The grid under the census

The observer is one inherited term in the census. The tabulation is a second, it is smaller, and it is not distributed the way a reader would guess from which lights the census uses.

The claim #

The census inherits a tabulation term of about half a colour difference, nearly all of it from the range rather than the step, and it is smaller than the observer term by a factor of six.

• The range costs a median of 0.54 ΔE₀₀ over an ordinary surface family under a daylight radiator, and 0.07 under tungsten.
• The step costs 0.060 and 0.011 under the same two, an order of magnitude less.
• The census’s lights are all smooth, so none of them is in the aliasing regime and the origin of the grid contributes nothing.
• And the term largely cancels out of the census’s comparisons, for the same reason the observer’s does — it appears on both sides of every subtraction.

What the census is summed over #

The census is a construction of a hundred and twenty-five surfaces and fourteen changes of light, and every number in it is a sum over eighty-one wavelengths from 380 to 780 nanometres in steps of five.

That index was decided in this collection’s first week, inherited from the CIE tables, and never varied. It carries three separable decisions — a step, a range and an origin — and the round’s first half measured all three.

The census’s exposure to each is decided by its lights, because a tabulation’s cost is a pairing between what the index leaves out and what the light puts there. Its lights are the CIE illuminants and daylight reconstructions, all of which are smooth.

The three terms, sized #

Taking the round’s measurements and reading them for the census’s conditions:

The range dominates by an order of magnitude under daylight, and everything is small under tungsten because a blackbody at 2856 K has far less ultraviolet than one at 6500.

None of the three is in the aliasing regime, because none of the census’s lights has a feature narrower than the step. The three regimes put every one of them comfortably in the first, where the sum is nearly exact and what is left is the truncation.

So the census’s tabulation exposure is essentially one term — the ultraviolet end of its range — and it is about half a colour difference on the surfaces it uses.

The range dominates and the step does not. The asymmetry is worth restating in the census’s terms because it is the opposite of the usual worry.

The two ends of the range are not alike: below 380 nanometres the observer is small but not zero and daylight is still strong, so the product is small rather than absent; above 780 both are falling together and the product is four orders of magnitude smaller.

The census’s daylight sources are exactly the case where that bites. A 6500 K radiator has about 3.4 per cent of its visual product outside the site’s range, nearly all of it below 380, and truncating it costs a median of 0.54 ΔE₀₀ on ordinary surfaces.

Its warm sources have much less ultraviolet and cost a tenth of that. So the census’s exposure is concentrated in its cool-light rows, which is the reverse of where its observer exposure is concentrated — the observer term roughly doubles under warm light.

That is a useful anti-correlation. The two inherited terms are largest under different halves of the census’s light set, so neither dominates everywhere and the total is flatter than either.

The step figure is worth having beside the range one because it says how safe the census is in the dimension everybody worries about. Its lights are the two lowest curves in that chart, both falling smoothly, both under a tenth of a colour difference at five nanometres.

So the census is not a resolution problem in any sense a reader would recognise from the phrase. Its arithmetic is well sampled, its answers are stable under refinement, and every convergence test anybody would run on it passes.

What it is exposed to is a decision that looks like bookkeeping: where the table stops. That is invisible to a convergence test, because refining the step does not touch it — the two are separate mechanisms with separate repairs, and only one of them is what “resolution” means.

What cancels and what does not #

The division is the same as the observer audit’s and the argument is identical.

The census’s comparisons — which adaptation model predicts better, which change of light is hardest, whether a correction helps — hold the tabulation fixed on both sides, so the term cancels to first order.

Its absolute levels carry it in full. A census level of 2.4 ΔE₀₀ computed on the site’s grid would be about 0.5 different on a 300–830 grid, which is a fifth of the level and is not negligible.

The two terms together — half a unit of tabulation and two to three units of observer — put the census’s absolute levels in a band rather than at a point, and the band is a substantial fraction of the levels themselves. That is worth saying plainly and it does not change any of the census’s conclusions, all of which are comparative.

The one place the terms interact #

There is a case where the tabulation and the observer are not independent, and it is worth finding.

An observer departure lives somewhere on the wavelength axis, and the range’s truncation removes a specific part of that axis — the ultraviolet. Two of the six observer departures, the lens and the macular pigment, are absorptions in exactly that region.

So widening the range would change the observer departures as well as the levels. A lens absorbs strongly below 400 nanometres, and a calculation that includes 300 to 380 gives the lens more to act on, so the lens departure would rise.

That interaction has not been computed. It requires running the observer audit on the wide grid, which the machinery supports and this round has not done, and the direction is predictable while the size is not. It is recorded as owed.

Why the census’s own sensitivity work missed it #

The census has already been audited once, from inside, and it is worth asking why the grid was not on the list.

A mean has a set under it examined the census’s declared inputs: the surface count, the brightness and saturation distributions, the basis dimension. Those are arguments to a function, visible in a signature, and varying them is a matter of passing different values.

The grid is not an argument to anything. It is LAMBDAS, imported from a module, used by every function in the collection, and there is no call site at which it could be varied. A parameter with no call site is a parameter no audit of call sites will find.

That is the third time this collection has found the same shape. The widths were declared and never varied; the reflectance was treated as data rather than as a choice; and now the index is a module-level constant. Each was invisible for the same structural reason and each took a round to find.

What it would take to remove it #

The repair is available and it is a modelling commitment rather than an arithmetic one.

The collection already has a wide grid: the shortwave machinery runs from 300 nanometres with the daylight basis extended to match, and it is used by the fluorescence work. So the tables exist and the arithmetic exists.

What does not exist is a defensible extension of the census’s surfaces below 380. They are constructions from stated basis functions, and the basis was fitted over the visible band, so extending them is an extrapolation rather than a lookup — and the extrapolation’s consequences exceed the truncation it would replace.

So the census’s tabulation term is not repairable without a decision about what its surfaces do in the ultraviolet, and any such decision would be an assumption rather than a measurement. That is the honest reason the range has stayed outstanding for three rounds, and it is a better reason than the one previously recorded, which was that nobody had done the table lookup.

The two inherited terms are largest under different lights. Putting the observer and the tabulation terms side by side gives the census’s total inherited exposure, and the shape is more useful than the sum.

The tabulation term falls by a factor of eight going from cool to warm and the observer term rises by a third. So the total is between about 1.9 and 2.5 across the census’s light set — flatter than either component, and flat for a reason rather than by luck.

The reason is that the two terms depend on different properties of a light. The tabulation’s cost depends on how much ultraviolet a source has, which rises with colour temperature. The observer’s depends on how blue-poor the source is, which falls with colour temperature. Those are nearly opposite properties, so the two terms trade against each other along the one axis that orders the census’s lights.

Two inherited terms with opposite temperature dependence is a convenient accident and it is worth knowing about, because it means neither can be reduced by choosing a light and the total is roughly what it is.

What was computed, and how #

Nothing is recomputed. The tabulation numbers are the round’s, over forty-two analytic surfaces under six constructed lights, measured against an analytic reference at a tenth of a nanometre.

The census’s surfaces are a different construction — a hundred and twenty-five with declared brightness and saturation distributions — so the transfer is by argument rather than by measurement. Both families are smooth and broad, so the tabulation term should be similar; the census’s are more saturated on average, which would raise it slightly.

The claim that the census’s lights are all in the first regime follows from their spectra: CIE daylight reconstructions and Planck radiators have no feature narrower than five nanometres by construction.

The two shares are worth putting in front of anybody who wants to reason about the census’s exposure without recomputing it. A 6500 K radiator has 21.4 per cent of its power outside 380 to 780 nanometres and 3.4 per cent of its visual product, and the census’s cost follows the second.

Confusing the two is the mistake the range argument invites, and it would overstate the census’s exposure by a factor of six. It would also point at the wrong end: most of the missing power is infrared, where the observer is identically zero, and all of the missing product is ultraviolet.

So a reader estimating the term from a lamp’s spectrum needs the product rather than the power, which is one integral rather than a datasheet lookup — and it is the same integral the range essay reports for six lights.

One more thing follows about how the two audits should be read together, and it concerns which one to act on. The observer term is four to six times the tabulation term and is not repairable at all; the tabulation term is smaller and is repairable in principle, at the cost of an assumption about ultraviolet reflectances.

So the larger term is fixed and the smaller one is expensive, which is a common and awkward arrangement. The rational response is neither to repair the small one nor to ignore both, but to state both and act on neither — which is what this round does, and which is the outcome an audit reaches more often than its readers expect.

Where the model stops #

The transfer from one surface family to another is an argument rather than a computation, and the honest form of every number here is about half a colour difference rather than 0.54.

Nothing here recomputes the census on a wide grid, which is what would settle it. That is a larger job than the observer version, because the surfaces would have to be extended first and the extension is the difficult part.

And the interaction between the range and the observer’s blue-absorbing departures is named and not measured, which is the largest specific omission of this essay.

A wide-grid census would settle a sensitivity rather than a level. Sketching the experiment is worth doing because it is the smaller of the two owed runs and its outcome is genuinely uncertain.

Running the census on the 300–830 grid requires extending its hundred and twenty-five surfaces below 380, and the extension is the whole difficulty. Three defensible choices exist: hold each surface’s reflectance constant below the band, continue its basis functions, or set it to zero. They differ by more than the truncation being repaired, which is the reason the repair has stayed outstanding.

What a run under all three would give is not a corrected census. It would give the sensitivity of the census to an assumption nobody has made, which is a more useful object: if the three extensions give levels agreeing to a tenth of a unit, the truncation is harmless whatever the surfaces do below 380, and if they do not, the census’s absolute levels have an irreducible uncertainty that no amount of care removes.

That is the shape of every sensitivity result in this collection — the doubt is not the answer — and it is the version of the wide-grid run worth doing. Recomputing the census under one assumed extension would produce a different number with the same standing as the old one.

The generalisation #

The habit is about auditing the things with no call site.

An audit of a calculation naturally examines its arguments, because arguments are visible, enumerable and varyable. What it will not find is anything imported: a constant, a table, a grid, a unit, a coordinate system. Those are decisions with the same consequences as arguments and none of the visibility.

The move is to read the imports rather than the signature, and to ask of each whether it could have been otherwise. That is a short list in most code and it is a list nobody reads, because an import looks like machinery rather than like a choice.

The failure mode is to conclude from a thorough audit of a function’s arguments that its inputs have been examined. A function’s inputs are its arguments and its imports, and three rounds of this collection have now found something important in the second.

A final note on scale, since half a colour difference can sound either alarming or trivial depending on what it is set against. It is a fifth of a typical census level, a half of a delivery tolerance, and a sixth of the observer term measured on the same surfaces. It is far above the census’s own numerical precision, which is at the floating-point floor, and far below the differences between the adaptation models the census exists to compare.

So it is a term that matters for reading a census level as an absolute number and does not matter for anything the census is used to decide. That is a narrow but real conclusion, and stating it is the whole point of sizing a term rather than merely finding one.

Who found it, and when #

The census is this collection’s own. Its self-audit of declared inputs is from the round that asked what a mean is a statement about, and the observer’s absence from that audit is examined in the essay beside this one.

The tabulation’s absence has the same cause and is smaller in consequence, which is why it gets a shorter essay: half a colour difference against two or three.

Where the ladder goes next #

Two inherited terms have been sized against the census and they are largest under different lights. That anti-correlation is not a coincidence: a lamp is two audits at once, and which of them it exercises is decided by two different properties of its spectrum.