A narrow table declares its own lines

A declared width buys a factor of two found that a colour table’s bandpass error — the error of storing a lamp and a sample as spectra tabulated through a slit — can be bounded if the lamp’s maker declares a few numbers about the lamp, the most important being the width of its narrowest feature. A narrow table can vouch for its own peaks then took one of the other declarations off the maker’s list: through a slit about as narrow as a lamp’s lines, each window’s highest point can be read from the table’s own entries, and the bound halved. One number was still the maker’s: the narrowest feature, 1.2 nanometres for a fluorescent tube’s mercury lines, without which the table could not say how much of a line the slit had flattened.

That essay closed on getting the last number from the table too. A 1.2-nanometre line seen through a one-nanometre slit makes a peak in the table three or four entries wide, and the ratios among those entries depend on the line’s width; a narrower line makes a sharper peak for the same area. So the narrowest line consistent with the entries is a measurement. The prediction was that a bound built on it stays valid at one nanometre and fails at five, and that the failure is sharp — somewhere between two and three nanometres, where the share of a 1.2-nanometre line a slit keeps falls through about a half.

The table names its width, until it cannot #

Measured to half a per cent, a one-nanometre table’s entries around each of the tube’s lines admit only line widths from 1.17 to 1.21 nanometres, and the truth is among them. A two-nanometre table admits 1.08 to 1.31. A three-nanometre table cannot rule out any width down to the narrowest tried, a fifth of a nanometre, for two of its three lines. The table’s own declaration — the narrowest width its lines admit — is 1.17 at one nanometre, 1.08 at two and 0.20 at three and five, and never above the truth. A bound built on it never falls below the error; at one nanometre it is within two per cent of the bound on the maker’s 1.2, at two within seven, and at three it is the global bound, where the maker’s number still bought a fifth.

• The prediction holds, sharply and where predicted, between two and three nanometres.
• “Fails” means useless, not invalid. A table that cannot see a line’s width admits the narrowest, and a bound built on the narrowest is safe and loose.
• A narrow table needs nothing from the lamp’s maker: at one nanometre its own declaration costs two per cent.
• The edge moves with the table’s precision — a fifth of a per cent pushes it past three nanometres, one per cent brings it under two.

Fitting a line to a few entries #

The tube is the collection’s analytic one, of the kind a lamp is not a blackbody described: three broad phosphor bands and four mercury lines, each a Gaussian 1.2 nanometres wide at half height, at 404.7, 435.8, 546.1 and 578.0. It is tabulated through a triangular slit of half-width one, two, three or five nanometres, on a grid of the same step — the arrangement the slit is what makes it legal established as the one a tabulation rule assumes.

Around each line the table rises into a peak, and the seven entries centred on it are fitted by the one shape the question allows: a single Gaussian line of some width, centre and height, seen through the same slit, standing on a background that may slope — the phosphor bed. For each width tried, from a fifth of a nanometre to twelve, the centre is scanned and the height and background solved by least squares; the best fit’s root-mean-square miss, as a share of the peak entry, says how well that width can explain the entries.

Through a one-nanometre slit the miss has a sharp minimum at the true width. A line of 1.2 nanometres fits the entries to a few parts in ten thousand; a line of 1.0 or 1.4 misses by more than a per cent. Through two nanometres the dip is broader and shallower. Through three and five the curve is flat from the narrowest width tried up to about two nanometres: every line narrower than the slit’s own width is fitted equally well, because the slit has reduced it to its own triangular shape.

Which widths each line admits #

A table’s entries are measured, and half a per cent is a reasonable precision for a spectroradiometer reading a line’s peak. The widths a line admits are those whose best fit misses by less than that. Through a one-nanometre slit, all three lines the table resolves — the blue, green and yellow; the violet line at 405 is too weak against the bed to count as a peak — admit 1.17 to 1.21 nanometres, within four per cent of the truth either way. Through two, about 1.1 to 1.3.

Through three the answer changes character. The yellow line still admits a range around its true width, 0.96 to 1.48, but the blue and green lines admit everything from 0.20 up to 1.5 and 2.0. Their entries are consistent with a line five times narrower than the real one, because at that slit a narrow line and a very narrow line leave the same three entries. Through five, the two lines that fit at all admit anything from 0.2 to several nanometres, and the blue line fits no width within half a per cent, because the phosphor band under it curves across seven entries thirty nanometres wide and a sloping background cannot follow it.

Three lines through three entries #

The degeneracy is visible. Through a three-nanometre slit the green line’s seven entries are fitted within half a per cent by a line 0.20 nanometres wide, by one 1.21 wide and by one 2.02 wide. Drawn between the grid points as the slit would read them, the three fits differ — the narrow line’s peak is sharper and its flanks steeper — but at the grid points themselves they agree, and the grid points are all a table has.

This is a grid is not a resolution in a particular form. The table’s resolution is set by the slit, and a feature narrower than the slit is described by its area and position alone; its width has been averaged away before the table was written. A narrow slit on a coarse grid would not help either, because the width lives in the ratios of neighbouring entries and a coarse grid has too few of them across a line.

The table’s own declaration #

The bound needs a width no line is narrower than, so the table’s declaration is the narrowest width any of its lines admits. At half a per cent that is 1.17 nanometres through a one-nanometre slit, 1.08 through two, and 0.20 — the narrowest width scanned — through three and five. The drop between two and three is as sharp as predicted: from nine tenths of the truth to a sixth of it in one step of the slit.

It is never above the truth at half a per cent or at one, which is the property that matters: a declaration wider than the real narrowest feature would let the bound assume the slit flattened lines less than it did, and the ceiling could fall below a window’s true peak. A declaration that is too narrow only makes the bound looser. The table’s consistency test errs, when it errs, on the safe side, because an unresolved line is consistent with every narrower line.

The precision moves the edge. Entries known to a fifth of a per cent pin the width more tightly and push the edge out: the three-nanometre table then declares 1.12 from its yellow line, though its other two lines fit no width at that precision. Entries known to one per cent pull it in: the two-nanometre table declares 0.92. At a fifth of a per cent the one-nanometre table’s declaration lands on 1.21, the scanned width just above the truth — a reminder that a declaration read from a table carries the resolution of whatever scan produced it.

Why the edge is where it is #

The prediction put the edge in the right place for a reason that turns out to be wrong. It reasoned from the share of a line the slit keeps at the nearest grid point, which was supposed to fall through a half between two and three nanometres. It does not: through a one-nanometre slit that share is already only 0.38, through two 0.27, through three 0.20, through five 0.13. It falls smoothly, with no half anywhere near the edge.

What decides the edge is how much of a table peak’s width is the line’s. A line seen through a slit makes a peak whose spread is the line’s own spread and the slit’s, added in quadrature: a 1.2-nanometre line contributes a variance of about a quarter of a square nanometre, and a triangular slit of half-width w contributes $w^2/6$. Through one nanometre the line is three fifths of the peak’s variance; through two, just over a quarter; through three, a seventh; through five, a sixteenth. The width can be read only while changing it changes the entries by more than their precision, and at half a per cent that stops being true somewhere between a quarter and a seventh.

That is also why the precision moves the edge. A table five times more precise can read a line that is a seventh of its peak’s spread; one twice as coarse cannot read a quarter. The edge is where the line’s share of the spread meets the table’s precision, and the slit only sets the first of the two.

What the bound costs #

The bound is the one the tables cannot bound what they discarded set up: Cauchy–Schwarz on the covariance two separately blurred tables throw away, with the lamp’s half capped by the narrowest feature it is declared to have. Only the declaration changes here. Built on the table’s own declaration, the bound never falls below the error, at any slit, and the ceiling it reads for each window is never below that window’s true peak. Its looseness — the bound over the actual error, a median over the earlier census’s sixty-eight notch filters — is ×130 at one nanometre against ×128 on the maker’s 1.2, and it still certifies twenty-one notches within one colour difference against the maker’s twenty-three. At two nanometres, ×139 against ×130.

At three nanometres the table’s own bound is ×116, the global peak’s ×115, where the maker’s width still bought ×91. The table has declared a width so narrow that the share of a line the slit keeps, by its own reckoning, is under four per cent, and a ceiling built on that is no ceiling. At five nanometres nothing is lost, because the maker’s width bought almost nothing there either.

A narrow slit shrinks the error, not the bound found that the bound’s looseness is mostly the lamp’s peak charged to every window. The last two essays have moved that peak and that width from the maker’s declaration into the table. What remains of the declaration at one nanometre is nothing: the table carries its own bound.

What an instrument maker should take from it #

A table measured through a slit at or below the lamp’s narrowest feature is self-describing. It carries its line widths in the ratios of its entries, its local peaks in their heights, and so everything a bound on its bandpass error needs. A colour engine handed such a table needs to know only the slit and the precision.

A table measured through a wider slit is not, and cannot be made so by analysis. The width is gone before the analysis begins. The engine then needs the maker’s number, or accepts the global bound — and the choice between them is a choice about whom to trust, not about how to compute.

And the precision has to be stated with the table. The same entries declare a usable width at half a per cent and a useless one at one, and a table without a stated precision declares nothing.

How the widths were read #

The tube is the collection’s: phosphor Gaussians at 545, 610 and 480 nanometres with widths 95, 75 and 60, and mercury lines of 1.2 nanometres at half height. Its table at slit w is the lamp through a triangular slit of half-width w on a w-nanometre grid, as the earlier census built it. A peak is an entry above both neighbours and above 0.4. At each peak, the seven entries centred on it are fitted, for each width from 0.2 to 12 nanometres in steps of four per cent, by H times a Gaussian line of that width and centre c through the same slit, plus $b_0 + b_1(\lambda - \lambda_0)$; c is scanned in fortieths of the slit across one slit either side, and $H$, $b_0$ and $b_1$ solved by least squares, keeping H positive. The miss is the root-mean-square residual over the peak entry. A line admits the widths whose miss is within the stated precision; the declaration is the narrowest admitted by any line. The bound is the earlier essay’s, with the declaration in place of 1.2 in the fraction of a line the slit keeps.

What this leaves out #

The lines are Gaussians. A real mercury line’s profile has wings — pressure broadening gives it a Lorentzian component — and a Gaussian fit to a line with wings would read it as wider than its core. That would push the declaration the unsafe way; the margin at one nanometre, 1.17 against 1.2, is small enough that it would need checking against a measured line.

The precision is a single number. Real noise is larger in the dim entries on a line’s flanks than at its peak, which would widen the admitted range at the flanks’ expense.

Only the tube is tested. A laser projector’s lines are a fifth of a nanometre wide, below every slit here — the narrow primaries a screen is a poor lamp found leaving holes in a display’s white; its table would declare the narrowest width scanned at every slit, and its bound would need the maker’s number or a slit narrower than any a colour instrument uses.

Still open: whether a table can say which of its peaks are lines #

The declaration here assumes every table peak is a line standing on a smooth bed, and the fit’s success is the test. A peak can also be two lines closer than the slit, or a narrow phosphor band, and either would be fitted as one wider line — the unsafe direction.

The calculation is this census on tubes whose lines are doubled — the 577 and 579 nanometre pair a real mercury lamp has, at 1.2 nanometres each — and on a tube with a narrow europium band near 611, with the question whether the fit’s miss flags them. The prediction is that a doublet two nanometres apart reads, through a one-nanometre slit, as a peak no single line fits within half a per cent, so the table refuses to declare from it; and that through two nanometres it fits as a single line about 2.5 nanometres wide, admitted and wrong. If so, a table’s own declaration is safe only where its slit resolves the lamp’s closest pair of lines, and the precision test is what says whether it does.

A measurement that cannot see a quantity admits every value of it #

The habit is about what an inference does at the edge of its data.

Asking which line widths a table’s entries are consistent with could have gone wrong in the dangerous direction — a table that cannot see a width might have reported one anyway, from the shape of its slit. It did not, because the question was asked as consistency rather than as a best fit: every width the entries could not distinguish was admitted, and the narrowest of them was the declaration. Where the table was blind, it said so by admitting everything.

The failure mode is to read a best fit as a measurement where the data cannot distinguish the answer from its neighbours. The best-fitting width through a three-nanometre slit was a number, and it was not information; the range of widths the entries admitted was the information, and it said there was none.