Two nanometres hide the yellow's notch from any fit

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 thread of essays has asked what such a table can say about lines narrower than its step. Each peak is fitted as a single Gaussian blurred by the slit; the widths that fit within a threshold say how narrow the line might be; the narrowest over every peak is the table’s declared width; and from it follows a ceiling on how high any line’s true peak can rise between the entries. A line’s wings widen the width a table declares found that declaration safe for ordinary lines.

A dense discharge breaks it. A reversed line is read as one line set out the mechanism: cooler atoms between the glow and the tube wall absorb at the centre of each line, notching it. A notch in a line is never refused found that a two-nanometre table reads a notched line as a single wider Gaussian, its notch averaged into the top. A reversed tube declares from its least reversed line reversed every line together and found the declaration set, at every depth, by the 578 nm yellow — the weakest of the strong lines, so the least reversed — and the ceiling built on it falling below the true peak from a column depth of two and a half behind a cool absorbing layer. A stricter fit declares a wider line tried a tenth of a per cent instead of half and made it worse: the yellow is never refused, and the stricter threshold only narrows the widths it admits from below.

A plain Gaussian cannot see a notch; a fit with a notch in it might. The proposal was to fit each peak as an emitted Gaussian with an absorbed Gaussian taken out of its centre, the notch’s depth free and its width tied to the column’s, and to call a line reversed when the fitted notch exceeds twice its own standard error. The prediction was that two nanometres leave too few entries across the yellow for the notch to be determined at any depth below five, while the one-nanometre table finds it from a depth of two.

Two nanometres find a spike; one finds the notch #

At two nanometres the best notch fit to the yellow peak carries a negative notch — a spike added to the Gaussian’s top — at every column depth from nought to five, unreversed table included. With noise, the readings a two-standard-error rule calls reversed are the same draws at every depth from nought to three, three of twenty at 0.1 per cent noise and six at 0.2, and each fits a notch as tall as the line itself. At one nanometre the fitted notch is nearly nought when nothing is reversed, past a fifth of the line from a depth of a half, and deeper at every deeper column, reaching 0.99 at five. Every noisy reading is called reversed from a depth of one, and half of them at a half — but so are seven and nine of the twenty unreversed readings. Calling a line reversed when its fitted notch passes a fifth of the line separates them exactly: none of the unreversed readings, all of them from a depth of one.

• The prediction’s first half holds, more strongly than stated. Two nanometres do not merely leave the notch uncertain; they leave the fit no way to tell a notch from its opposite.
• The second half comes early. One nanometre finds the notch at a depth of one, not two.
• The proposed rule is the weak part. Two standard errors from seven entries and four fitted numbers are not two standard errors.

The two curves go in opposite directions from the first point. The one-nanometre curve starts at −0.02 — no notch, to within the fit’s own error, on an unreversed line — and rises steadily with the column: 0.22 at a depth of a half, 0.38 at one, 0.62 at two, 0.79 at three, 0.99 at five. The two-nanometre curve starts at −4.2 and rises towards nought, never crossing it. A negative notch of four times the line’s height is not a measurement of anything in the lamp; it is the fit using its spare number to reshape a Gaussian that would otherwise be the wrong shape for the entries.

What the notch fit is #

The plain fit has three numbers at each trial width and centre: the height of a slit-blurred Gaussian and a sloping base under it. It is solved by least squares on the seven table entries centred on the peak, at every width the census scans and every centre within a slit of the peak, and kept at the smallest residual. That is the census’s fit exactly, and on the clean tables it reproduces the census’s own least miss at both slits and every depth to twelve places.

The notch fit adds one regressor: a second slit-blurred Gaussian at the same centre, 0.6 nanometres wide — the cool column’s width, half the line’s — entering with its own coefficient. The ratio of that coefficient to the line’s height, sign reversed, is the notch’s depth as a share of the line. Four numbers are solved by least squares from seven entries at each trial, and the notch’s standard error follows from the residual and the inverse of the design, with three degrees of freedom left over.

The model is not the lamp’s true shape. A reversed line is the emitted profile multiplied by the column’s transmission, which is exponential in the column’s depth; subtracting a Gaussian is the first-order version of that, right when the column is thin and progressively wrong as it thickens. The test of the fit is not whether it reproduces the lamp exactly but whether it finds a notch where there is one and not where there is none.

Why two nanometres cannot tell a notch from a spike #

Through a slit two nanometres wide, a line 1.2 nanometres across and a notch 0.6 across both arrive as nearly the same shape: the slit’s own triangle, slightly rounded. Their blurred profiles differ only in the last few per cent of their shoulders, and the seven entries, two nanometres apart, sample those shoulders at three points on each side. To the fit, the notch regressor is very nearly the line regressor with a different height. Adding it gives the fit a way to move the effective width of its Gaussian without changing the width parameter, and the residual is smallest when it uses that freedom to make the top slightly sharper — a negative notch — while the width parameter grows to nearly two nanometres, wider than the line.

That is not a property of the noise; it is visible on the clean table. It is the same trade one slit, two requirements priced for an instrument: a wide slit is kind to a line and blind to a notch, and the table cannot be read for what the slit removed. And it means a fitted notch depth on a two-nanometre table carries no information about reversal in either direction. The census’s plain fit already showed the symptom from the other side: its least miss at two nanometres stays between 0.013 and 0.033 per cent at every depth, so close to perfect that no threshold could refuse it.

The readings two standard errors call #

On the two-nanometre table the rule calls three of twenty readings reversed at 0.1 per cent noise when nothing is reversed, and the same three at depths of a half and one; four from one and a half to three, six at four, ten at five. At 0.2 per cent noise it calls six of twenty at every depth from nought to four. The count does not grow with the column until the column is very deep, and the readings called are the same ones — the same noise draws — at every depth. What decides whether a reading is called is its noise, not its notch.

The called readings make the point plainly. Every one of them fits a notch of depth almost exactly one — a notch as tall as the line, which subtracts the whole Gaussian and leaves only the base. In those draws the noise happens to make the seven entries fit a nearly flat base a little better than any Gaussian, and the fit finds the degenerate solution in which line and notch cancel; the cancellation makes the notch’s coefficient large and well-determined, so its standard error is small and the rule fires. The uncalled readings scatter well below nought — spikes again, as on the clean table — rising towards nought as the column deepens. Only at depths of four and above does a reading occasionally fit a positive notch short of one, and those few are as likely to be noise.

So the prediction’s consequence holds in its strongest form: a two-nanometre table cannot reveal a reversed lamp by this fit, or by any fit that adds a notch to the line, because the slit has made the two shapes the same to seven entries. The declaration’s safety on a two-nanometre table rests on knowing the lamp from somewhere other than its table.

One nanometre reads the notch #

A narrow slit shrinks the error, not the bound found a one-nanometre slit buying less than expected for a colour engine’s bound. For reading a notch it buys everything. Through a one-nanometre slit the line and its notch arrive as different shapes — the line’s shoulders lie outside the slit’s triangle, the notch’s inside — and the seven entries, a nanometre apart, span the line with points to spare. The fitted depth becomes a measurement. On the clean table it is −0.02 when nothing is reversed and climbs at every deeper column. With noise each depth gives a tight cloud: at 0.2 per cent the unreversed readings scatter between −0.14 and +0.13; at a depth of a half, between 0.12 and 0.32; at one, between 0.29 and 0.47; and from there the clouds rise and narrow towards one.

Every noisy reading is called reversed by the two-error rule from a depth of a half — sooner than the prediction’s two. But the rule also calls seven of the twenty unreversed readings at 0.1 per cent noise and nine at 0.2: a third to nearly a half of lines with no notch at all.

Why two standard errors are too few #

A standard error estimated from a fit’s own residual is honest when the residual is mostly noise and there are enough of them to estimate it. Here there are seven entries and four fitted numbers, three degrees of freedom. With three degrees of freedom the estimated noise level itself varies by a factor of two or three from reading to reading, and a reading whose residual happens to be small has a small standard error on every coefficient, including a notch of nought point nought something. Two standard errors under those conditions is a far weaker test than it sounds: the distribution of the ratio has heavy tails, and the fit is also choosing its width and centre to minimise that same residual, which shrinks it further.

The ratio on the unreversed readings at 0.2 per cent noise runs from −99 to +22. A threshold high enough to exclude them would also discard the half-depth readings in which the notch is plainly there. The rule that works is on the depth, not on the ratio.

A rule on the depth #

Calling a line reversed when its fitted notch passes a fifth of the line’s height calls none of the twenty unreversed readings at either noise, eleven of twenty at a depth of a half, and all twenty at every depth from one. A fifth is not tuned to the noise: the unreversed readings’ largest fitted depth is 0.05 at 0.1 per cent noise and 0.13 at 0.2, and the reversed readings’ smallest from a depth of one is 0.33 and 0.29. The gap between the two clouds is wide enough that any threshold between about 0.14 and 0.28 does the same job.

That rule matters for the declaration. Behind the same cool column, the one-nanometre table with the plain fit declares from the yellow at depths of a half, one and one and a half and its ceiling falls short of the true peak at all three — by one per cent at a half, eight per cent at one and a half — before it refuses the yellow outright from a depth of two. The depth rule calls the yellow reversed in every reading wherever the ceiling falls short by more than one per cent, and in half the readings at the one depth where it falls short by one. On a one-nanometre table a reader with a notch in the fit would decline to declare exactly where declaring was unsafe.

The plain fit, set beside it #

The one-nanometre table’s plain fit already notices the notch, in its own way, much as a table can refuse a pair it cannot resolve found narrow tables refusing a split yellow they could not read as one line: its miss climbs from 0.048 per cent on the unreversed table to 0.11 at a half, 0.23 at one, 0.71 at two and 3.8 at five. At half a per cent it refuses the yellow from a depth of two; at a tenth of a per cent it would refuse from a half. The two-nanometre table’s plain fit never moves from a few hundredths of a per cent.

The two readers of the one-nanometre table do different jobs. The plain fit can refuse, which says that the line is not a Gaussian; the notch fit measures, which says how deep the notch is. A refusal gives no number to correct the ceiling with, and the earlier essays found that a refused yellow leaves the table declaring nothing at all. A fitted depth of 0.6 at a column depth of two is the start of a correction: the reversed line’s true peak, ahead of the notch, is a computable amount above the entries. Whether that correction can be made safely is a separate question, and a declared width buys a factor of two is a reminder of what is at stake: a declared number is what lets a colour engine bound its own error at all.

How the tables were read #

The lamp is the collection’s mercury tube with every line behind its own absorbing column, the 546 nm green line’s optical depth at the column’s centre set from nought to five in steps of a half and every other line’s in proportion to its strength, so the yellow’s own depth is 0.46 of the green’s. The column is a Gaussian 0.6 nanometres wide, half the lines’ 1.2. Each table is the lamp through a triangular slit whose half-width equals the step, integrated at a tenth of a nanometre. At the yellow peak both fits use the seven entries centred on it, every width the census scans from 0.6 to 3 nanometres in one per cent steps, and centres within a slit of the peak in fortieths of a slit. Noise is relative, Gaussian and independent at each entry, twenty seeded readings at each depth and level.

What the census does not reach #

The notch width is given. The fit ties the notch to the column’s true width, 0.6 nanometres. A reader of a real table does not know it, and a free notch width adds a fifth number to seven entries.

One line. The yellow is the line that sets the declaration, so it is the line that matters; the blue and green, more deeply reversed, are refused by the plain fit at one nanometre already.

A first-order notch. The subtracted Gaussian is the thin-column approximation, and its fitted depth is a share of the fitted line, not the column’s optical depth. The mapping from one to the other is monotone here but not computed.

Still open: whether a fitted depth can repair the ceiling #

On a one-nanometre table the notch fit returns the yellow’s notch as a depth that climbs with the column. A notched line’s true peak sits above what the entries show by an amount set by that depth, and a ceiling raised by it might stay safe where the plain declaration fails.

The calculation is the one-nanometre table behind the cool column at every depth, the yellow fitted with its notch, the ceiling built from the fitted line’s height and width — the notch taken back out — and compared with the reversed lamp’s true peak in every window, clean and in the twenty noisy readings. The prediction is that the repaired ceiling stays above the true peak at every depth to three, because the fitted line underneath the notch is wider and taller than the plain fit’s, and that past three the first-order notch underestimates a thick column’s trough and the repair falls short again. If it holds to five, a one-nanometre table carries enough to declare safely behind a reversed lamp, and only two-nanometre tables need the lamp declared from outside.

A fit’s spare number goes somewhere #

The habit here is about what a fit does with a parameter the data cannot pin down.

Adding a notch to the fit was meant to give it a way to see one more thing. On the two-nanometre table it gave it a way to do something else: reshape the Gaussian it already had, by an amount and in a direction that minimised the residual and had nothing to do with reversal. The notch coefficient was not left undetermined in the sense of being large and uncertain; it was determined, precisely, as the answer to a different question. And on the noisy readings, where noise sometimes made a flat base fit best, it was determined as the answer to a third.

The failure mode is to read a fitted parameter by its name rather than by what the data can distinguish it from. The test is to fit a case where the true value is known to be nought — here, the unreversed table — and see what the fit returns. On the one-nanometre table it returned −0.02, and the parameter meant what it was called. On the two-nanometre table it returned −4.2, and it did not.