A declared width buys a factor of two

The tables cannot bound what they discarded left a colour engine with a choice between a bound that cannot be computed and a bound that is useless. Cauchy–Schwarz with the true window variances never fails and sits a median fifteen times above the error, but it needs the fine spectra the tables were made from. Cauchy–Schwarz with the variances estimated from the tables can be computed and is not a bound: it falls below the error on six of a tube’s sixty-eight notches and twenty-two of a laser’s, on the line, where the error is largest. The one valid computable bound — Bhatia–Davis, with the lamp’s peak declared — is a median 196 times loose.

That essay named the missing number. A blurred table does not carry how narrow the structure inside its window was, the covariance depends on it, and the width of a lamp’s narrowest feature is a number a measurement could record. It predicted a bound taking that width as an argument would land between fifteen and two hundred, “closer to the first for a tube and further for a laser, because the laser’s declared width would be so small that its worst-case variance approaches the peak’s.”

The bound is straightforward to write. Its answer is the right size and the wrong way round.

One number, a factor of two, and the lamps the other way about #

With the width of its narrowest feature declared, the bound under a fluorescent tube sits a median 86 times above the error instead of 196 — a factor of 2.3 — against the true-variance bound’s 15. Under a three-laser projector it sits at 14, which is the true-variance bound’s own looseness of 15.

• The prediction was inverted. The tube ends up six times looser than the true bound and the projector reaches it, not the other way round.
• It never fails at a declaration the lamp meets: nought of sixty-eight notches under each lamp, at every declared width from the true one downwards.
• It fails badly at one the lamp does not meet. Declaring forty nanometres on a tube whose lines are 1.2 wide puts the bound below the error on 47 of 68 notches, and on the projector 64 of 68.
• Nothing in the calculation can tell the two cases apart. The arithmetic is the same; only the declaration is false.
• It bounds and it does not track. Its rank correlation with the error is 0.27 under the tube and 0.44 under the projector, and its looseness spans a factor of six thousand across the census.

What a declared width forbids #

The Bhatia–Davis bound is loose because of the lamp it is a statement about. Given only a window mean and a peak, the largest variance belongs to a two-point lamp: all the light at the peak over part of the window and nothing over the rest, switching instantly between them. No lamp does that — not a mercury line, not a laser, not a phosphor. But nothing in the declaration forbids it, so the bound has to allow for it.

A narrowest feature is exactly the declaration that forbids it. A lamp whose features are no narrower than twenty nanometres cannot put its light into a spike inside a five-nanometre window, and its largest possible variance there is much smaller.

The three panels have the same kernel-weighted mean, which is all the table says, and variances of 3.20, 1.91 and 0.11. At 0.6 nanometres the worst lamp is a comb of spikes spaced 4.3 nanometres apart, each reaching the declared peak, with a flat pedestal beneath it to make up the mean. At twenty it is a single broad hump that cannot even reach the peak without overshooting the mean.

That comb is the part the earlier essay’s prediction missed. A narrow feature is not one narrow feature. Nothing in a declaration of 0.6 nanometres says the lamp has one line in this window; a window ten nanometres across will hold a dozen such features, and a comb of them approaches the two-point lamp that Bhatia–Davis was already allowing for. So a very narrow declaration buys almost nothing, and the bound only tightens once the declared feature is a substantial fraction of the slit.

It is worth being clear that the comb is not a strawman put up to make the bound look bad. A real fluorescent tube has four mercury lines in the visible range and a real triphosphor tube has three more emission peaks over them; a real multi-line discharge has more. Nothing in the lamps computed here rules out two features inside one five-nanometre window, and a lamp has a waveform is a reminder that what a lamp is doing at any instant is further from a single smooth curve than its time-averaged spectrum suggests. A bound has to admit the lamps that exist, and it happens to admit some that do not.

Where it lands, on both lamps #

Both curves are flat across the narrow declarations and fall away above about five nanometres. Under the tube the bound reads ×83 at 0.2 nanometres, ×86 at 1.2 — the true line width — ×64 at 5 and ×47 at 10. Under the projector it reads ×14 across the whole narrow range. The lamp’s own line width sits in the flat part on both, which is why the factor is two rather than ten.

The projector’s result needs its own sentence, because it looks like a success and is not the one the prediction claimed. Bhatia–Davis is already nearly tight for the projector, at ×30 against the tube’s ×196, and the reason has nothing to do with line widths. A three-laser spectrum is nearly dark over most of the census’s wavelengths, so the table value in most windows is small, and the quantity Bhatia–Davis bounds — the mean times the peak minus the mean — is small with it. The projector’s bound was good before the width was declared. The width took it from ×30 to ×14 and the same factor of about two applies.

So the answer to the question is: one extra number, on both lamps, buys about a factor of two. That is a real improvement on the only valid computable bound there was, and it is not the order of magnitude that would have made the bound usable as a number rather than as a warning.

The census, three bounds at once #

The three rows per lamp are in the order the information is in them — the fine spectra, the tables plus two declared numbers, the tables plus one — and each loosens as the information falls. What the figure adds to the numbers is the spread: every row runs over more than three decades, and the ranges overlap almost completely. A notch where the true-variance bound is ×5 and a notch where the declared bound is ×5 are different notches, and neither bound can say which.

None of the six rows contains a failure, because each is either a true-variance bound or a declaration the lamp meets. That is the whole of the improvement over the table estimate, and it is not nothing: the tables cannot bound what they discarded found that the estimate failed exactly on the notches where the error was largest, which is the worst possible place for an error bar to be wrong.

The improvement also has a shape worth naming, because it is the shape a declaration always produces. The table estimate was a guess at a quantity: it read a slope and a curvature off three neighbouring table entries and inferred the variance a smooth function would need to produce them, which is a statement about the lamp that the data does not support. The declared-width bound is a constraint on a set: it makes no claim about what this lamp does inside the window and only about what any lamp meeting the declaration could. The first can be wrong about a particular lamp; the second can only be wrong about the declaration. A grid is not a resolution is the essay about the first mistake in its original form — reading a table’s step as though it were the instrument’s width — and this is the same repair one level up.

A bound is only as honest as the number it was given #

A declaration is an input, and an input can be wrong.

Left of each lamp’s own line width nothing fails. Right of it the failures climb: one notch at ten nanometres for the projector, three at ten for the tube, and by forty nanometres 47 of 68 and 64 of 68. The bound has not changed and the arithmetic has not changed. What changed is that the declaration stopped being true.

That is a different kind of fragility from the table estimate’s. The estimate failed because it was reading a quantity out of data that did not contain it, and no amount of care with the tables would have fixed it. This fails only when somebody says something false, and a lamp maker has an obvious incentive to declare a comfortable width — a wider declaration produces a tighter error bar, and the error bar is what a customer reads.

A lamp switched on is not the lamp measured is the reminder that a declared number describes a lamp under conditions, and a lamp’s line width is not fixed: a discharge line broadens with pressure and with current, and an LED emitter broadens with temperature. So a declaration would have to name its conditions, and a lamp running hot in a luminaire would be outside its own declaration in exactly the direction that breaks the bound.

It holds, and it does not track #

The last property is the one that decides what the bound is for.

Every point is above the diagonal and the cloud is not a line. The rank correlation between the bound and the error is 0.27, and the looseness runs from ×5.4 on one notch to ×33,631 on another. Under the projector the correlation is 0.44 and the range is ×2.1 to ×9,810.

A bound that holds tells an engine that its answer is within some distance of the truth. A bound that tracks would tell it which of its answers to distrust — which would let a colour engine flag the notches that need a better measurement and let the rest through. This does the first and not the second, and the difference is the difference between a specification and a diagnosis.

The reason it cannot track is the same reason it is loose. The bound is the worst case over every lamp consistent with the declarations, and the actual error is one lamp’s covariance, which depends on where the structure happens to sit relative to the sample’s. Two notches with identical tables and identical declarations can have covariances of opposite sign; the tables cannot bound what they discarded showed that directly, with two tubes whose lines differ in width at the same power giving nearly one table and very different errors.

How the bound was computed #

The lamp’s window variance is maximised over a family of combs of Gaussians of the declared width: a period from the declared width up to four windows, a phase across one period, and an amplitude that is the largest keeping the lamp under its declared peak and its kernel-weighted mean at or below the table’s value. What is left of the mean is a flat pedestal, which adds nothing to the variance. The sample’s half of the Cauchy–Schwarz product is still the estimate from its own table, which is the arrangement the question asked for.

So this is a maximum over a family rather than a proof over every admissible lamp, and the census is where it is tested: it did not fall below the error on any of the 136 notches at a declaration either lamp meets. That is a result about these two lamps and this sample family, not a theorem, and the essay says so rather than calling it a bound without qualification.

Pulling the mean out of the search is what made it affordable. The amplitude and the variance both scale with the table’s value at a fixed ratio of peak to mean, so the search over periods and phases runs once per pair of those rather than once per table entry — two seconds instead of six and a half minutes, with the same numbers.

What this does not settle #

The sample’s variance is still an estimate. A sample with structure narrower than its own slit would break this bound the way the table estimate broke, and nothing here tests that, because the census’s samples are notches of known shape. A declared narrowest feature for the sample is the symmetric repair and it is a stranger thing to ask a dyer for than a lamp maker.

One slit width, one sample family. Everything is through a five-nanometre slit on each table, and one slit, two requirements found that the lamp and the sample want different widths. The bound’s looseness depends on the ratio of the declared feature to the slit, so a one-nanometre table would move all of these numbers and the tube’s declaration would move from the flat part of the curve into the falling part.

The error is in colour differences and the bound is in tristimulus values. Each bound on the three channels is turned into a colour difference by taking the largest CIEDE2000 at the corners of the box it allows, which is a conservative step: the true error vector is somewhere inside that box and generally not at a corner. Part of every looseness quoted here belongs to that conversion rather than to the bound, and how much is not separated. The error on a gap is not the errors at its ends is the essay about that kind of compounding.

And the comb family is a choice. A different family — asymmetric features, features of unequal height, a feature partly outside the window — could reach a larger variance and would make the bound larger and safer. The direction of that error is the right one for a bound and it is still an assumption.

Still open: whether the slit can be declared instead #

The bound is loose because a feature narrower than the slit can be packed into it. That suggests the repair is at the other end: rather than asking a lamp maker to declare how narrow the lamp gets, ask the instrument to be narrow enough that the question stops mattering.

The calculation is this census with the slit as a variable and the declaration held at each lamp’s true line width. The prediction is that the bound’s looseness falls roughly as the ratio of declared width to slit width rises past about a half, because that is where the comb stops fitting — which is the same threshold a second slit buys a quarter found for the separable arrangement, arriving from the error rather than from the bound.

If that is right, the two repairs are one repair. A one-nanometre slit makes a tube’s 1.2-nanometre lines a comparable feature, the comb collapses to a single hump, and the declared-width bound should approach the true-variance bound without anybody measuring a fine spectrum. If it is not right — if the looseness stays near two orders of magnitude at every slit a real instrument has — then no arrangement of declarations produces a usable error bar, and the honest output of a bandpass calculation remains the arrangement of the measurement rather than a number attached to its result.

A worst case is a statement about a set, and the set is wider than the example #

The habit is about which lamps a declaration admits.

The prediction this essay tested reasoned from one lamp: a laser’s line is very narrow, a very narrow line has nearly all the window’s light at the peak, so the worst case approaches Bhatia–Davis’s. Every step is true of one line. The bound is not about one line — it is about every lamp meeting the declaration, and a declaration of 0.2 nanometres admits a comb of fifty of them, which is closer to the two-point lamp than one line is.

The move is to ask what the least lamp-like member of the admitted set looks like before estimating where a bound will land. A declaration of a narrowest feature constrains the features and says nothing about how many, and the number is what decides whether the constraint bites.

The failure mode is reasoning about a worst case from the typical member of the set. The typical lamp meeting a 0.2-nanometre declaration is a laser. The worst one is a comb nobody would build, and the bound is a statement about that one.