The departures are larger than the tolerance

Four departures, four numbers, and the number they should be compared against is not each other. It is the tolerance a specification is written in.

The claim #

Every one of the four departures exceeds the tolerance the industry writes, on ordinary material, and none of them is named in a specification.

• A tight delivery tolerance is around 1 ΔE₀₀, and a critical one — a brand colour, a repeat order — can be 0.5.
• The wavelength index costs 6.98 on a coated printing paper, and is the one departure a standard does name.
• The range costs 6.70 on the same paper, and no standard names it because it is inside a computation.
• The place index costs 1.96 on a pigmented plastic and 12.65 on marble.
• The direction index costs 1.00 on an eggshell paint at a window, and 3.01 under a lamp.
• And the smallest departure on the most opaque material — coated paper through an aperture, at 0.53 — is still half a tight tolerance, from an effect nobody records.

What a tolerance is for #

A tolerance says how close two colours have to be before somebody accepts them as the same. It is used in two ways that get confused: as a decision rule in production, and as a statement of agreement between two parties who measured separately.

The second use is where these numbers bite. Two tolerances do not meet in a tolerance, and this collection has already measured what a delivery chain’s own stages cost: the profile’s interpolation is 0.13, the rendering intent is 0.00 and the room is 4.30, with the largest being the one nobody controls.

The departures in this round sit alongside those and are of the same order or larger, with one difference: the delivery stages are known and budgeted, and these are not. A budget with three items in it is not conservative if there are seven.

The comparison, set out #

Against a tolerance of 1.0 ΔE₀₀, the round’s four ordinary cases stand at 6.98, 6.70, 1.96 and 1.00 — that is, at seven times, seven times, twice, and exactly the tolerance.

Against a critical tolerance of 0.5, all four are between two and fourteen times it.

And the departures are not the only unrecorded terms. The difference between two apertures — the disagreement two laboratories would have measuring the same sample at four and eight millimetres — is 1.00 ΔE₀₀ on a pigmented plastic and 5.90 on marble. The difference between a viewing booth and a window on a glossy paint is about 2.9. Neither of those is a departure from the model; each is a disagreement between two correct measurements.

A specification that names an illuminant, an observer and a number has named three things and left four unnamed, and the four unnamed ones are collectively larger than the number.

Why the tolerance is not simply wrong #

A reasonable response is that tolerances are met in practice, and industries do deliver colour to one ΔE₀₀ every day, so the departures must somehow not be biting.

They are not biting, and the reason is instructive: they are held constant rather than removed.

A print shop measures its sheets on one instrument, with one aperture, in one geometry, under one condition. Every departure is present in every reading and identically so, so every difference between readings has the departures cancelled out. A tolerance on a difference is therefore safe even when the underlying numbers are all wrong by several ΔE₀₀.

That works exactly as long as both readings come from the same arrangement. It fails at every handover — between a supplier and a customer, a proof and a press, a laboratory and a factory, a measurement and a specification — and a handover is precisely where a tolerance is being used as a statement of agreement rather than as a decision rule.

So the industry’s practice is not naive. It is a system that has learned to compare like with like, and whose failure mode is exactly the case where like cannot be arranged.

What a specification would have to name #

Four departures give four fields, and two of them already exist in some industries.

The measurement condition — M0, M1, M2, M3 — fixes the lamp and therefore the wavelength index. It is in the graphic arts standards and it is used.

The geometry — 45°/0°, d/8° with the specular included or excluded — fixes part of the direction index. It is universal in colour measurement and it addresses the instrument’s half of the departure, not the room’s.

The aperture is recorded as a hardware note where it is recorded at all, and no tolerance is stated relative to it. It should be: an aperture and a tolerance together are a statement, and either alone is not.

The field — what the illumination’s directional structure is where the result will be looked at — appears nowhere. A viewing booth is a luminaire and a specification that names a booth has named a field implicitly; one that says “D50” has named a spectrum and left the geometry open.

The fifth field is the one no industry has: the wavelength range of the computation. It is invisible because it is not a property of any instrument.

What was computed, and how #

Every number here is one already computed elsewhere in the round, quoted in one unit and set beside a tolerance rather than beside another departure.

The tolerances are the ones this collection has used throughout: 1.0 ΔE₀₀ as a tight delivery tolerance and 0.5 as a critical one. Both are conventions from practice rather than measurements, and a threshold is not a unit — the number 1.0 is not a just-noticeable difference and never was, which this collection has taken apart at length.

The two-aperture and two-field disagreements are computed as differences between two readings rather than as departures from a model, which is deliberate: they are what two laboratories would actually see, and they are the numbers a specification is trying to bound.

The one arithmetic caution is that these cannot be added. Two of the four partly cancel, by more than the smaller of the two, so a budget summing them overstates the total by a factor that has to be computed pair by pair.

Where the model stops #

Every departure’s size depends on the sample, and the four ordinary cases were chosen to be unremarkable rather than representative. A laboratory measuring opaque matte unbrightened material has all four at or near zero.

The tolerances are stand-ins. Real specifications are more elaborate — a tolerance is a region rather than a radius, often an ellipsoid with different semi-axes, sometimes a probability. Comparing a scalar departure with a scalar tolerance is a simplification in the direction of clarity.

And nothing here says the departures should be removed. For most of them the practical answer is to hold them fixed and declare them, which costs a field in a form rather than an instrument.

The two numbers a handover needs #

Turning the argument into something usable takes two numbers rather than a new theory, and both are available today.

The first is the condition set: which lamp, which geometry, which aperture, which field. Three of those four have an accepted vocabulary and the fourth does not, so the practical form is to record the aperture in millimetres beside the reading and to state the viewing arrangement in words.

The second is the disagreement the conditions allow, which is what this round computes. Two laboratories on the same sample at four and eight millimetres differ by 1.00 ΔE₀₀ on a pigmented plastic; two rooms differ by about 2.9 on a glossy paint; two measurement conditions differ by up to 7 on brightened paper. A specification that names a tolerance smaller than the disagreement its own unstated conditions allow has written a number that cannot be met by two parties independently, however careful either of them is.

That is the sharp version of the argument and it does not require any of the departures to be removed. A tolerance is only meaningful relative to the conditions it holds fixed, and the conditions this round measures are all currently held fixed by accident rather than by agreement.

Why the largest departures are the safest #

There is an inversion in the table worth pointing out, because it looks like a paradox and is not.

The two departures with the largest numbers — the wavelength index at 6.98 and the range at 6.70 — are the two least likely to cause a dispute, because both are already fixed by convention for anybody who follows one. A print shop on M1 and its customer on M1 have the same wavelength factor, and two computations on the same grid have the same range.

The two with the smallest numbers are the dangerous ones, because nothing fixes them. Two instruments with different apertures are both correct and both compliant, and nothing in either report says they differ.

So the risk in a handover is not the size of a departure but whether a convention holds it still, and that is exactly the ordering the round’s instrument essay arrives at from the other direction: the departures cheapest to remove are the ones somebody already removed.

The generalisation #

The rule worth carrying is that an unnamed condition is an uncertainty of the size the condition can vary by, and that size is usually not estimated because the condition is not thought of as a variable at all.

The test is mechanical. List everything about a measurement that could have been done differently by a competent person. For each, compute what the difference would be. Anything above the tolerance belongs in the specification; anything below can be left to practice. The list is longer than a specification’s fields, always, and the gap between the two is what handover disagreements are made of.

This collection has run that test twice before in different variables. The unit a difference is quoted in is worth about a factor of two on every published quantity, and it is never named. The set an average is taken over is the most elastic input in the collection, and it lives in a function signature. Both are conditions that nobody thought of as variables, and both turned out to be worth more than the numbers they qualified.

Both of those are one departure acting on one sample. The reason four of them cannot simply be added into a budget is that only two of the collection’s six published quantities can be pushed through at all.

What the four are departures of is one decision rather than four, and the identity that ties them together is checked rather than assumed.

What this collection’s own numbers look like under the same test #

The round’s departures are not the first unnamed conditions this collection has measured, and putting all of them in one column is uncomfortable reading.

The unit a difference is quoted in is worth about a factor of two on every published quantity here, after the change of scale is removed. The set an average is taken over carries an elasticity between 0.49 and 0.99, which is larger than any declared width in the collection. The basis an adaptation is diagonal in reorders the census. And now four more.

None of those is named in a specification either, and most are not named in a paper. The pattern across four rounds of auditing is consistent enough to state as a rule: the quantities that move a published number most are the ones that were chosen once, silently, and never appear in its statement.

A tolerance is the place where that becomes somebody’s problem rather than a methodological remark, because a tolerance is a promise between two parties. Everything above is a list of ways in which two careful parties can keep the promise and still disagree.

Doubling the aperture recovers about half, and it is predictable #

The two-aperture disagreement is quoted for two materials and the two are the same number in disguise.

1.00 ΔE₀₀ of a 1.96 departure on the plastic is 51 per cent; 5.90 of 12.65 on marble is 47. The materials differ by a factor of six in the departure and agree to four points in the share, which says the disagreement is not a separate measurement but a fixed fraction of the departure itself.

The fraction is predictable. The share of a translucent sample’s light that an aperture misses falls as about the 0.87 power of the aperture radius, so doubling from four millimetres to eight leaves 0.548 of the residual and removes 45 per cent of the departure. Against the measured 51 and 47, that is agreement to within six points on a law fitted to a completely different pair of numbers.

Which makes the aperture disagreement computable rather than measurable. A laboratory that knows a material’s departure at one aperture knows its disagreement with any other aperture without measuring it: about 45 per cent for a doubling, and one minus the ratio of the two radii raised to the −0.87 power in general.

It also settles the most opaque case, which the essay leaves at half a tight tolerance. Coated paper’s 0.53 departure at four millimetres implies a four-against-eight disagreement of 0.24 ΔE₀₀ — which is half a critical tolerance rather than half a tight one, on the material chosen to be the least translucent in the table, between two instruments both of which are correct.

The single largest departure exceeds the whole published budget #

The delivery chain’s three stages are quoted at 0.13, 0.00 and 4.30, which sum to 4.43 ΔE₀₀.

The wavelength index alone, at 6.98, is 1.58 times that total and 1.6 times its largest item. So a budget with three items in it is not conservative if there are seven is an understatement of its own case: it is not conservative if there is one, because the largest unbudgeted term is larger than everything budgeted put together.

That reframes the round’s relationship to the earlier one. The delivery budget’s headline was that the room dominates the chain by a factor of thirty over the largest device-side term, and that conclusion survives — the room really is thirty times the profile. What does not survive is any reading of the budget as a total. The chain’s largest known term is smaller than two of its unknown ones, and both of the unknowns act on the sample rather than on the viewing, so neither is downstream of the room.

Ranked by what a handover actually risks #

The inversion the essay names — that the largest departures are the safest — can be made into an ordering, and the ordering is short enough to be the practical output of the whole round.

A departure that a convention holds fixed contributes nothing to a handover between two parties who follow that convention. What is left is:

The two largest numbers in the round drop out of the ranking entirely, and what is left at the top is an aperture — the departure the essay’s own claim list puts third. Against a tight tolerance of 1.0, the aperture alone allows two correct laboratories to differ by six times it on marble, three times it on a plastic, and a quarter of it on coated paper.

So the round’s actionable finding is narrower and firmer than four departures exceed the tolerance. It is that one unnamed condition, with a known and computable size, is the largest handover risk in colour measurement that no standard addresses — and that its size can be predicted from the material’s diffusion length without a second measurement.

Who found it, and when #

Every industry that has met one of these has written a condition for it, and always after a dispute rather than before.

Geometry designations came out of the 1930s, when instruments first disagreed about gloss. The M-conditions came out of 2009, when brightened stock made proofs and press sheets disagree. Aperture advice came out of the plastics and textile industries in the middle of the century, and stayed as advice because it never became contractual.

The pattern is that a condition becomes a field in a specification when money changes hands across it. That is a reasonable way for standards to develop and it has one predictable failure: a condition that only matters at a handover nobody makes yet is never written down, and the first person to make that handover discovers the gap at their own expense.

The asymmetry between a rule and a promise #

The essay’s argument comes down to a distinction that is easy to state and easy to lose.

A tolerance used as a decision rule — accept this batch, reject that one — is safe under an unnamed condition, provided the condition is the same for every reading. The departures are all present and all identical, so they cancel out of the comparison, and a shop can hold a tenth of a ΔE₀₀ on its own instrument indefinitely.

A tolerance used as a promise — this will match the specification when the other party measures it — is not safe, because the other party’s conditions are not the first party’s. Every departure that cancelled in the first use appears in full in the second.

The same number is written on both, and a specification does not say which it is. That is the gap the four departures live in, and it is a gap in the use of a number rather than in the number — which is why no amount of care in either laboratory closes it, and why the remedy is a field in a form rather than a better measurement.

Where the ladder goes next #

The measurement that would turn this into something actionable is a round-robin: one set of translucent samples, several laboratories, apertures and geometries recorded. Such studies exist in dentistry and in plastics and they report disagreements of several ΔE₀₀; what they do not do is attribute the disagreement to the individual conditions, because attribution needs the same sample measured under each condition rather than one measurement per laboratory.

The cheaper thing is a convention, and it is available now: report the aperture beside the number. It costs a column, it is already known to whoever took the reading, and it converts an unexplained disagreement into an explained one — which is the whole of what a measurement condition ever does.