independently and the best is reported, which is exact rather than approximate over a set with three members. In every profile point and under every bound, the answer takes the largest count offered, which is the honest signal that this parameter is doing what the box’s walls were doing before physics arrived.

Where the model stops #

The wall is a Gaussian and a real reflectance is not. A single symmetric band is a caricature of a pigment’s absorption, which has a tail on one side, sometimes a second band, and a substrate underneath it. The family is a family of caricatures and its extremes are the extremes of the caricature.

The floors are declared. Nothing here quotes a measured band width for a named pigment, and the physics is named rather than computed: no vibronic progression is modelled and no spectrum of any real colourant appears. What is computed is the response of the answer to the floor, which is the part that does not depend on the floors being right.

The observer is this collection’s own. The centre wavelength runs to the short end because the eye’s sensitivity collapses there, and where it collapses depends on the lens, the macular pigment and the short-wavelength cone — three things that differ between people by a good deal. A worst case computed for a seventy-year-old’s lens would put the band somewhere else.

And the search maximises one number. The worst case is worst for an adapted observer averaged over this collection’s reflectance family, and a different test set would find a different wall. That is the same objection the census’s own construction faces one essay over, and it has the same answer: perturb the construction and see whether the conclusion moves.

The generalisation #

The pattern is stateable in one line and is the phase’s central one in its clearest form.

A worst case is a property of a constraint set, so the useful thing to report is not the number but the function. How the answer depends on the tightest bound anybody was willing to state is a curve, and a curve invites a reader to substitute their own bound; a number invites them to quote it.

The second half is about which bound to look for. The instinct is to bound the thing that is obviously extreme — here, how saturated a wall is allowed to be. That bound turns out never to bite, because the worst wall is dark rather than colourful. The bound that binds is on the band’s width, which nobody would have nominated.

The distinction between the three kinds of bound is worth carrying past this subject, because they behave differently under scrutiny. An arithmetic bound is a decision to stop searching and cannot be defended at all. A physical bound can be argued about but not moved: a vibrational quantum is what it is. A bound on what is manufactured is the least stable of the three, because it is a statement about an industry at a date — and it is the one that reads as most concrete.

And the third half is the methodological one. Profiling one parameter with the others optimised is a more reliable instrument than searching all of them under a moving constraint, because it converts a boundary problem into an interior one. The first version of this essay’s central curve was wrong, and it was wrong in a way that looked like a result.

Who found it, and when #

The broadening of electronic transitions by vibrational coupling is Franck’s and Condon’s, from the 1920s, and the resulting band shapes are the subject of every textbook on molecular spectroscopy. That lanthanide f–f transitions are anomalously narrow because the 4f electrons are shielded by filled 5s and 5p shells is equally old and is why neodymium glass filters are wavelength standards.

None of that is a colour science result and all of it is a constraint on colour science. The gap this essay fills is not in either literature: it is between them, where a colour-science search over a family of reflectances meets a chemical fact about how narrow a reflectance feature can be, and nobody had put the two in the same calculation.

Where the ladder goes next #

Three of the four parameters are now bounded and the fourth is bounded by the observer. What remains unbounded is the fifth, which is not continuous at all: the number of times the light bounces, which the search treats as a free integer and always takes to the largest value offered.

A room does not offer an integer. What a room does to the light in it is a geometric sum over every number of bounces, weighted by how much light survived each one — and that is bounded by the walls reflecting less than everything, which is not a choice anybody makes.