The conditions are the result

A round that reports twenty numbers and ten conditions will be remembered for the numbers. That is the wrong way round: the numbers depend on choices and the conditions do not.

The claim #

An audit’s conditions are worth more than its sizes, because a size is a measurement of an example and a condition is a statement about a mechanism.

• Every size in this round depends on three choices: a sample, a light and a construction. Change any and the number changes, sometimes by an order of magnitude.
• Every condition holds under all of them. Eight are exact to the floating-point floor and they are exact under every light and every sample tried.
• A condition is falsifiable in a way a size is not. A wrong size is a number somebody disagrees with; a wrong condition is a claim that fails a test.
• And a condition is usable. It says where to look for a fault, which sample cannot report one, and what change would remove a term.

The ten, stated plainly #

The round’s conditions, in the order the audits found them.

A spectrally flat sample has the same colour on every wavelength grid — every step, every range, every origin, every slit width — because the sample’s sum and the white’s sum are proportional.

A spectrally flat sample has the same colour for every observer, at any age, any field size, any macular density, any cone density and any peak, for the same reason.

A stimulus with one wavelength in it has the same colour for every observer, because it too is a scalar multiple of its own white.

An observer differing from another by a gain on each cone is the same observer, when the white is divided out in those cones.

Three curves spanning one space are one observer, when the basis change is undone before anything nonlinear happens.

A light with no feature narrower than the tabulation step costs nothing to sample, whatever the sample.

A light with nothing outside the tabulated range costs nothing to truncate, whatever the sample.

A Lambertian surface returns the same light in every direction, so a radiosity solver is exact on one.

A uniform illumination field reads any surface’s own reflectance, by reciprocity, whatever the surface.

And a directional solver with its lobe removed is the radiosity solver, exactly.

A condition is worth more than a size, for three reasons. Three reasons, and the third is the practical one.

A size depends on choices and a condition does not. The macular departure is 1.71 ΔE₀₀ on a red pigment under a 6500 K radiator and 4.47 under tungsten and 0.26 on the flattest member of the surface family. The condition that it vanishes on a flat sample holds under every one of those.

A condition is falsifiable sharply. A size can be disputed by disputing the example; a condition either holds at the floating-point floor or does not, and this round has a case where a proposed condition turned out to be a limit rather than an identity and had to be reported as 13.0 ΔE₀₀ instead of zero.

And a condition tells a reader what to do. A size says how much trouble there is; a condition says where it is not, which is what a diagnosis needs. Knowing that a disagreement about a neutral cannot be an observer or a grid disagreement eliminates most of the chain in one reading.

The diagnostic, assembled #

Putting the conditions together gives a procedure that costs a few measurements and settles a great deal.

Measure a flat neutral. If it disagrees, the fault is in the white point, the illuminant declaration or the arithmetic — it is not the grid, not the observer and not the scene’s finish, because all three vanish on it exactly.

Slide the tabulation grid. If the answer moves, the light has structure the grid cannot hold and no downstream repair will help.

Change the quadrature rule. If the answer moves, most of what was being called a sampling error is the truncation at the ends of the range.

Compare a matt and a glossy reference. If a scene result moves, the Lambertian assumption is load-bearing in that geometry.

Each of those is a comparison that exploits a condition, each takes minutes, and each eliminates a whole class of cause. A condition is a diagnostic in the same way a conservation law is, and for the same reason: it says what cannot have happened.

The distributions bottom out where the samples are flattest, which is the identity approached rather than imposed, and the floor is a property of the family rather than of the observers. A family containing an exactly flat member would have a floor of exactly zero.

Those two figures are the same condition seen through two audits, and putting them beside each other is the clearest demonstration that the condition is about the sample rather than about either audit’s subject.

The first shows six observer departures whose smallest members are the flattest surfaces. The second shows six tabulation errors that are identically zero on a perfectly flat one. Neither audit knew about the other’s mechanism and both bottom out at the same place.

That is what a condition looks like when it is real: it holds for reasons the audits do not share, and an artefact of one audit’s machinery would not appear in the other’s.

Why the sequences matter as much as the identities #

A condition on its own says a quantity is zero somewhere. A sequence away from it says what the quantity is proportional to, which is a much stronger statement and is where the round’s practical recommendations come from.

The observer departure is exactly linear in the sample’s excitation deviation from its adapting white — to two parts in a hundred over the whole walk, in four combinations of departure, light and sample. That linearity is what licenses an observer allowance computed from a sample’s own reflectance, which is the round’s one actionable proposal.

Without the sequence, the identity would be a curiosity about grey cards. With it, the identity is the intercept of a line whose slope is the departure’s size, and every point on the line is the same phenomenon.

The same holds for the tabulation: the step’s error scaling linearly rather than quadratically is what identified it as an endpoint term, and the identification came from the shape of a sequence rather than from any single value.

A zero plus a slope is a model; a zero alone is an anecdote.

Recomputing the conditions under a second light is the cheapest available test of the classification, and it separates the two kinds cleanly. An identity that held only under one illuminant would be a coincidence of that illuminant’s spectrum; all eight hold under both, at the same floor.

The two limits change, because they are measurements of a mismatch between two bases and a mismatch depends on what is being looked at through them. An identity does not move when the experiment changes and a limit does, which is a better test than any threshold on the numbers themselves.

The condition that turned out to be a limit #

The round’s most instructive result about conditions is the one that failed, and it is worth putting in front of the successes.

Two of the observer conditions — a gain on each cone, and a change of basis — are exact in the observer’s own cone space. Imposed in CAT16’s cone space, which is what this collection and most of the field compute in, they leave 13.0 and 8.11 ΔE₀₀ standing.

Those are not small numbers and they are not errors. They are the cost of imposing a condition in coordinates other than the ones it holds in, and finding them required computing the condition twice rather than once.

The general lesson is that a condition carries its coordinate system with it, and the coordinate system is usually implicit. An identity proved in one basis and applied in another is not an identity; it is an approximation of unknown quality, and the quality here is thirteen colour differences.

That is the sharpest thing the round has to say about how to use its own results.

How to tell an identity from a limit #

The classification is mechanical and this collection has had to make it three times.

An identity is at the floating-point floor: 10⁻¹³ or below for a quantity of order one, which is what a few hundred floating-point operations leave of an exact zero. A limit is small and above the floor, and its smallness is a fact about the example rather than about the mechanism.

The distinction cannot be made from the number alone at first sight, because both are small. It is made by pushing the condition: an identity stays at the floor as the condition is imposed harder, and a limit falls along a sequence.

The previous round had to make that distinction five times, where three of eight conditions were identities and five were limits, and each limit was reported with the sequence its residual falls along. This round’s ten split eight to two, and the two are the coordinate-system cases above rather than genuine limits.

A small number is not evidence of an identity; a floor is, and the difference is a claim about mechanism rather than magnitude.

The ten are a checklist a specification can be run against. There is a way of using the ten that is more direct than the diagnostic above and it is worth stating, because it turns the round into something a practitioner can carry.

Each condition names a situation in which a term is absent. So a specification, a workflow or a calculation can be checked against the list, and each condition that holds removes a term from the budget.

Are the samples near-neutral? Then the observer term and the grid term are both absent, and a tolerance can be tight.

Is the light smooth on the scale of the tabulation? Then the step and the origin contribute nothing, and only the range remains.

Is the light free of ultraviolet? Then the range contributes nothing either, and the tabulation term is gone entirely.

Are the surfaces matt? Then the scene solver is exact and the directional term is absent.

A workflow satisfying all four — neutral samples, a smooth LED source, matt surfaces — carries essentially none of this round’s terms, and such workflows exist: press-side density control on a matt stock is nearly one. A workflow satisfying none of them carries all of them, and that is a saturated sample on a gloss substrate under a warm narrowband lamp, which is a shop window.

The lamp in the shop decides turns out to be an understatement: the shop fails three of the four conditions at once.

What the conditions do not say #

Three honest limits on how far the conditions can be pushed.

They are about matching, not appearance. A flat sample is the same colour for every observer in the sense that every observer’s coordinates agree. Whether it looks the same is a question about neural processing, and nothing here bears on it.

They are exact for idealisations. A real ceramic tile is flat to a few tenths of a per cent and its residual is a small multiple of its own departure from flatness. The identity is exact for the idealisation and the approach to it is steeper than linear in the published unit.

And they are conditions on this collection’s constructions. The observer identities are proved for observers built from one pigment template; a set of tabulated fundamentals would satisfy them equally, since the derivations never use the curves’ shape, but that has not been checked against a table.

What was computed, and how #

Each identity is asserted at 10⁻¹⁰ ΔE₀₀ and the measured values are between 10⁻¹⁵ and 4 × 10⁻¹³. Each assertion is written so that it could fail, and several did during development — the gain condition failed at 36 ΔE₀₀ until the adaptation was moved into the observer’s own cone space, which is how the coordinate-system finding was made.

The two limits are reported as limits with their sizes rather than as failed identities, which is the classification the previous round established.

The sequences are computed by mixing a sample towards a flat reflectance in five or six steps, and the linearity is asserted to two per cent in the excitation distance while the colour difference is asserted to be non-linear by at least ten per cent — a pair of assertions in opposite directions, which is what distinguishes a claim about the eye from a claim about the unit.

One more property of the list deserves stating. The four questions above are about the situation rather than about the calculation, so they can be answered by somebody who has never seen this collection’s machinery. That is the whole value of publishing conditions: a mechanism stated as a condition is usable by anybody who can recognise the situation, and a mechanism stated as a number is usable only by somebody with the same example.

Where the model stops #

The conditions are proved for the linear part of the calculation and nothing downstream of tristimulus values is covered. An appearance model’s nonlinearity, a gamut mapping’s clipping and a halftone’s exponent all sit outside them.

The identities are verified numerically rather than proved symbolically. The derivations above are short enough to be checked by hand and this collection asserts rather than proves, which is its standing habit and is a weaker form of evidence.

And ten conditions is what this round found, not what exists. A departure with no fuller model to compare against has no computable condition, and several are named and not measured.

The generalisation #

The habit is about which half of an audit to publish loudest.

An audit produces sizes and conditions, and sizes are what get quoted. They are concrete, they compare against tolerances, and they answer the question a reader arrives with. They are also the half that does not travel: a size is a measurement of one example under one construction, and a reader with a different example has to redo the work.

Conditions travel. They are statements about mechanism, they hold under substitution, and they can be used as diagnostics by somebody with none of the machinery that produced them.

The failure mode is to treat the conditions as the technical apparatus behind the numbers. They are the result and the numbers are the demonstration, and a round remembered for its numbers has been remembered for its least portable part.

A last note about the ten as a set. They are not independent: the first two are the same algebraic fact applied to two different variable objects, and the sixth and seventh are two halves of one statement about a light. Counted as mechanisms rather than as statements there are perhaps five, and counted as consequences of the pairing there is one.

That collapse is worth resisting when reporting and worth performing when thinking. A reader needs the ten, because each names a different situation they might be in; an author needs the one, because it is what makes the next audit cheap.

Who found it, and when #

The practice of reporting a mechanism’s conditions rather than only its size is old in physics and unusual in colour measurement, where the standard output of an investigation is a number with a test set attached.

Within this collection the habit dates from the round that audited the model equation, which found that each of its four departures vanished under two stated conditions and reported the eight rather than only the four sizes. This round follows it and adds the coordinate-system caveat, which that round did not need because all of its conditions were imposed in one basis.

Where the ladder goes next #

An audit’s last obligation is the list of what it could not do. What this round could not reach is longer than what it reached, and every item on it is named rather than estimated.