The notch fit declares the true width, and the true width runs out

A lamp’s spectrum is published as a table — power every nanometre or two, each entry the lamp seen through a slit as wide as the step — and a run of essays has asked what such a table can promise about lines narrower than its step. Each peak is fitted as a Gaussian blurred by the slit; the narrowest width any peak admits is the table’s declared width; and from it follows a ceiling, an upper bound on how high a line’s true peak can rise between the entries. A declared width buys a factor of two found that bound tight and safe for ordinary lines.

A dense discharge breaks it. Cooler atoms between the glow and the tube wall absorb at the centre of each line and notch it, which a reversed line is read as one line set out and a notch in a line is never refused followed to a cooler layer narrower than its line, and a plain Gaussian reads a notched line as a single wider one. A reversed tube declares from its least reversed line reversed every strong line together and found the declaration set by the 578 nm yellow, the least reversed, and the ceiling built on it failing as the column deepened; on a one-nanometre table it failed before the table started refusing lines. A stricter fit declares a wider line found that no threshold on a plain fit helps.

Two nanometres hide the yellow’s notch from any fit put the notch into the fit — an emitted Gaussian less an absorbed one at its centre — and found the one-nanometre table reading the notch from a depth of a half, rising to 0.99 of the line at five, where the two-nanometre table could not tell a notch from a spike. Its closing section turned that into a proposal. If the fit finds the notch, it also finds the line underneath: declare that line’s width, and build the ceiling on it. The prediction was that the repaired ceiling stays above the reversed lamp’s true peak at every depth to three, because the line under the notch is wider and taller than the plain fit’s, and fails past three, where a first-order notch underestimates a thick column’s trough.

The repair finds the true width, and the true width holds to two #

On the one-nanometre table, every line behind a cool column half its width, the notch fit at the yellow peak returns an emitted line between 1.192 and 1.204 nm wide at every depth from nought to five — within two hundredths of the tube’s own 1.2 nm core — and within six hundredths in twenty noisy readings at 0.2 per cent. The plain declaration’s ceiling falls below the true peak from a depth of a half, and from two the table refuses every line. The ceiling built on the notch fit’s width stays above the true peak in every window to a depth of two, and from two and a half falls below it, to 0.956 at two and a half and 0.775 at five, in the window at 434 nm beside the reversed blue line. The ceiling built on the true 1.2 nm core does exactly the same, within a hundredth at every depth. On the two-nanometre table the core’s ceiling is safe to three and fails from three and a half.

• The repair is as good as a repair can be. It recovers the line’s true width, and a declaration cannot do better than the truth.
• The prediction’s depth is too generous by one step. Safe to two, not three, on the table the repair needs.
• What fails past two is the ceiling’s rule. A ceiling computed for an unnotched line of the right width still undershoots a notched one.

The four curves begin together near 1.01 to 1.08 with no column, every declaration safe. The plain declaration’s curve dips below one at a depth of a half and stops at two, where the table refuses every line and so declares nothing. The notch fit’s curve and the true core’s rise slightly to a depth of one and a half and fall from two: 1.015 at two, 0.956 at two and a half, 0.907 at three, 0.775 at five. The two lie on top of each other throughout, the largest gap between them under a hundredth. The two-nanometre table’s core curve sits higher, safe to a depth of three, and crosses below one between three and three and a half.

What the notch fit returns #

A plain fit to a notched line chooses the wider Gaussian that best matches the flattened top, so the plain declaration widens as the column deepens: 1.16 nm with no column, 1.24 at a half, 1.34 at one, 1.46 at one and a half, and then none at all, because the fit’s misses pass the threshold at which a peak is refused. A declared width wider than the truth makes the ceiling assume a broader line, whose table entries sit closer to its peak, and the ceiling comes out too low; that is why the plain declaration fails as it widens.

The notch fit’s emitted width does not widen. It sits between 1.192 and 1.204 nm at every depth, the census’s own grid of trial widths stepping one per cent at a time around the tube’s 1.2 nm, because the absorbed Gaussian takes the notch and leaves the emitted one to fit the line’s flanks, which the column does not touch. With 0.2 per cent noise in the entries the widths scatter between 1.15 and 1.25 nm and still centre on the core. The fit has, in effect, read the discharge’s own line width through the reversal. That is more than the prediction asked of it: the closing section expected the line under the notch to be “wider and taller than the plain fit’s”, and it is narrower — it is the right one.

The true width is not enough #

The natural reading of a repair that recovers the truth is that the problem is solved, and the first figure says otherwise. The ceiling on the notch fit’s width and the ceiling on the true core fail together from a depth of two and a half, because the ceiling’s rule is the same for both. It takes the entries round a window and the declared width and asks how high a Gaussian of that width could rise between them — a factor computed once for an unnotched line under the slit. A notched line is not a Gaussian. Its highest point is on a shoulder beside the notch, and its entries are pulled down in the middle by an amount the factor does not include.

Read window by window, the repaired ceiling at a depth of two and a half sits comfortably above the true peak across most of the table and dips below it in one window. The lowest is at 434 nm, 0.956 of the true peak, beside the blue line at 435.8 nm. At shallower depths, moved by the handle, the lowest window is at 406 nm beside the violet line and stays above one; from a depth of two the blue line’s window takes over, and from two and a half it sinks below one with every deeper column. The yellow, whose fit supplied the width, is never the window that fails. Each strong line is reversed by the column in proportion to its own strength, and the blue’s notch is deeper than the yellow’s at every depth, so a ceiling rule that is just adequate for the yellow is short for the blue.

On the two-nanometre table the same true width gives a ceiling safe to three. The wider slit averages more of each line into each entry, which softens what a notch does to the entries round it, and the factor computed for an unnotched line stays adequate for longer. The one-nanometre table that can see the notch is also the table on which the notch most undermines the ceiling, which leaves a table-maker with a real trade between resolving a reversal and bounding it.

The lines do not fail in the order of their reversal #

Followed past two and a half, the repaired ceiling does not fail in one window and stay failed there alone. The windows beside the strong lines go under one after another. The blue window, at 434 nm, crosses below the true peak at two and a half; the green, at 545, at three and a half, reading 0.985; the violet, at 406, only at five, reading 0.991. The yellow’s window never does: it reads 1.52 at a depth of one and a half and is still 1.29 at five.

The natural guess is that each window fails when its own line’s notch passes some depth, and the tube’s line strengths refuse it. Each line’s column is the green’s depth times that line’s strength over the green’s: the blue’s is 0.77 of the green’s, the yellow’s 0.46, the violet’s 0.23. So the blue window fails when its own line is reversed to a depth of 1.9, the green’s at 3.5 and the violet’s at 1.15, while the yellow’s survives its own line at 2.3. The green, the most reversed line in the tube, is the second to fail; the yellow, more reversed than the violet at every depth, never fails at all.

What does decide the order is not settled by this calculation, and the candidates are the two things the ceiling’s rule leaves out besides the notch. One is where the table’s entries fall against each line’s centre: a line whose core sits between two entries has its top read on its flanks, which a notch lowers less than its centre. The other is what lies under the line — the phosphors’ broad bands under the blue and the green, almost nothing under the violet — since a ceiling taken from the largest nearby entry includes the background and a notch removes only line. The width transfers from the yellow to every line, because the column does not change a line’s width; the ceiling’s rule does not, because the column changes each window’s entries by an amount that depends on more than the line’s own depth.

Noisy tables keep the repair #

A real table carries noise, and the notch-fit essay found its fitted depths scattered by it. The width is steadier than the depth. At 0.1 per cent noise all twenty readings give a safe ceiling at every depth to two and none from two and a half; at 0.2 per cent between sixteen and twenty are safe to two, and none after. The four unsafe readings at 0.2 per cent with no column have fits that strayed to the narrow end of the scatter, near 1.15 nm, which makes the ceiling assume a sharper line than the tube has and come out low; they are the noise’s, not the reversal’s.

So the repair carries over to a measured table with the reliability the notch fit itself has. Where the earlier essay’s rule on the fitted depth — a notch deeper than a fifth of the line — tells a table it is looking at a reversed lamp, the fitted width gives it a declaration that is safe to a column depth of two. Past two, no declaration based on one width and the ceiling’s unnotched rule is safe, and the honest course is the one the plain fit takes by accident: refuse.

What a lamp’s table can promise #

A one-nanometre table of a reversed lamp can now declare its true line width, from the notch fit at its least reversed peak, and the ceiling on that width is safe to a column depth of two. That closes the window the earlier essays found, in which the plain declaration was unsafe but the table had not yet started refusing: from a half to two, the repaired declaration is safe where the plain one was not. From two and a half, the table should refuse even though it knows the width, because its ceiling’s rule does not.

That is a narrower promise than the prediction’s three, and a more interesting one. It separates two things a declaration does — say how wide the lines are, and bound how high they rise — and finds the first solved and the second not. A table that wants to bound reversed lines past a depth of two needs a ceiling whose rule includes the notch.

How the declarations and ceilings were computed #

The tube and tables are the reversed-line census’s: every strong mercury line of a triphosphor tube behind its own cool absorbing column, the green line’s column at depth τ and every other line’s at τ times its strength over the green’s, each column half the width of its line; the tables sampled through a triangular slit as wide as their step, one or two nanometres. The notch fit is the earlier essay’s: at the yellow peak, on seven entries, a slit-blurred Gaussian less a slit-blurred notch 0.6 nm wide at the same centre on a sloping base, solved by least squares at every trial width and centre and kept at the smallest residual. Its width is the emitted Gaussian’s.

The plain declaration is the narrowest width admitted over every peak at the census’s half-per-cent threshold. A ceiling on a width is the census’s: in every window of the table, the largest entry nearby divided by the factor by which a slit-blurred Gaussian of that width can fall below its own peak, or the pair rule where two entries share a line, against the reversed lamp’s true maximum within a slit of the window. A ceiling is safe when that ratio is at least one in every window. The noisy readings are the earlier census’s twenty seeded tables at each noise.

What this does not reach #

One column width. The columns are half their lines’ width; a column as wide as its line notches more broadly and may move both the repair and the depth at which the ceiling fails.

A fit at one peak. The width comes from the yellow alone and is applied to every line, as a declaration is; the blue line’s own notch fit, at the window that fails, is not used.

A Gaussian line. The tube’s lines are Gaussian cores here; a line with wings carries more of its power away from the centre, which a line’s wings widen the width a table declares found widening the plain declaration.

Still open: whether a ceiling with the notch in it holds past two #

The ceiling fails past two because its rule assumes the line it bounds is a Gaussian under the slit. The notch fit already measures each line’s notch depth — the earlier essay found it on the one-nanometre table from a depth of a half — and a ceiling could take it into account: the factor by which a notched line’s peak exceeds its entries, rather than an unnotched one’s.

The calculation is the notch fit run at every strong peak of the one-nanometre table, the ceiling’s factor recomputed in each window for a slit-blurred Gaussian of the declared width with that peak’s fitted notch, and the repaired ceiling set against the true peak at every depth and in the noisy readings. The prediction is that a notch-aware ceiling stays safe to a depth of four, failing only where the blue line’s notch passes nine tenths of its height and the shoulders of the line are too narrow for seven entries to place; and that in noisy readings it fails sooner, because the notch’s depth is less steady than the width. If it holds to five, a one-nanometre table carries everything needed to bound a reversed lamp; if it fails at three, a table should refuse a reversed lamp past the depth at which its notch is first read. A ceiling that knows the notch holds to the deepest column makes that calculation.

A repair can be right and still not be enough #

The habit here is about what it means for a fix to work.

The proposal was a repair to the declaration, and judged as a declaration it is flawless: it returns the tube’s true width at every depth, clean and noisy. Judged by what the declaration was for — a ceiling that holds — it holds one step less far than predicted. The temptation is to blame the repair, adjust its fit, and try again. The comparison that settles it is to give the ceiling the truth and see whether the truth does better.

The failure mode is to judge a repaired input by the downstream result without checking what the downstream result would be given the true input. The test is to run the true input through the same calculation. Here the true width fails exactly where the repaired one does, which puts the remaining failure in the calculation — the ceiling’s assumption about line shape — and not in the input any further work on the fit could improve.