A departure is not a unit

The previous round built an instrument and it worked well: six published quantities, recomputed under six colour-difference formulae, with each formula calibrated onto the published one’s scale so that only real disagreement survived. This round tried to point the same instrument at a different kind of choice, and it did not fit.

The claim #

A unit is a function applied to the answers and a departure changes the object, so the two need different instruments and reach different distances.

• A unit can be swapped at the end of any computation. Every quantity in the previous round’s inventory took one.
• A departure has to be plumbed through every stage from the sample to the tristimulus, and each stage has to be willing to accept the departed object.
• Two of the six quantities admit a departure — the adaptation census and the metameric pair — because both take reflectances and a light.
• Four do not, and the four reasons are different rather than being one obstacle.
• And one departure cannot be expressed in the census’s interface at all, because its whole content is that the sample does not have a reflectance.

What the previous round’s instrument was #

That round’s machinery holds an inventory of six published quantities: a change of light after an observer has adapted, a camera profile’s error, the gap between two standard observers, a metameric pair under the lamp that breaks it, the same image on two papers, and an observer two seconds into a new room.

Each entry knows how to recompute itself with a different colour-difference formula substituted, and each is required to reproduce its own source file’s published number to the last bit when the published formula is used. That last requirement is what makes the inventory trustworthy — a re-implementation that has drifted gives a plausible table and measures nothing.

The instrument works because a unit is a metric on the answers. Every one of the six quantities ends in a colour difference, so every one of them has a place where a different function can be applied, and nothing upstream needs to change.

Why a departure will not go through the same hole #

A departure is not applied at the end. It changes what the sample is, which means the change enters at the first stage and has to survive every stage after it.

Each stage is a function with a signature, and a signature is a statement about what kind of object the stage accepts. A function taking a list of reflectances will accept a departed reflectance — a list of numbers is a list of numbers — but only if the departure can be expressed as a reflectance.

That is the whole of the boundary. Three of the four departures produce a reflectance: an aperture reading is a reflectance, a glossy surface’s directional-hemispherical reflectance is a reflectance, and a truncated integral leaves the reflectance alone. The fourth does not, and cannot, because a fluorescent sample’s radiance factor is a different curve under every light.

So the census can be handed three of the four and refuses the fourth, and the refusal is written into the code rather than discovered at run time: asking for a departed set of reflectances under the wavelength index throws, with a message saying that there is nothing to hand it.

The four that are out of reach, one reason each #

The camera profile. Its fit is over a chart whose patches are declared as reflectances, and a departed chart is a different chart rather than the same chart measured differently. The profile fitted to it would be a profile for another object, and comparing the two would measure the difference between two charts. That is a real question and it is a different one.

The observer gap. The quantity is the difference between two colour-matching functions on one set of surfaces. It is a property of two kernels; a departure acts on the sample, and the sample here is a common factor that mostly cancels. There is nowhere for the departure to enter that changes the answer.

The paper change. The two papers are measured curves, and what a departure would change is what those measurements meant. Substituting a departed curve replaces a measurement with a model, which is a different kind of substitution from the one being audited.

The settling observer. The clock is in the observer. No departure of the sample reaches it.

Four quantities, four different obstacles, and the pattern is that a departure needs the sample to be a live variable at the point where it enters. In these four the sample is a fixed input, a common factor, a measurement, or absent.

What the two reachable ones say #

Both give the same answer and it is a negative result worth having.

Replacing every one of the census’s hundred and twenty-five test surfaces with what a four-millimetre aperture would report moves each of the fourteen rows’ residuals by at most 5.0 per cent, and by 4.6 on the row the collection quotes most. Replacing them instead with what a glossy version of the same surface returns moves them by at most 9.3 per cent.

Those are small, and the reason is structural rather than lucky. Both departures change the reflectance, and the census’s residual is what an observer’s gain leaves after a change of light — a quantity computed from ratios of tristimulus values before and after. A change to the reflectance appears in both terms of the ratio and largely cancels.

So: a departure that does not depend on the light is very largely absorbed by an observer’s own gain. The one departure that does depend on the light is the one the census cannot be given, and that is not a coincidence either — it is the same property seen twice.

What the gain absorbs, and what it does not #

The cancellation account is right about one of the two quantities and the wrong way round about the other.

Applying the aperture departure to every surface and asking separately what it does to the unadapted change and to the residual: the total moves by 0.73 per cent on average and 1.44 at worst, and the residual by 4.12 per cent on average and 4.75 at worst — five and a half times more.

So the departure really is largely absorbed, and it is absorbed before the gain rather than by it. A change to the reflectance appears in both contexts of the ratio and cancels there, which is why the total barely moves at all. Whatever survives that cancellation then lands entirely on the residual, which is a small difference between two nearly equal quantities — so a departure worth one per cent of the change is worth four per cent of what is left of it.

The gloss departure behaves differently, and the difference confirms the reading. Its total moves by 5.59 per cent and its residual by 6.78 — a ratio of 1.21 rather than 5.66. Gloss adds a flat term to every reflectance, and a flat term does not cancel between two lights nearly as completely as a spatial kernel does, so it moves the total substantially and the residual only a little more.

Two departures, two mechanisms, and the same conclusion about size. Both leave the census’s numbers within ten per cent of where they were. But the aperture’s smallness comes from a cancellation upstream of the observer, and its residual is proportionally the more disturbed of the two — which is the reverse of what absorbed by an observer’s own gain suggests, and it matters to anybody reading a residual as though a small departure could only make it smaller.

What was computed, and how #

The reachability judgement is written down rather than computed, which is deliberate and is the file’s own weak point. It is a claim about interfaces, and an interface is a thing a person reads.

What is computed is the two entries that claim to be reachable. Both are pushed through the collection’s own census function by handing it a different set of reflectances, using the same code path the published numbers came from — so a discrepancy would be a discrepancy in the surfaces rather than in the machinery.

And the refusal is computed too. The function that produces a departed set of surfaces throws for the wavelength index, and the assertion behind the exposure figure requires that it throw. A boundary that is asserted is a boundary the build maintains; a boundary that is only described in prose drifts the first time somebody adds a case.

Where the model stops #

The four unreachable entries might be reachable with more work. Each obstacle is an interface rather than a law, and an interface can be widened. What the round claims is that widening four of them is a larger job than the audit was, which is itself the finding: an audit of a structure costs more than an audit of a convention, by about the ratio of a rewrite to a substitution.

And two reachable entries are a thin instrument. The previous round had six quantities and could say that every one of them moved by a factor of two. This round has two, and can say that both move by under ten per cent. The second statement is weaker and it is weaker in a way the first could not have warned about.

What a wider interface would have to look like #

The four unreachable quantities are unreachable because of what their functions accept, and it is worth asking what a version that accepted a departure would take instead.

The answer is the same in all four cases: a sample would have to be a description rather than a curve. A function taking reflectance: number[] cannot be handed a fluorescent sample; a function taking sample: { kind, coefficients, kernel?, matrix? } can, and can decide what to do with each kind.

That is an ordinary piece of software design and it is not free. Every stage between the sample and the answer would have to know about the richer type, which means the change propagates exactly as far as the departure does — and the propagation is the cost the audit ran into.

It also has a real benefit beyond this round. A sample that carries its own description can say which variable its reflectance is in, which is a convention this round found missing and worth up to eight ΔE₀₀ the moment two curves are combined. One type change would close two gaps.

The negative result, stated plainly #

The round’s two reachable entries agree and the agreement is the most useful thing in this essay, so it is worth separating from the discussion of interfaces.

Replacing the census’s surfaces with what an aperture reports moves nothing by more than 5.0 per cent. Replacing them with what a room reports moves nothing by more than 9.3. Both are small, and both are small for the same structural reason: the census reports what an observer’s gain leaves, and a gain applied after a change to the reflectance largely absorbs it.

That result is worth having in both directions. It says the collection’s adaptation work is robust to two of the four departures, which is reassuring. And it says the census is a poor instrument for detecting them, which is a warning: a quantity that barely moves under a departure cannot be used to measure one.

The previous round had the same shape of warning and phrased it as ratios surviving better than levels. A residual is a difference of two things computed the same way, so a common-mode change cancels — which makes it a good statistic for what it measures and a blind one for everything that enters both terms.

The generalisation #

The transferable point is a hierarchy of what an audit can vary, ordered by how far into a computation the change enters.

A metric applied to the answers is the easiest: swap the function at the end and nothing upstream cares. That is a unit, a colour-difference formula, a summary statistic.

A parameter inside a stage is next: multiply it by 1.25 and re-run. That is what the elasticity round did, and it reaches every declared number in a computation.

A set the computation averages over is harder, because it has no magnitude and needs several instruments to say anything about it.

An object is hardest, because it changes what every stage is being handed. That is where this round is, and the cost shows up not as difficulty in the arithmetic but as stages that refuse the new object.

The practical consequence: when planning an audit, ask where the change enters. A change entering at the end is an afternoon. A change entering at the beginning is a rewrite of everything downstream, and the honest budget is the number of stages between the entry point and the answer.

The identity the four rest on and the interaction between them are the two things a list of four numbers cannot carry, and both are measured.

The last of the four is not an inventory at all. Each departure has two factors, and either factor being empty makes the departure vanish — which is a statement that can be checked rather than swept.

Why the boundary is worth publishing #

An audit that reaches two of six quantities could be reported as reaching two, with the other four left unmentioned. Publishing the boundary instead is a deliberate choice and it has two justifications.

The first is that the four refusals are informative, and each in a different way. One says the sample is a fixed input; one says the sample is a common factor that cancels; one says the sample is a measurement rather than a model; one says the sample is not involved at all. Those are four different facts about the collection’s own structure, and none of them would have been visible without trying.

The second is that a stated boundary is a testable claim about the next round. If later work reaches one of the four, the reason it could not be reached here becomes a thing that changed rather than a thing that was wrong — and this collection’s habit is that a deferral rests on a measured number rather than an assumed one.

The alternative — reporting the two successes and describing the instrument as general — is the failure mode this collection has written about under other names. A fit can be exact and empty; an instrument can be precise about the two things it reaches and silent about the four it does not.

Who found it, and when #

The general practice has a name in the social sciences — multiverse analysis, computing a result under every defensible combination of the arbitrary choices inside it — and its standard finding is the one three rounds of this collection have reproduced: the construction moves the answer more than the sampling does.

What the multiverse literature says less about is that the choices are not all the same kind, and that the cost of varying one depends on where it sits. A paper reporting a multiverse over analytic choices is usually reporting a multiverse over choices made near the end, because those are the ones a script can loop over. The choices made at the beginning — what counts as an observation, what the object of study is — are the expensive ones and are usually reported as assumptions rather than swept.

That is the same boundary this essay describes, in a field with no spectra in it.

The cost of the audit, by stage #

A last observation about expense, since this essay is about what an audit can afford.

The unit audit’s cost was in re-implementation: six quantities recomputed under six formulae, each required to reproduce its own source. That is bounded work — six times six — and every hour of it produced a number.

This round’s cost was in acquisition. Two of the four departures needed physics this collection did not have, and neither could be quoted: a diffusion kernel and a microfacet lobe both had to be implemented, cross-checked against their own closed forms, and then checked against each other where they shared a constant. Most of the round was spent before any number existed.

That ratio is the practical difference between auditing a convention and auditing an object, and it is worth carrying into planning. An audit of a convention is priced per quantity; an audit of an object is priced per model, and the second price is paid whether or not the audit finds anything.

Where the ladder goes next #

The obvious next audit is the one this round has now demonstrated to be expensive: the observer as three curves. It is an object rather than a parameter, so it enters at the beginning, and every stage downstream would have to accept a different kind of observer.

There is a cheaper version of it available, and the round’s own experience says to take the cheap version first. Where a published alternative exists, an audit is a substitution; where it does not, an audit is a construction. The observer has published alternatives — ten-degree functions, the physiological fundamentals, a population — and this collection has all three already, which means the substitution is available and only the plumbing is missing.