The darkest wall anybody sells

A bound on what a paint can be is only useful if it binds, and the one everybody would name does not.

The claim #

Three nested bounds on a painted wall give two answers, because the third — a ceiling on how saturated a paint is allowed to be — never binds. The worst wall for an adapted observer is dark rather than colourful.

• The three are nested by construction: the arithmetic box, then a reflectance a molecule could produce, then a paint somebody sells.
• They give 28.4, 21.9 and 21.9 ΔE00. The second and third are the same number to within the search’s own noise.
• The purity ceiling is slack by a wide margin. Declared at 0.6, the worst physically admissible wall reaches 0.52; under the loosest bound it is 0.18.
• What binds instead is the band’s width — the constraint a reader would name last — and the fact that a reflectance cannot exceed one.
• And the mechanism is worth more than the number. What breaks an adapted observer is a wall that takes most of the light away and returns a narrow band of what is left, which is a deep bottle-green rather than a signal orange.

Three bounds, nested #

The worst change of light a painted room can produce has no maximum inside a box, so the number a search returns is a number about the box. Bounding the band’s width by what a molecule can do gives the bound a mechanism and does not remove the dependence.

This essay is about the third bound, which is of a different kind again. The first is arithmetic — a range typed into a file, defensible only as a decision to stop searching. The second is physical — a vibronic progression on a molecular transition, arguable but not movable. The third is commercial, and is a statement about an industry at a date.

A commercial bound sounds like the weakest of the three and is in one sense the most concrete: a paint shop’s range is a finite set of things somebody can buy. It is also the least stable, because next year’s range is different, and a conclusion resting on it has a shelf life the other two do not.

What the bound says #

An architectural paint is a pigment at a loading a roller can carry, on a white base, sold to cover a wall. It reaches a fraction of the excitation purity — how far a colour’s chromaticity reaches from the white towards the spectral locus — that a physically admissible reflectance could.

The ceiling is declared at 0.6, which is generous. A strong yellow or a saturated red reaches into the high sixties or low seventies; most of a range sits well below half. Declaring it generously is deliberate: a bound that does not bite at 0.6 does not bite at 0.5 either, and the finding is stronger for having given the constraint every chance.

The worst wall under the physical bound has a purity of 0.52. It is inside the ceiling and stays inside at every band width the profile covers. The two searches — with the ceiling and without — return the same wall and the same residual to a part in ten thousand, which is the numerical signature of a constraint that is never active.

Why the worst wall is dark #

The mechanism is the interesting half and it is not obvious in advance.

A change of illumination hurts an adapted observer in proportion to how badly a diagonal gain in the observer’s own basis fails to undo it. A gain can move the three channels independently and nothing else; what defeats it is a change that redistributes light within a channel’s band, because a single multiplier per channel cannot represent that.

A dark wall with a narrow band does exactly that. It removes most of the light everywhere and returns a spike, so the surviving spectrum is concentrated in a region much narrower than any cone’s sensitivity, and the three cone responses end up in a ratio no diagonal can restore.

A saturated wall is a different object. High purity means the chromaticity is far from the white, which needs a reflectance strongly biased towards one end of the spectrum — but not necessarily a narrow one. A saturated red paint reflects broadly above 600 nanometres, and a broad shift is close to what a gain is for. Saturation is a statement about a chromaticity and narrowness is a statement about a spectrum, and it is the second that a gain cannot follow.

So the intuition fails because it is reasoning about the wrong quantity. The eye’s problem is bandwidth, not distance from white.

What binds instead #

Three constraints hold the answer under the physical bound, and none of them is the one a reader would nominate.

• The band-width floor, at forty nanometres — the narrowest an organic pigment’s absorption can be.
• The requirement that a reflectance not exceed one, which the answer reaches: the searched wall has a base of 0.09 and a depth of 0.91, summing to almost exactly one.
• The short-wavelength end of the box, which is not a fact about paint at all but about where the observer stops seeing.

The second is worth a note because it is the one constraint that is neither declared nor arguable. base + depth ≤ 1 is what a reflectance is, and the box the previous round used does not enforce it — a base of 0.85 with a depth of 0.9 is inside that box and describes a wall reflecting 1.75 of the light that falls on it. Adding the constraint alone leaves the answer exactly where it was, to 1.9 × 10⁻¹³, because the search wants a dark wall anyway.

Checking that is what distinguishes a bound that changes an answer from one that tidies a box, and it had to be checked rather than assumed: a search finding its optimum in an unphysical corner would have invalidated everything downstream of it.

Tightening the bounds makes the wall more saturated, not less #

The two purity figures are quoted a line apart and the relation between them is the opposite of what a reader would expect. Under the loosest bound the worst wall reaches a purity of 0.18; under the physical one it reaches 0.52.

Relaxing the constraints therefore lowers the saturation of the answer, by a factor of 2.9. The ceiling declared at 0.6 is slack by 0.08 where the search is most constrained and by 0.42 where it is least — so the constraint that is supposed to cap saturation is furthest from binding exactly where the search has the most freedom to be saturated.

That inversion is a stronger version of the essay’s argument than the argument makes. It is not merely that the objective happens not to want saturation; it is that removing every constraint but the box takes the answer further from the purity ceiling, which is the behaviour of a quantity the objective is actively indifferent to. A constraint that binds less as the feasible region grows is one the objective is not pushing against at all.

The mechanism follows from the essay’s own account. Under the loosest bound the band width has no floor, so the search takes a very narrow band at the short-wavelength end — where the observer’s sensitivity is small, so most of what reaches the eye is the base’s broadband contribution and the chromaticity stays near the white. Under the physical bound the band cannot be narrower than forty nanometres, so it has to be deep instead, and a deep band returns proportionally more of its own colour. Narrowness and saturation trade against each other, and the objective wants the first.

One constraint accounts for the whole difference #

The three answers are 28.4, 21.9 and 21.9, and the arithmetic of that is worth stating explicitly because it is the essay’s cleanest result.

The physical bound costs 22.9 per cent of the arithmetic answer and the commercial bound costs nothing. Since the only thing the physical bound adds is a forty-nanometre floor on the band’s width — the other two constraints it carries being the reflectance ceiling, which is inactive, and the observer’s own band, which is inactive — the entire difference between the loosest and tightest bounds is one number.

So the ladder of three bounds is really a ladder of one. That is a much sharper way to put the essay’s generalisation than the constraint a reader would nominate is not the one that binds: on this problem there is exactly one binding constraint anywhere in the hierarchy, and everything else in the three predicates is decoration.

It also says what a fourth bound would have to do to matter. Anything that does not touch the band width will return 21.9, whatever else it constrains — which is a prediction the same search could check in a few seconds against any candidate constraint somebody proposes.

Two statements about the reflectance ceiling that do not sit together #

The section on what binds makes two claims about base + depth ≤ 1 and they are hard to hold at once.

The answer sits on it exactly: a base of 0.09 and a depth of 0.91, which sum to 1.00. And adding it changes nothing, to 1.9 × 10⁻¹³, because the search wants a dark wall anyway.

If the search freely chose a depth of 0.91 with a base of 0.09 when nothing stopped it going further, then the two summing to exactly one at two decimal places is a coincidence with a prior of about one in a hundred. If instead the depth was capped by the constraint, the constraint is active and imposing it cannot be a no-op.

There is a third possibility and it is the likely one: the parameterisation already enforces the constraint, with the depth expressed as a fraction of what the base leaves available, so a search that never violates it cannot be moved by adding it. In that case the 1.9 × 10⁻¹³ is a check that the predicate agrees with the parameterisation rather than a check that the optimum is physically admissible, and the essay’s own gloss — because the search wants a dark wall anyway — is not what the number demonstrates.

Which of the three it is matters for the sentence built on it: checking that is what distinguishes a bound that changes an answer from one that tidies a box. If the parameterisation was already tidy, the check distinguishes nothing, and the reassurance that a search was not finding its optimum in an unphysical corner has not been earned — it has been assumed one level down.

The repair is one line: report the base and depth from a search run with the predicate genuinely removed, and see whether the depth exceeds 1 − base. If it does not, the essay’s gloss is right and the coincidence is real. If the parameterisation makes that impossible to run, that is the answer.

What was computed, and how #

The purity is computed against D65’s white point rather than the equal-energy white this collection’s dominant-wavelength machinery defaults to. That default is right for a question about a chromaticity in the abstract and wrong for a question about a surface under a light: purity is measured from the white the surface is being seen under, and a wall lit by daylight that is neutral against equal energy is not neutral.

Getting it the other way round would have shifted every purity here by a few hundredths and would not have changed the conclusion, since the ceiling is slack by 0.08. That is worth saying because it is the kind of detail that would matter if the answer had been close.

The three bounds are three predicates over the same four parameters, and the same search is run under each — so a difference between two answers is a difference in the feasible region rather than in the method. Where two of the three return the same answer, they return it from two independent simplex runs, which agree to a part in ten thousand: that is the noise floor of the search and is why the two are reported as the same number rather than as 21.911 and 21.910.

What a catalogue would add #

It is worth being clear about what a real paint range would contribute if one were consulted, because the answer is not a better number.

A catalogue is a finite set of reflectance curves, so a search over it is an enumeration rather than an optimisation and its answer is the worst member rather than a supremum. That is a genuinely different object: it has no boundary to sit on, it cannot be extrapolated, and it changes when the catalogue does.

It would also be a measurement, which this collection would then be quoting. Measurements are quoted here without apology and everything downstream is computed, and a range of reflectances would sit firmly on the quoted side. The site’s fourth invariant divides measurements — quoted without apology — from everything downstream, which is computed. A range of measured reflectances would sit on the quoted side and would make every number below it partly a statement about somebody’s spectrophotometer.

The version here avoids that at the cost of being a caricature, which is the usual trade and is stated rather than hidden. What it buys is that the response to the bound is computed rather than quoted, so a reader with a catalogue can read their own answer off the curve — which is the same move the ellipses’ missing error bar needed one field over.

Where the model stops #

No real paint appears anywhere in this essay. The wall is a Gaussian band on a base, the purity ceiling is a declared number, and no manufacturer’s range was consulted. What is computed is the response of the answer to the bound, which is the part that does not depend on the bound being right.

And a purity ceiling is a poor model of a paint range even as a declared input. A range is a finite set of pigments with particular spectra, not a convex region of chromaticity space — so the real constraint is more like this wall is not made of anything anybody mills, which is a statement about a catalogue rather than about a number.

The finding survives that objection in one direction only. If the true commercial constraint is tighter than a purity ceiling of 0.6 in some direction the ceiling does not describe — a narrowness limit, say, since real pigments are broad — then it might bind after all, and it would bind through the band width rather than through the saturation. That is the physical bound again, arriving from the commercial side, which is a satisfying place for the argument to end up.

What a paint shop would have to sell #

It is possible to run the argument in reverse and ask what a paint would have to be to make the purity ceiling the binding constraint, and the answer explains why it never is.

A purity of 0.6 with a band forty nanometres wide requires a wall that returns almost nothing outside its band — a base near zero — because purity measures distance from the white and any broadband component pulls the chromaticity back towards it. So a high-purity narrow-band wall is a very dark wall, and the search is already choosing dark walls for a different reason.

The two requirements therefore pull in the same direction rather than against each other, which is why one of them can never bind while the other is active. A constraint only binds when it opposes the objective, and here the saturation ceiling and the objective agree about what a bad wall looks like — up to a point the objective never reaches, because what it actually wants is a narrow band and a dark base rather than a chromaticity far from the white.

The generalisation #

The result is a small one about intuition and a larger one about method.

The constraint a reader would nominate is not usually the constraint that binds, and this is the third time in this phase it has happened: the camera’s dye widths were bound by a colorimetric residual rather than by throughput; a display primary’s boundary is held by realisability rather than by gamut coverage; and a wall’s worst case is decided by bandwidth rather than by saturation.

The pattern behind all three is the same. A nominated constraint is chosen because its mechanism is easy to state, and the mechanism being easy to state is unrelated to its being tight. What decides a boundary is where a level set happens to run, which is arithmetic rather than storytelling.

Setting the four kinds side by side is the clearest summary this thread produced. An arithmetic bound cannot be defended. A physical bound can be argued about and not moved. A commercial bound is concrete and perishable. And a bound that falls out of a mechanism — a convergent series, a reflectance no greater than one — is the only kind that needs nobody’s agreement. A search whose answer sits on the fourth kind has found something; on any of the other three it has found a constraint.

The method that catches it costs almost nothing: write every candidate constraint as a cost and measure whether it is active. A constraint that is never active is a sentence that can be deleted from a specification, and knowing which ones those are is worth as much as knowing which ones bind.

Who found it, and when #

That the reflectances of real colourants occupy a small part of what is physically admissible is old and well quantified: the optimal colours Schrödinger described in 1920 and MacAdam computed the limits of in 1935 bound what any reflectance can do, and the Pointer gamut, published in 1980, is the empirical statement of what real surfaces actually reach. Both are much larger than what a paint shop sells.

Neither is a constraint on bandwidth, which is the gap this thread has been working in. An optimal colour is a step function — the sharpest possible reflectance — and the MacAdam limit is computed from those step functions, so the limit already assumes the one thing a pigment cannot do. A bound that says no real colourant is that sharp is a different and more restrictive statement than a bound on what a reflectance can reach.

Where the ladder goes next #

Three bounds have been tried on a wall and the answer is that the physical one binds and the commercial one does not. The same question asked of a device rather than a surface has a different answer and a more useful one, because a device’s constraints are all real: somebody has to build it.

A display’s primary is chosen for four things at once, and the four hold its boundary in different directions — with one of them, again, never binding at all.