A stricter fit declares a wider line

A lamp’s spectrum is published as a table: power at every nanometre or two, each value the lamp seen through a slit as wide as the step. A table of a discharge lamp hides how narrow its emission lines are, and how high their true peaks, because the slit spreads each line across several entries. A thread of essays has built the means to recover them. Each peak in the table is fitted as a single Gaussian blurred by the slit; the widths the fit accepts within a stated threshold — half a per cent of the peak’s height, chosen at the start — say how narrow the line might be; the narrowest of these over every peak is the table’s declaration; and from a declared width a ceiling follows on how high any line’s true peak can be. A line’s wings widen the width a table declares found the declaration safe for lines with Lorentzian wings.

Then the lines were reversed. In a dense discharge, cooler atoms in front of the hot ones absorb at the centre of each line, notching it. A reversed line is read as one line and a notch in a line is never refused found a two-nanometre table reading a reversed line as a single wider Gaussian, its notch averaged away by the slit. A reversed tube declares from its least reversed line reversed every line together, each in proportion to its strength, and found the two-nanometre table admitting every reversed line at half a per cent and declaring a width — set by the 578 nm yellow, the least reversed — whose ceiling falls below the true peak from a column depth of two and a half behind a cool absorbing layer, to ×0.69 at depth five.

The repair that suggests itself is a stricter fit. A notched top is not quite a Gaussian, and a threshold tight enough should notice. The prediction was that at a tenth of a per cent the two-nanometre table refuses the green and blue lines behind the cool column from a depth of about three, admits the yellow to four, and between them declares unsafely from the yellow, as the one-nanometre table had done; and that a threshold that strict would also refuse unreversed lines on a table read from a real, noisy instrument — so that a two-nanometre table could not be made safe for a reversed lamp by fitting it harder.

The refusals arrive; the declaration gets worse #

At a threshold of a tenth of a per cent, behind the column half as wide as the lines, the two-nanometre table refuses the 435.8 nm blue line from a depth of three and the 546.1 nm green from four, as predicted. It never refuses the 578 nm yellow, at any threshold, depth or column, and the yellow goes on setting the declaration. But a stricter threshold declares a wider width at every depth — at depth two and a half, 1.41 nm at half a per cent, 1.48 at a fifth, 1.51 at a tenth — and the ceiling built on a wider width is lower. Every stricter threshold lowers the ceiling at every depth behind both columns, and moves the depth from which it falls below the reversed lamp’s true peak earlier: from two and a half to two behind the cool column, from four and a half to four behind the full one. On the unreversed tube’s table with noise, a tenth of a per cent refuses nothing until the noise reaches half a per cent; there it refuses the yellow — the declaring line — in sixteen readings of twenty.

• The prediction’s refusals are right to the depth. A tenth of a per cent sees the blue and green notches.
• Refusing lines does not make the declaration safe, because the line that declares is the one the notches spare.
• A stricter threshold makes it less safe. It discards the narrow widths a loose fit admitted, which were the ones that made the ceiling high.
• The strict threshold is too strict only for a noisy table. Up to a fifth of a per cent of noise it refuses nothing unreversed.

Each solid line is one threshold behind the cool column, each dashed line behind the full one. The three thresholds stack in the same order at every depth, the strictest lowest. Behind the cool column the ceiling crosses below the true peak at a depth of two and a half for half a per cent, and at two for a fifth and for a tenth, and by depth five the strictest threshold’s ceiling is ×0.66 of the peak, against ×0.69 for the loosest. Behind the full column the crossing moves from four and a half to four. A stricter threshold does what the loose one did, earlier.

Why stricter means wider #

A threshold accepts every width whose best fit misses the table by less than it. For a well-sampled line the accepted widths form a range around the best width, and the declaration is the range’s lower end — the narrowest width the data cannot rule out. Tightening the threshold shrinks the range from both ends, and the lower end moves up. At depth zero, the unreversed tube, the declaration rises from 1.07 nm at half a per cent to 1.16 and 1.18 at the two stricter thresholds; at depth two and a half from 1.41 to 1.48 and 1.51. The ceiling is built on the declared width: a narrower assumed line means a higher possible true peak, and the narrowest width the table cannot exclude is the one that makes the ceiling safe. A stricter threshold throws that width away.

So the stricter fit’s tighter judgement is applied in the wrong direction for a ceiling. It is right that the data rule out the narrow widths more firmly at a tenth of a per cent — if the line really were the Gaussian the fit assumes. The reversed yellow is not. Its notched top is fitted best by a wider Gaussian than the line’s emitted width, so the narrow widths a loose threshold still admitted were closer to the truth than the best fit, and the strict threshold discards exactly them.

Which lines each threshold refuses #

The refusals are where the prediction put them. At half a per cent behind the cool column the table refuses nothing at any depth; at a fifth it refuses the blue only at depth five; at a tenth it refuses the blue from depth three and the green from four. Behind the full column, where each notch is as wide as the line and deeper in effect, a fifth of a per cent refuses blue and green from three and a half and a tenth from two and a half and three. The yellow is never refused. Its column is under half the green’s, because the yellow line is the weakest of the three strong lines and a column’s depth scales with the line’s strength, and its notch stays shallow enough to pass every threshold up to a tenth of a per cent at every depth.

That is the structural reason no threshold repairs the table. The declaration is the narrowest width any admitted line allows, and refusing the deeply reversed lines leaves the shallowly reversed one to declare. A table can refuse a pair it cannot resolve found refusal a useful signal where a line’s shape departs grossly from one Gaussian; a shallow notch is the departure a refusal cannot see, and it is the one that sets the width.

The noisy table #

The prediction’s second half was that a threshold strict enough to catch a reversal would also refuse unreversed lines on a table read from a real instrument, whose entries carry noise. The census reads the unreversed tube’s two-nanometre table twenty times with each entry scattered by a relative noise of a tenth, a fifth, a half and one per cent, and fits each reading as before.

The fit absorbs most of the noise. It has five free numbers — the Gaussian’s height, centre and width and a sloping background — against seven table entries, so most of a random scatter is taken up by small shifts in those numbers rather than left as misfit. At a fifth of a per cent of noise the median miss stays under a tenth of a per cent at every line: 0.058, 0.033 and 0.046 per cent at the blue, green and yellow. At half a per cent of noise the yellow’s reaches 0.12, and at one per cent 0.23.

Up to a fifth of a per cent of noise, no line is refused at any threshold. At half a per cent of noise the tenth-of-a-per-cent threshold refuses the yellow in sixteen readings of twenty, the green in six and the blue in one; at one per cent, the yellow in every reading. Half a per cent of threshold refuses nothing at any noise tried. The prediction’s claim holds only for a noisy instrument, and its size is a property of the instrument rather than of the threshold: a good spectroradiometer reading a strong line well above its noise floor sits at the quiet end, where the strict threshold is harmless.

What the noise does pick out is the line that matters. The first unreversed line a strict threshold refuses as noise grows is the yellow, the same line that declares when the others are reversed. On a reversed lamp read by a noisy instrument, a tenth of a per cent would refuse the blue and green for their notches and the yellow for its noise, and declare nothing — which is at least safe.

What a table’s reader should take from this #

Fitting harder does not make a two-nanometre table of a reversed lamp safe. It refuses the lines whose reversal it can see and declares from the one whose reversal it cannot, a little wider than before, and the ceiling falls sooner. The prediction reached the right conclusion — that no threshold rescues the table — for half the right reason: the threshold is not too strict for real data, at least for a quiet instrument; it is pointed the wrong way for a ceiling.

The useful repair lies elsewhere. The one-nanometre table refused every strong line before its declaration could fail for long; the two-nanometre table’s problem is that its slit averages a notch into a top a Gaussian fits. A reader who must use a two-nanometre table of a lamp that may be reversed should treat the declared width as a lower bound on how wide the line is and not trust a ceiling built on it, whatever threshold the fit used.

A threshold is a statement about the instrument #

The collection’s thresholds have always carried an assumption about who reads the table. One slit, two requirements found an instrument with one slit unable to serve both a line lamp and a notched sample, and priced the compromise; a narrow slit shrinks the error, not the bound found a narrower slit improving what a colour engine computes from a table without improving what it can promise about its own error. The fit threshold is a third such choice. Half a per cent was chosen as a margin above what a well-sampled Gaussian line misses by — a few hundredths of a per cent — with room for a real instrument’s noise. The census confirms the room: up to one per cent of noise, half a per cent refuses no unreversed line.

A tenth of a per cent spends that room. It is safe on a quiet instrument and on a table computed rather than measured, and on those it catches the deeply reversed lines. What it cannot do is catch the shallow reversal that sets the declaration, and what it does do — narrow every admitted range from below — is the opposite of what a ceiling needs. A declared width buys a factor of two found how much a colour engine gains from a lamp declaring its narrowest feature; that gain rests on the declaration being the narrowest width the data allow, and a stricter fit makes it less so.

How the tables were read #

The lamp is the reversed-tube census’s: the tube’s lines as Voigt profiles, every line behind a column of its own atoms whose optical depth at the line’s centre is the green line’s depth τ scaled by the line’s strength over the green’s, the column’s profile as wide as the line’s emission or half as wide. Its table is the lamp through a triangular slit of half-width two nanometres, integrated at a tenth of a nanometre, at every two. Each peak is fitted over its seven nearest entries by a Gaussian blurred by the slit, with its centre scanned over the slit’s width and a straight background, at widths from 0.6 to 3 nm in one-per-cent steps; a width is admitted if its best fit’s root-mean-square miss is under the threshold times the peak’s own entry. The declaration is the narrowest admitted width over the admitted peaks, and the ceiling the reversed-tube census’s, built on the declared width and set against the reversed lamp’s true peak in every window. The noisy readings scatter each entry of the unreversed tube’s table by a seeded Gaussian of relative size 0.1, 0.2, 0.5 or 1 per cent, twenty readings at each, fitted the same way.

What the census does not reach #

Gaussian fits. A fit with a notched profile — two Gaussians, one negative — could recognise a reversal directly; it has more numbers than seven entries comfortably determine, and whether it would be stable is untested.

Uniform relative noise. A real instrument’s noise grows towards a table’s weak entries and is not independent from one entry to the next; correlated noise is absorbed less readily by a sloping background.

One slit. The one-nanometre table’s behaviour under a stricter threshold — earlier refusals, and the same narrowing of the admitted range — is not recomputed here.

Still open: whether a notch-aware fit refuses the yellow #

A Gaussian fit cannot see a shallow notch, and the yellow’s notch is shallow. A fit that models a reversed line as an emitted Gaussian with an absorbed Gaussian taken out of its centre has one or two more numbers — the notch’s depth and width — and could in principle find the yellow’s notch where a plain Gaussian finds only a wider line.

The calculation is this census with each peak fitted by that reversed profile as well as by a plain Gaussian, the notch’s depth free and its width tied to the column’s, and a line called reversed when the reversed fit’s notch is more than twice its own standard error. The prediction is that two nanometres of slit leave too few entries across the yellow for the notch to be determined — its standard error larger than any depth the census runs below five — while the one-nanometre table finds it from depth two. If so, a two-nanometre table cannot reveal a reversed lamp by any fit, and the declaration’s safety rests on knowing the lamp from somewhere other than its table.

A stricter test is not a safer one #

The habit here is about what tightening a test buys.

A threshold that admits fewer fits sounds like a more careful one, and for the question a fit answers — is this line a Gaussian? — it is. But the declaration was never the answer to that question. It was the narrowest width the table cannot rule out, a quantity that grows as the test tightens. Tightening the test made the answer to the fit’s question more reliable and the declaration’s derived quantity less safe, because the two pull in opposite directions at the lower end of the admitted range.

The failure mode is to tighten a test without asking which way its result is used. A test used to admit is made safer by tightening; a test whose admitted set is used as a bound is made safer by loosening at the end the bound is taken from. The census’s ceiling is taken from the narrow end, and the narrow end is what tightening discards.