A ceiling that knows the notch holds to the deepest column

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 built that ceiling, and the essays since have tried to break it.

The reversed tube breaks it. In a dense enough discharge, a cooler layer of the same atoms stands in front of the hot ones and absorbs each line’s centre, leaving a notch — a reversed line is read as one line by a plain fit, as a wider line than it is, and a notch in a line is never refused by the declaration. Two nanometres hide the yellow’s notch from any fit then fitted the yellow peak with a notch in the model, and found a one-nanometre table reading it from a depth of a half.

The notch fit declares the true width, and the true width runs out used that fit to declare the tube’s true 1.2 nm core width and built the ceiling on it. It held to a depth of two and failed from two and a half, in the blue line’s window — and the true width failed at exactly the same depths. The declaration was right; the ceiling’s rule was not. The rule assumes the line it bounds is a Gaussian under the slit, and a notched line is not.

The closing section proposed the obvious repair: the fit already knows each notch’s depth, so build each window’s ceiling for a line carrying that notch. The prediction was that a notch-aware ceiling stays safe to a depth of four, failing where the blue line’s notch passes nine tenths of its height and the line’s shoulders are too narrow for seven entries to place; and that in noisy readings it fails sooner, because a notch’s depth is less steady than a width. The alternatives were safe to five, in which case a one-nanometre table carries everything needed to bound a reversed lamp, or failing at three, in which case a table should refuse a reversed lamp past the depth at which its notch is first read.

Safe to five, clean and noisy #

With each window’s ceiling built for its nearest line carrying a notch — the yellow notch’s depth from the notch fit, read as an optical depth −ln(1 − d) and scaled to every line by its strength — the one-nanometre table’s ceiling stays above the reversed lamp’s true peak in every window at every depth from nought to five; its lowest window stands 0.9 to 29.5 per cent above the peak. Built on the true notches it stands 0.8 to 13.3 per cent above. In each of the notch census’s twenty noisy readings at 0.1 and at 0.2 per cent, at every depth, the ceiling on that reading’s own fitted notch stays above the true peak too, the lowest at 0.6 per cent. With the fitted depth used directly as the optical depth, the ceiling holds to four and fails from four and a half. The tightest window moves from the blue line, where the unnotched ceiling failed, to the weak violet line at 404.7 nm.

• The prediction is beaten. The ceiling holds at the deepest column the census runs, not just to four.
• Noise does not undo it. Four hundred and forty readings, none below the true peak.
• How the fit’s depth is read decides the result. The fit returns a first-order depth; read as such, the ceiling fails a column short of the deepest.

The four curves agree to a depth of one and a half and part after it. The unnotched ceiling crosses below the true peak between two and two and a half and keeps falling, to 0.78 at five. The three notch-aware ceilings climb instead — a ceiling that knows the notch grows more cautious as the notch deepens — and only the first-order one turns down again, crossing the true peak between four and four and a half. The fitted ceiling runs above the true-notch ceiling at every depth past one, by a widening margin, and the reason for that margin is the subject of a section below.

What a notch does to the factor #

The ceiling rests on two factors. For any line of the declared width placed anywhere, the nearest table entry reports at least a share $k$ of the line’s true value at any wavelength within half a step, and two flanking entries summed report at least a share $k_2$; so the true value can be no more than the entry over $k$, or the pair’s sum over $k_2$, and the ceiling is the smaller. A notch lowers both factors. With no notch, $k$ is 0.376 and $k_2$ 0.954; at a notch of optical depth three, 0.331 and 0.620. The pair’s factor falls furthest, because the pair’s strength was seeing both sides of a line’s single top, and a notched line has two tops with a trough between them that the flanking entries straddle.

A smaller factor is a higher ceiling — the entry divided by a smaller share — so the notch-aware ceiling rises exactly where the line has a notch, by as much as the notch hides. The unnotched ceiling divides by the unnotched factors everywhere, and at a notched line those overstate how much of the line the entries saw; it is too low by the ratio of the two factors, which is the failure the earlier essay found.

Window by window #

Across the table the ceiling stands far above the true value in the bed between lines — ratios above two are drawn at two — and dips at each line, where the entries are closest to telling the truth. At a depth of five the unnotched ceiling falls below the true peak at the blue line, 0.78, and at the green, 0.90; the notch-aware ceiling lifts both windows well clear. Its lowest window is now at the violet line near 404 nm, a weak line whose notch is shallow — 0.23 of the green’s depth — and whose window the ceiling therefore treats almost as unnotched. There, at 1.11 on the true notch and 1.29 on the fitted one, the margin is smallest, but the violet line’s own notch is too shallow to threaten it.

So the blue line’s failure was never about the blue line. It was about the factor: any line notched deeply enough fails an unnotched ceiling, the blue line was the first to cross because it is strong and its window has no help from a neighbour, and once the factor knows the notch every line’s window is safe.

How the fit’s depth must be read #

The notch fit models a reversed line as an emitted Gaussian less a notch Gaussian scaled by a depth d — a subtraction, the first-order form of the absorption it stands for. A real column absorbs as exp(−t·n), where t is the optical depth at the notch’s centre, and the two agree at the centre when 1 − d = exp(−t). Read that way, the fitted depth is within a tenth of the true optical depth to a column of two, the yellow’s notch at 0.97 against 0.92. Past two it runs ahead: 1.56 against 1.38 at three, 2.28 against 1.83 at four, 4.25 against 2.29 at five, as d approaches one and −ln(1 − d) steepens.

Read directly as the optical depth, d can never pass one, and the yellow’s notch at a column of five is read as 0.99 where it is 2.29. Scaled to the green and blue lines by strength, that understates their notches by more than half, the factors come out too large, and the ceiling falls below the true peak from four and a half. The difference between a ceiling safe to five and one that fails is a logarithm, applied to a number the fit already returns.

The safety is partly an overstatement #

Built on the true notches, the ceiling’s margin over the true peak grows to 13.3 per cent at a depth of three and settles near 11 per cent beyond. Built on the fitted notch it keeps growing, to 29.5 per cent at five. The extra margin is the fit’s overstatement of a deep notch, and an overstated notch errs safe: a deeper assumed notch means smaller factors and a higher ceiling. The ceiling is therefore safe at five for two reasons at once, and only one of them is that it knows the notch. The other is that a first-order fit read through a logarithm exaggerates exactly the notches that matter.

That matters for what can be claimed. The true-notch ceiling is the honest statement — a ceiling built on the right notch holds to the deepest column, with a tenth to spare. The fitted ceiling is a usable one, safe and more cautious than it needs to be at depth, and its caution is not a property anyone designed.

Noisy readings #

The prediction expected noise to undo the repair sooner, because a notch’s depth scatters more than a width does. At 0.1 and 0.2 per cent noise every one of the twenty readings, at every depth, builds a ceiling above the true peak, the lowest 0.6 per cent above it, with no column at all, where there is no notch to matter. The noisy readings’ worst ceilings track the clean table’s a little below it, because noise scatters the fitted depth both ways and the readings that read it shallower give lower ceilings; but at no depth does the scatter reach the margin. At depth, where the notch matters, the logarithm’s overstatement gives the margin room the noise never uses.

What the declaration now has to carry #

The declaration began as one number. A reversed tube declares from its least reversed line found the whole table’s width set by whichever peak admitted the narrowest Gaussian, and a stricter fit declares a wider line found that no fit threshold made a plain Gaussian refuse a notched line. The repair here does not make the declaration one better number. It makes the declaration a description of each line’s shape: a width, shared, and a notch depth, read at one peak and carried to the others by their strengths.

That has a cost the census does not charge. Carrying the yellow’s notch to the blue and green lines assumes every line is reversed by the same column, in proportion to its strength — true of the census’s tube, where one cool layer covers every line, and an assumption about any real one. A tube whose blue line passed through more cool gas than its yellow, because the column is not uniform, would have a deeper blue notch than the scaling says, and the blue window would get a ceiling built for a shallower notch than it has. The margin the fitted ceiling carries at depth — a sixth to nearly a third from a column of three — is room for some such error, and how much is not computed.

There is a simpler fallback the earlier essays pointed to and this one makes less necessary. A table can refuse a pair it cannot resolve found a table able to decline a declaration it cannot support; a table that reads a notch it cannot fit — a depth past what its slit can place — could decline in the same way. With the notch in the ceiling, the one-nanometre table never needed to.

What a table maker should take from this #

A one-nanometre table can bound a reversed lamp. It needs to fit each strong peak with a notch, read the notch’s depth as an optical depth, and build each window’s ceiling for a line carrying that notch. Done that way, the ceiling held at every column the census runs, on clean readings and on noisy ones.

The ceiling’s rule is part of the declaration. A declared width alone was not enough — the true width failed where the fitted one did — and the earlier essays’ repair of the width was a repair of the wrong number. What the table must declare, for a lamp that reverses, is the width and the notch.

Read a first-order parameter as what it approximates. The notch fit’s depth is a subtraction standing in for an exponential, and the table’s safety at depth rests on converting it back.

How the ceilings were built #

The lamp is the census’s reversed tube: four mercury lines with 1.2 nm cores on a broad phosphor bed, every line behind a cool column half its width, the green line’s optical depth from nought to five and each other line’s in proportion to its strength. The table is that lamp through a triangular slit of half-width one nanometre, sampled every nanometre. The ceiling’s factors are found as the earlier ceilings’ are, by placing the line at every offset within three slit widths and three line widths, in steps of a twelfth of the smaller, and taking the least share any entry, or any flanking pair, reports of the line’s true value within reach — now for a 1.2 nm Gaussian times exp(−t·n), n a Gaussian half as wide. With no notch they are the earlier factors exactly. Each window takes the factors for its nearest line if that line lies within reach, its optical depth the yellow notch’s scaled by the lines’ strengths, and the unnotched factors otherwise. The yellow notch’s depth comes from the truth, from the notch census’s fit at the yellow peak read as −ln(1 − d), or from d itself; the noisy readings are the census’s own twenty at each noise.

What this does not reach #

Strengths known. The yellow notch is scaled to the other lines by their true strengths; a table would have to estimate each line’s strength from its own entries, and a reversed line’s entries understate it.

One column width. Every column is half its line’s width, the census’s cool layer. A column as wide as the line — the hot gas absorbing its own light — gives a broader, shallower notch, and the factors would need recomputing for it.

One slit. The two-nanometre table cannot read the notch at all, and is not repaired here.

Still open: whether a two-nanometre table can borrow the notch #

Two nanometres of slit leave too few entries across the yellow for the notch fit to read it, and the two-nanometre table’s core-width ceiling held to three and failed from three and a half. A lamp measured twice — once finely, once coarsely, as a manufacturer’s datasheet and a laboratory’s survey often are — could carry the notch from the fine table to the coarse one.

The calculation is the ceiling’s factors recomputed for a two-nanometre slit, and the two-nanometre table’s ceiling built on the notch the one-nanometre table’s fit reads, at every depth, against the true peak. The prediction is that it holds to five as the one-nanometre table’s does, because the factors are a property of the line and the slit and the borrowed notch is the same notch; and that its margin is wider, because a broader slit hides more of a notch and its factors fall further. If it fails, a coarse table’s ceiling depends on more than the line’s shape — on where its few entries fall against the notch — and a borrowed notch is not enough. A coarse table can borrow a notch, but not a phantom makes that calculation.

Repairing the rule, not the input #

The habit here is about where a failure lives.

The earlier essays treated the ceiling’s failure as a problem of its input: a declared width too wide, then a width read with the notch in the fit. The last of them found the true width failing exactly where the repaired one did, which is the evidence that the input was never the problem. The rule that turned a width into a ceiling assumed a shape the lamp did not have.

The failure mode is to keep improving the number fed to a calculation that is wrong for the case. When the best possible input fails at the same place as a poor one, the input is not the failure. The test is to feed the calculation the truth and see whether it holds — the true width did not, and the true notch did — and to fix whichever part the truth could not rescue.