A narrow table can vouch for its own peaks

A narrow slit shrinks the error, not the bound found out where a declared-width bound on bandpass error is loose. The bound is Cauchy–Schwarz on the covariance two separately blurred tables discard, the lamp’s standard deviation in each window times the sample’s. The sample’s half is estimated from its own table and is harmless. The lamp’s half is bounded by a declaration of three numbers: the table’s value in the window, the width of the lamp’s narrowest feature, and the lamp’s peak. And the peak is the weak one — a single value for the whole spectrum, the top of a fluorescent tube’s green mercury line, which the bound has to allow for in every window, including windows four hundred nanometres from any line. Narrowing the slit made that worse rather than better, because a narrower window has a smaller mean beside the same peak.

That essay measured how much the peak cost by cheating: it replaced the tube’s peak with each window’s own highest point, read off the fine spectrum, and the bound at a one-nanometre slit went from 253 times the error to 81. A colour engine does not have the fine spectrum. Its closing section proposed that it does not need it. A table read through a slit about as narrow as the lamp’s features is nearly a fine spectrum already, and a feature that cannot be narrower than the declared width cannot be blurred away: the slit keeps some computable fraction of its height, so the table’s entries near a window say how tall the lamp can be inside it.

This essay computes that fraction, builds the ceiling from it, and puts the ceiling in the bound.

Half the looseness, for no extra declaration #

At a one-nanometre slit on a one-nanometre grid, a 1.2-nanometre feature keeps at least 0.38 of its height at the nearest grid point and at least 0.95 at the two flanking points together. The ceiling each window reads from its neighbouring table entries is never below the tube’s true height there, and sits a median 2.05 times above it, where the tube’s global peak sat 19.6 times above. With that ceiling in place of the peak, the bound under the tube goes from ×253 to ×128, against the oracle’s ×81, and certifies 23 of 68 notches within one colour difference where the global peak certified none.

• The ceiling is valid by construction, for any lamp built from features no narrower than the declared width, and it never fell below a window’s true peak on either lamp at any slit.
• It is worth something only when the slit is comparable to the feature. At five nanometres the slit keeps an eighth of a 1.2-nanometre line, the ceiling is above the tube’s own peak wherever a line could hide, and the bound is the global one to within two per cent.
• A peak known ten times better tightens the bound by two, because the standard deviation a declaration allows grows roughly as the square root of the peak.
• The projector gains nothing. Its lines are five times narrower than the narrowest slit, so the table keeps too little of them to read, and every lit window holds a line whose height is near the global peak anyway.

How much of a feature a slit has to keep #

The whole construction rests on one inequality, and it is worth seeing where it is weakest.

A slit blurs a feature, and the blur can only take so much of it away. A triangular slit one nanometre in half-width averages each grid point’s neighbourhood with weights falling from one at the centre to nothing a nanometre away. A Gaussian feature 1.2 nanometres wide survives that averaging at about three quarters of its height if it sits on a grid point. The question the ceiling needs answered is harder: for a wavelength anywhere between two grid points, how small can the nearest grid point’s reading be, compared with the lamp’s value at that wavelength?

The answer is found by trying every placement. It is smallest when the wavelength is half-way between grid points and the feature sits just beyond it, so that the wavelength is on the feature’s rising flank and the nearest grid point is on its far tail. There the feature stands at 0.28 of its height and the grid point reports 0.10 — a fraction of 0.38. No placement does worse, and a wider feature does better, so 0.38 is a guarantee for every feature at least as wide as the declaration.

Because the nearest grid point depends on the wavelength alone, not on the feature, the guarantee survives adding features. A lamp that is a sum of such features, with a smooth continuum under them, has a value at any wavelength no more than its nearest table entry divided by 0.38. That is the step that lets a statement about one feature become a statement about a lamp.

The same argument works for the two grid points either side of the wavelength, taken together. Whatever the placement, a feature that one of them sees poorly the other sees well, and their sum keeps at least 0.95 of the feature’s value between them. A window’s ceiling is the smaller of the two readings, since each is valid.

Both fractions fall as the slit widens past the feature. For the tube’s lines the pair reading keeps 0.95 at one nanometre, 0.57 at two and 0.25 at five; for the projector’s 0.2-nanometre lines it keeps a fifth at one nanometre and a twenty-fourth at five. Once the slit is several times wider than the feature, the fraction goes roughly as the feature’s width over the slit — the feature is a sliver of the window and the table reports it as one. That is the regime a grid is not a resolution described for a single table: a line narrower than the slit is recorded as its area, and its height is gone.

The ceiling, window by window #

At a one-nanometre slit the ceiling follows the lamp. Across the phosphor continuum it runs at about twice the continuum’s height — the price of the guarantee, which has to allow for a line hiding at the worst position between two grid points. At the green line it rises to meet the tube’s peak two or three windows early and falls two or three windows late. At the yellow line it rises to about 1.9, above the line’s true height of 1.5 but well short of the green line’s 2.8, which is where the global declaration would have put it. Over the whole spectrum the ceiling sits a median 2.05 times above each window’s true peak.

At five nanometres it does not follow anything. The table’s entries are blurs of the lines over ten-nanometre windows, the slit keeps an eighth of a 1.2-nanometre line, and a table value of 0.35 in the continuum is compatible with a line of height 2.8 anywhere in the window. So the ceiling reaches the tube’s peak everywhere in the stretch shown, and the dots — the table itself — sit uselessly on the continuum while the lines stand three times taller than any of them. A five-nanometre table does not know where its lamp’s lines are, which is the finding the tables cannot bound what they discarded started from.

What the ceiling does to the bound #

The table’s peak buys a factor of 1.98 at one nanometre, 1.43 at two, 1.26 at three and 1.02 at five. At one nanometre it closes more than half of the distance on a logarithmic scale between the global bound’s ×253 and the oracle’s ×81; at five it closes none, because it knows nothing the global peak does not. The gain falls at every step, and it falls to nothing where the fraction a slit keeps falls below about a sixth.

The distance that remains to the oracle is the guarantee’s cost: a ceiling of twice the continuum is still twice the continuum. And the distance from the oracle to the true-variance bound, ×81 against ×13, is not the peak’s at all. It is the width doing what a width does — allowing a feature where the lamp has none, in every window whose continuum is flat.

Why ten times better buys two #

The ceiling is ten times nearer each window’s true peak than the global peak is — ×2.05 against ×19.6 — and the bound improves by a factor of two. That looks like a disappointing return and it is exactly the return the declaration’s arithmetic promises.

The variance a declaration allows grows roughly with the peak, so the standard deviation grows roughly with its square root. The worst lamp is a feature reaching the peak on a floor that makes up the window’s mean, and a taller feature needs a narrower share of the window to keep the mean, so its variance goes up in proportion to its height. At one nanometre a peak of 20 times the mean allows a standard deviation 3.4 times that of a peak of twice the mean — close to the square root of ten. And the bound is a sum of standard deviations, not variances.

So information about a peak is worth its square root in a Cauchy–Schwarz bound, and that is a statement about the inequality rather than about lamps. A factor of ten in the ceiling becomes about three in each window’s standard deviation, and the windows that dominate the bound — those next to the notch, where the sample’s own standard deviation is large — are the ones near the line, where the ceiling and the global peak are nearly the same. The two effects together make the two.

The first notches certified #

Every notch moves down by about a factor of two, and that is enough to cross a line that matters. With the tube’s global peak the bound at a one-nanometre slit was between 1.3 and 3.7, and no notch was certified within one colour difference; 24 of 68 were certified within two. With the table’s peaks it is between 0.7 and 2.2: 23 notches are certified within one colour difference and 65 within two.

That is the first time in this sequence of censuses that a bound computable from the tables and three declared numbers has said something a colour specification could use. It is still a bound on errors that are under a quarter of a colour difference throughout — the notch whose bound is 2 has an error of perhaps 0.02 — so it is still loose by a factor of a hundred. But the question it answers is not how large the error is. It is whether a colour engine that cannot see the fine spectrum can promise a tolerance, and at one nanometre, for a third of these notches, it can promise one.

The projector, where it changes nothing #

Under the three-laser projector the table’s peak and the global peak give the same bound at every slit, to the digit, and for two reasons that happen to agree. The first is the fraction. A 0.2-nanometre line keeps at most a fifth of itself in a one-nanometre table and a twenty-fourth in a five-nanometre one, so a window’s ceiling — its table entries divided by that fraction — is always above the projector’s own peak, and the cap puts it back there. A slit five times wider than the feature vouches for nothing, which is the tube’s five-nanometre result arriving at one nanometre.

The second is the lamp. A laser spectrum is dark except around three lines, so every window with light in it contains a line, and the three lines’ heights are within a factor of 1.6 of each other. The median lit window’s own peak is within a sixth of the projector’s tallest. There was little for a local reading to improve even if the slit had kept enough to read it. A declared width buys a factor of two found the projector’s declared bound close to its true-variance bound for the same reason: a lamp that is all peaks declares its peak nearly correctly for every window by declaring it once.

What an instrument maker would state #

Tables measured through a slit no wider than the lamp’s narrowest feature carry their own peaks, to within the fraction computed here, and a colour engine can read them. The one number a lamp still has to supply is its narrowest feature’s width, which a declared width buys a factor of two argued a datasheet could carry. The peak, which it could also carry but only as one number for the whole lamp, becomes redundant — the table states it better, window by window.

The threshold is the ratio of slit to feature, and it is the same threshold this sequence of censuses has kept finding from different directions. One slit, two requirements found a line lamp wanting a slit wider than its lines so the grid could see them. A second slit buys a quarter found the separable arrangement losing most when the lamp’s features were narrower than its slit. Here a table vouches for its peaks once the slit is within a factor of two of the feature, and for nothing at a factor of four.

And the bound stays a warning light rather than a measurement. It halved; it did not become tight, and nothing in it tracks which notch is worst. A narrow slit shrinks the error, not the bound argued that at a one-nanometre slit the arrangement of the measurement, not any bound, is what makes the colour trustworthy. That is still true. What has changed is that the bound can now confirm it for some notches rather than for none.

How the ceiling was built #

The slit is triangular, of half-width w, on a grid of step w from 380 to 780 nanometres; the lamps are the collection’s fluorescent tube, with mercury lines 1.2 nanometres wide on a phosphor continuum, and three-laser projector, with lines 0.2 nanometres wide. The fraction a slit keeps is found by placing a Gaussian of the declared width at every position in steps of a twelfth of the smaller of the width and the slit, over three slit widths and three feature widths either side of a grid point, and taking the smallest ratio of the nearest grid point’s blurred value — or the two flanking points’ blurred values summed — to the feature’s value at every wavelength within half a step. Features one and a half and three times wider were checked and keep more.

Each window’s ceiling is the smaller of the largest of the three table entries whose points can be nearest to it divided by the nearest-point fraction, and the larger of the two adjacent pairs’ sums divided by the pair fraction, capped at the lamp’s declared peak and floored at the table’s own value. It replaces the declared peak in the declared-width variance of a declared width buys a factor of two, and everything else in the census — sixty-eight notches under each lamp, the sample’s variance from its table, the corner conversion to CIEDE2000 — is as in a narrow slit shrinks the error, not the bound.

What this leaves out #

The fraction is found by search. It is the smallest ratio over a fine grid of placements, not a closed-form minimum, and a placement between grid positions could in principle do slightly worse. The ceiling was checked against every window’s true peak on both lamps at every slit and was never below it, which is evidence the search is fine enough and not proof.

Features are assumed Gaussian. A real mercury line is closer to a Voigt profile, with wider wings, which keeps more of itself at a distance and so makes the fraction larger; a feature with steeper sides than a Gaussian could make it smaller. The declaration would then have to state the shape as well as the width.

And the ceiling assumes a non-negative lamp. It is a sum of features no narrower than the declaration on a non-negative continuum. Every real lamp’s spectrum is non-negative; a measured table with noise is not, and a negative entry next to a line would lower the ceiling below the truth. A measured table would need its noise stated before its ceiling could be trusted.

Still open: whether the width can be read from the table too #

The ceiling takes one number from the lamp — its narrowest feature — and reads the rest from the table. At a slit comparable to the feature, the table carries evidence about that width as well: a line of width 1.2 through a one-nanometre slit makes a peak in the table of a particular shape, three or four entries whose ratios depend on the line’s width and position. A narrower line makes a sharper table peak for the same area.

The calculation is an inverse: for each table peak, the narrowest feature consistent with the entries around it, and the question whether a bound built on that inferred width stays valid. The prediction is that it does at one nanometre and fails at five, for the same reason the ceiling does — a five-nanometre table records a line’s area and nothing of its shape — and that the failure is sharp rather than gradual, somewhere between two and three nanometres, where the fraction a slit keeps of a 1.2-nanometre line falls through about a half. If it holds, a table measured through a narrow enough slit declares everything its own bound needs, and a colour engine needs nothing from the lamp’s maker at all.

A guarantee is worth its square root #

The habit is about pricing a better input by what the bound does with it, before buying it.

The ceiling was a real improvement in knowledge: each window’s peak known ten times better, for nothing, from data already on hand. It bought the bound a factor of two, because the bound is a sum of square roots of quantities that scale with the peak. Any Cauchy–Schwarz bound behaves like that, and so does any error budget that adds standard deviations: better knowledge of a scale enters as its square root.

The failure mode is to expect a bound to improve in proportion to what was learned. It improves in proportion to how the learned quantity enters it — and here, as in most bounds built from variances, that is its square root.