Every worst surface sits on a declaration

A worst case is only a fact about the world when the thing that stops it is. Two rounds of this collection have bounded a painted wall and got real answers; bounding the objects in front of the wall gives an answer that is a reading of a configuration file.

The claim #

The supremum of the adaptation residual over the test surfaces is not a property of surfaces. It is a reading of two declared numbers, and the one constraint in the file that is about paint is slack everywhere.

• All fourteen worst surfaces sit exactly on the L¹ modulation bound and exactly at the top brightness level. Every constraint active, on every row, with no interior structure anywhere.
• That is the opposite of what bounding the wall gave. There the residual turns over at a band width of 6.0 nanometres, because a band that narrow returns too little light to move the white — which is physics.
• The excitation-purity ceiling never binds. The most saturated surface the region admits reaches a purity of 0.459 against a declared ceiling of 0.6.
• So imposing it changes nothing at all, on any row, and a bound that changes nothing is indistinguishable from a bound that is absent unless somebody checks.
• The right report is a conditional one: the worst case the declared region contains, with the declaration printed beside it.

Two worst cases, two shapes #

This collection has now searched for a worst case twice, over two different objects, and the two answers have opposite characters.

The wall. What is the worst change of light a painted room can produce — a search over a wall’s centre wavelength, band width, depth and base reflectance, under three nested bounds. The answer has interior structure: the residual turns over at a band width of about six nanometres, because below that the band returns too little light to move the adapting white, so a narrower paint is a less disruptive paint. That turnover is a fact about physics, it was found by profiling one parameter with the others optimised, and it corrected a claim the round before this one had made about the family having no worst case.

The surfaces. What is the worst object for a given change of light — a search over a brightness and two modulation depths, over the region the test set is drawn from. The answer has no interior structure at all. Every one of the fourteen searches terminates at the corner: level 0.540, which is the region’s declared top, and |c₁| + |c₂| = 0.700, which is the region’s declared L¹ bound.

Fourteen for fourteen. The search is not finding a worst surface; it is reading off where the region stops.

Why the two differ #

Because the two objects are bounded by different kinds of thing.

A wall’s parameters are bounded by what a pigment can do. A band narrower than about ten nanometres needs an absorber with a transition that sharp, and the essays that established those floors argue them from the physics of electronic transitions in dyes and pigments — the lanthanides being the exception that proves the rule, and not being paint. Because the bound is physical and the physics also decides the objective, the objective has structure inside the bound and the answer can be interior.

The test surfaces’ parameters are bounded by nothing of the sort. The reflectance basis is a cosine series, its two modulation coefficients are dimensionless numbers, and the L¹ bound of 0.7 exists so that the surfaces stay inside [0, 1] without clamping — which is a numerical requirement, not a physical one. There is no reason a real surface should have |c₁| + |c₂| ≤ 0.7 rather than 0.8 or 0.5. And the residual grows monotonically with modulation depth for every change of light, so the maximum is wherever the modulation is allowed to stop.

A monotone objective on a declared box has its maximum at the box’s corner. That is not a finding about adaptation; it is arithmetic. The finding is that this is the situation, that nothing said so, and that the number was about to be published as a worst case.

The constraint that does not bind #

There is one bound in the same file that is about the world, and it is worth measuring rather than assuming.

The excitation-purity ceiling of 0.6 is this collection’s statement of how saturated a wall can be — a limit about a market rather than about physics, since the argument is that nobody manufactures or sells paint beyond it. It is one of the three nested bounds the wall search runs under and it does real work there.

Applied to the test surfaces it does none. The most saturated surface the region admits has an excitation purity of 0.459, found by search over the whole region rather than over the lattice, against a ceiling of 0.6. The ceiling is slack everywhere, by a comfortable margin, and imposing it on the test set changes not one of the fourteen suprema — checked, on the row with most to lose, to six decimal places.

The reason is the basis. Two cosines of one and two half-cycles across the visible band cannot make a narrow, deep feature; the deepest thing they can do together is a broad lobe, and a broad lobe has modest excitation purity however deep it goes. The family’s own smoothness is a tighter constraint on saturation than any market is, and no one had noticed because the two constraints had never been in the same picture.

What a slack bound is worth #

Nothing, and the reason it is worth saying so out loud is that a slack bound and an absent bound are indistinguishable from the code.

The painted bound is implemented, it is in the function’s signature, it is applied on every evaluation, and it returns the same fourteen numbers as the unbounded version. An argument that cited it — the worst case for a paintable surface is 3.65 ΔE*₀₀ — would be a true sentence whose stated reason is not the reason. The purity ceiling is not what makes 3.65 the answer; the L¹ bound is.

So the check is an equality rather than an inequality. The assertion requires the two ladders to agree exactly, which is what a slack constraint guarantees and what a binding one would break. If a future change to the basis let the surfaces reach purity 0.6 — a third basis function with a narrower feature would do it — the assertion fails, and the failure is the right notification.

That is the general pattern this collection keeps arriving at from different directions: an assertion that has never rejected anything proves nothing, and a constraint that has never bound is the same object in a different costume.

What to report instead #

The worst case, with the declaration printed beside it. Not the worst surface costs 3.65 ΔE*₀₀ but the worst surface this region contains costs 3.65, and the region is bounded at a modulation sum of 0.7 and a brightness of 0.54, neither of which is measured. The second sentence is longer and is the one that is true.

And the ratio rather than the level, wherever the argument allows — the conclusion the quadrature argument reaches independently. The worst-to-mean ratio survives the declaration almost entirely — moving the region’s reach scales the mean and the worst case together — so the statement this change of light is four times worse for its worst object than for an average one is a statement about the change of light, while its worst object costs 1.48 is a statement about a boundary.

That distinction is now the rule for every worst case this collection publishes over a set, and it has the same shape as the rule about levels and orderings that came out of the quadrature argument: the comparative quantity is the robust one, and it is usually the one the argument actually needs.

What would give a real bound #

Three candidates, none of them cheap, and naming them is more useful than pretending the question is closed.

Energy conservation is already there and is not enough. The surfaces are required to stay in [0, 1] without clamping, which is what the L¹ bound implements — so the declared bound is the physical one for this basis, and the arbitrariness is in the basis rather than in the number. A different basis would have a different admissible region and a different corner.

Smoothness is the real constraint and is not implemented, which is also what decides how many dimensions the family really has. Real reflectance spectra are smooth because absorption bands have finite widths, and a bound on how much spectral curvature a surface may have would be a bound from the world. It is also exactly the constraint that would interact with what a fourth dimension costs, since the expensive fourth shapes are the ones with the most curvature.

And a measured collection would bound it empirically, by containing what it contains. That is the honest answer and this site does not have one, which is the absence this collection keeps arriving at from every direction.

The L¹ bound is not the physical one either #

The section on what would give a real bound concedes one thing to the declaration: the surfaces are required to stay in [0, 1] without clamping, which is what the L¹ bound implements — so the declared bound is the physical one for this basis. The two declared numbers do not support that.

A surface here is a level times one plus a sum of two cosines, and both cosines are at +1 at the short end of the band, so the sum of their coefficients is attained. At the region’s top level of 0.540, the requirement that a reflectance stay at or below one permits

|c₁| + |c₂| ≤ 1/0.540 − 1 = 0.852

and the requirement that it stay at or above zero permits 1.000. The binding physical limit is 0.852 and the declared bound is 0.700 — 82 per cent of it.

Put as reflectances: at the declared corner a surface runs from 0.162 to 0.918, and at the energy-conservation corner it would run from 0.080 to 1.000. The declared region stops a comfortable distance short of touching either end of the physical range.

So the arbitrariness the essay locates in the basis is also in the number. A second declaration is hiding inside the one the essay clears, and it is the one every worst case in the collection sits on: raising the L¹ bound to what energy conservation actually allows would move all fourteen suprema, because the objective is monotone up to whatever the bound is set to.

That does not weaken the essay’s conclusion; it sharpens the arithmetic behind it. The right report is still the conditional one, and the condition now has a size attached: the region reaches 82 per cent of the way to the physical wall, and the last 18 per cent is unaccounted for.

The purity ceiling cannot bind, not merely does not #

The measurement is that the region reaches an excitation purity of 0.459 against a ceiling of 0.600 — 76 per cent of it, with a slack of 0.141. The essay treats that as a comfortable margin at the current settings and flags a third basis function as the change that would close it.

There is a stronger statement available without a third basis function. Raising the L¹ bound all the way to its physical limit scales the modulation by 1.217; if purity rose in proportion, the region would reach 0.559 — still short of 0.600. So on this basis, at this level, the ceiling cannot be made to bind by any admissible setting of the bound it is competing with. The additive-versus-market constraint is not slack because the number happened to land there; it is slack because a two-cosine family cannot get to a purity of 0.6 while remaining a reflectance.

The linear-scaling assumption is the weak part and it is checkable in one search: report the region’s maximum purity at the energy-conservation corner rather than at 0.700. If it comes out above 0.6 the stronger claim fails and the essay’s version is right; if below, the ceiling is not merely inactive but unreachable, and the check the essay installs — the region cannot reach the purity ceiling — is guarding against a change of basis and nothing else.

The ratio’s top end checks against two other essays #

The worst-to-mean ratios are quoted as running between 1.9 and 4.0, and the top of that range can be recovered from numbers published elsewhere in the collection, which is worth doing because it ties three independent computations together.

The macular row’s supremum over this region is 1.478 and its published mean is 0.368. Their ratio is 4.02 — the top of the stated range, on the row with the most concentrated distribution in the census, which is exactly the row that should carry it.

That is a three-way consistency check across essays that compute different things: a search over the region, a mean over the lattice, and a ratio quoted as a range. All three agree, which is the best available evidence that the region, the lattice and the census are describing one object rather than three.

It also makes the essay’s own recommendation concrete. This change of light is four times worse for its worst object than for an average one is a sentence about the macular pigment; 4.02 is the number, and it survives the declaration because both halves of it scale together when the region’s reach moves. The level it is a ratio of does not.

Where the model stops #

The search is over three parameters of one basis. Nothing here bounds what a worst case would be over surfaces of a different construction, and the clamped realistic family gives systematically lower numbers — so the region’s corner is not an upper bound over surfaces in general, only over this family.

Nor does the essay establish that the L¹ bound is wrong. It establishes that it is undefended, that the answer is a monotone function up to it, and that the answer is therefore whatever it is set to. A defence would be a statement about real surfaces’ modulation depths, which is a measurement nobody here has made.

What the wall search got right #

Worth being explicit about, because the contrast is the essay’s whole argument and it would be unfair to leave the wall search as merely the thing that worked.

Its answer is interior because its bounds are physical and because the search was set up to find an interior answer. The width profile fixes one parameter and optimises the other three, fourteen times, rather than searching all four under a floor — which converts a boundary problem into an interior one and is the reason the turnover at six nanometres is visible at all. The first version of that search did it the other way and reported a lower worst case at the narrowest floor than at three times it, which is not a shape any function has.

So an interior maximum is partly a property of the problem and partly of the method, and the surface search here would not have been rescued by better method: profiling one of its three parameters with the other two optimised gives a monotone curve rising to the bound, on every row, which is the honest report of a monotone objective.

The distinction matters for reading the two results together. The wall’s six-nanometre turnover is a fact about pigments. The surfaces’ corner is a fact about a file. Both were found by the same search machinery, and only the setup and the bounds differ.

Who found it, and when #

The distinction between an active and an inactive constraint is the whole of Karush–Kuhn–Tucker, and the practical corollary is standard: an optimum on a constraint reports the constraint, and an optimum in the interior reports the objective. Any optimisation text says so, and any engineer who has watched a solver return the upper bound of a design variable has learned it once.

Its arrival here is the second half of an argument the round before this one started and stated as an open thread: what is not bounded is the set of surfaces the residual is averaged over, and a worst case is as much a statement about that set as about the wall. The thread was written down correctly. What it did not anticipate is that the answer would be entirely a statement about the set, with no interior structure at all — which is a stronger conclusion than the thread expected and a less satisfying one.

What the check now asserts #

The repair is a check rather than a number, and it has three parts because the finding has three.

That every worst surface is on both bounds. Fourteen searches, fourteen corners, asserted as a count of interior answers being zero. If a future basis or a future objective produced an interior maximum, that would be a real change in what the region means and is worth being told about.

That the region cannot reach the purity ceiling. A search over the whole region for its most saturated member, compared against the declared ceiling. This is the assertion that would fail if a third basis function with a narrower feature were added — which is exactly the change that would make the ceiling start to bind, and exactly the change somebody might make for an unrelated reason.

And that imposing the ceiling changes nothing. Checked on the row with most to lose, to six decimal places, rather than on all fourteen — because a slack constraint guarantees the rest and re-deriving the guarantee costs two minutes of search per run. The spot check is what keeps the guarantee from being an argument nobody tests.

Together those three are the difference between the purity ceiling is slack as a sentence in an essay and as a property of the code. The sentence goes stale silently; the check does not.

Where the ladder goes next #

The set has been audited from four directions and one thread of the round is closed. The other opens somewhere else entirely: a claim about confusion points that rested on a declared width, and a unit that does not.