A coarse table can borrow a notch, but not a phantom

A lamp’s spectrum is published as a table, each entry the lamp seen through a slit as wide as the step, and a table can promise a ceiling — an upper bound on how high a line’s true peak rises between its entries — from the width of the narrowest line it holds, the promise a declared width buys a factor of two first made. For a reversed tube, every line notched by a cooler column of the same atoms, the promise needs more than a width. A ceiling that knows the notch holds to the deepest column built each window’s ceiling for a line carrying the notch the one-nanometre table’s own fit read, and found it safe at every column depth to five, clean and in 440 noisy readings.

The two-nanometre table has no such fit to call on. Two nanometres hide the yellow’s notch from any fit found its slit leaving too few entries across the yellow line for any fit to place a notch, and the notch fit declares the true width, and the true width runs out found its ceiling, built for an unnotched 1.2 nm line, failing from a column of three and a half. A coarse table is not exotic: many published lamp spectra are tabulated every two or five nanometres, and the same lamp is often measured finely somewhere else, by its maker or a laboratory.

The proposal was to carry the notch across — to build the coarse table’s ceiling for lines carrying the notch the fine table read. The prediction was that the borrowed notch holds the two-nanometre ceiling to five, as the fine table’s own does, with a wider margin, because a broader slit hides more of a notch and its factors fall further. The alternative was that the coarse ceiling depends on more than the line’s shape — on where its few entries fall against the notch — so that a borrowed notch is not enough.

The borrowed notch holds, and one phantom does not #

Built for the notch the one-nanometre fit reads, the two-nanometre table’s ceiling stays above the reversed lamp’s true peak in every window at every column depth from nought to five, its lowest window between 8.4 and 42.4 per cent above the peak; built for the true notch, between 8.4 and 25.3; with no notch it fails from three and a half. From a column of one, its margin is wider than the one-nanometre table’s at every depth. Borrowed from each of the fine table’s 440 noisy readings, the notch keeps 439 coarse ceilings safe; the 440th, at no column at all, takes a notch of 0.133 the fit read in noise and sets the ceiling at 0.993 of the true peak. Under the two-nanometre slit the flanking pair’s factor rises with a shallow notch, by ten per cent at an optical depth of 0.4, before falling. Borrowing only notches deeper than 0.2 — the depth rule an earlier essay found separating reversed lines from unreversed ones — keeps every one of the 440 safe; borrowing those more than twice their standard error lets the phantom through.

• The prediction holds on the clean table. Safe to five, with the wider margin it expected.
• The alternative is wrong, and something else is right. Where the entries fall does not break the borrowed ceiling; a notch that is not there does.
• A coarse table needs a stricter notch than a fine one. The same phantom left the fine table’s own ceiling safe.

Two things in the picture need explaining, and both come from the same place. At shallow columns, half to one, the coarse ceiling with no notch stands well above the one with the notch — 1.20 and 1.23 against 1.09 and 1.15 — so at those depths knowing the notch makes the coarse ceiling tighter, not looser. And at a column of five the borrowed ceiling climbs to 1.42 while the true-notch one turns down, the fine fit’s overstatement of a deep notch carried across.

What a notch does under a wide slit #

A ceiling divides a table’s entries by the least share of a line’s true value they can report — for a single entry, and for the two entries flanking a wavelength summed — and takes the smaller result. Under a one-nanometre slit both shares fall as a notch deepens: a notched line has two tops, and the entries between them see the trough. Under a two-nanometre slit the pair’s share first rises, from 0.567 with no notch to 0.627 at an optical depth of 0.4, and only then falls, below its unnotched value between depths of one and two.

The reason is the slit’s width against the line’s. A two-nanometre slit averages over more than the 1.2 nm line, and its weakest moment — the pair’s least share — is when the line sits between two entries with its narrow top falling where neither sees it fully. A shallow notch lowers that top and widens the line’s shoulders a little; the two wide entries, which saw only the shoulders anyway, now see a larger share of what is there. So the coarse ceiling for a shallow notch divides by a larger share, and is lower, than the ceiling for no notch. Only a deep notch, whose trough the entries cannot avoid, lowers the share and raises the ceiling as on the fine table.

That is correct arithmetic, and it is what makes the coarse table sensitive. A notch the ceiling assumes but the lamp lacks lowers the coarse ceiling for nothing, and if it lowers it far enough it lowers it below the true peak.

A notch read in noise #

The case that fails is the one with nothing to find. With no column in front of the lamp the fine fit should read no notch, and on the clean table it reads −0.02. In twenty readings carrying 0.2 per cent noise it reads between −0.14 and +0.13: noise in seven entries is enough to put a small notch in or take one out. Borrowed as read, a negative notch is no notch and changes nothing. A positive one lowers the coarse ceiling in proportion, and the deepest, 0.133, lowers it to 0.993 of the true peak — the one unsafe ceiling in 440.

The fine table survived the same reading. Its own ceiling on the phantom notch stayed above the true peak, because under its slit a shallow notch barely moves the pair’s share — up by under one per cent — while the single entry’s share falls, and the ceiling stays put. The phantom is harmless where it was read and harmful where it is carried, and that is a reason to treat a borrowed parameter more strictly than a native one.

Which notches to borrow #

The census already has two rules for calling a notch. Two nanometres hide the yellow’s notch from any fit called a line reversed when its notch exceeded twice its own standard error, and found that rule calling unreversed lines in noise; it also found a depth threshold of 0.2 separating reversed from unreversed readings cleanly. Borrowing only notches deeper than 0.2 keeps every one of the 440 coarse ceilings above the true peak, the lowest at 1.084 — the unnotched ceiling with no column. Borrowing those over twice their error does not: the phantom notch of 0.133 is twenty-one standard errors deep, because a fit on nearly noiseless entries reports a small error for whatever it finds.

The depth rule costs nothing at the depths where the notch matters. At a column of half, the yellow’s true notch reads near 0.22, and in some noisy readings it falls under 0.2 and is not borrowed; those ceilings revert to the unnotched rule, which at that depth is safe and looser. From a column of one every reading’s notch is past the threshold.

Window by window at the deepest column #

At the deepest column the coarse table’s failures are the fine table’s, at coarser grain: the unnotched ceiling falls below the true peak at the blue line, 0.91, and dips at the green. The borrowed notch lifts both windows well clear, and the lowest window moves to the weak violet line near 402 nm, at 1.42 — the same migration the fine table showed, for the same reason: the violet line’s notch is the shallowest in the tube, and its window is the one the notch helps least.

The coarser table pays in margin #

The prediction’s second half — a wider margin — holds at every depth from one, and the size of the difference is the price of the coarse slit. With the same notch, the two-nanometre ceiling stands 15 per cent above the true peak at a column of one where the one-nanometre ceiling stands 3, and 42 against 29 at five. A ceiling is a promise about what can hide between entries, and entries two nanometres apart leave more room for a 1.2 nm line to hide than entries one apart. The coarse factors are smaller at every notch depth past the shallow ones, so the coarse ceiling divides by less and sits higher.

That margin is not waste in the way the fine table’s extra margin at depth was. There, the fitted notch’s overstatement added caution the true notch did not need. Here, most of the margin is there with the true notch too — 25 per cent at its widest — because it is the slit’s, not the fit’s. A table maker choosing between publishing a lamp every nanometre and every two is choosing, among other things, how loose a promise about its peaks the table can make, and for a reversed lamp the coarse promise is half again as loose.

This is the family of trade-offs the census has measured before. A table can refuse a pair it cannot resolve found a table declining to read a width where no single line fits its entries — a refusal it can make only where its slit resolves the misfit — and a reversed line is read as one line found the reversed line taken for a single, wider line by tables at both slits. In each case the coarse table had less to go on and either promised less or promised wrongly. A borrowed notch lets the coarse table make the fine table’s kind of promise, at the coarse table’s price: it does not learn to see the notch, it is told about it, and the telling has to be checked.

What a table maker should take from this #

A coarse table can bound a reversed lamp if someone else reads the notch. With the notch from a one-nanometre measurement of the same lamp, the two-nanometre ceiling held at every column the census runs, with a margin of a twelfth or more.

Carry a parameter across only when it is called. A notch in a line is never refused found a two-nanometre table never refusing a line notched by a cool column, however deep the notch — the coarse table cannot tell a notch is there. The borrowing rule faces the mirror problem: it must refuse the notches that are not there, and the coarse table cannot tell that either. A notch the fine fit reads but would not call reversed is noise, and the coarse table’s arithmetic is sensitive to exactly that kind of shallow notch. The depth rule is the right gate; the standard-error rule, which measures how sure the fit is of what it read rather than whether what it read is large enough to matter, is not.

The direction of a correction depends on the instrument. On a fine table a notch always raises the ceiling; on a coarse one a shallow notch lowers it. The same borrowed number pushes two tables’ promises in opposite directions, and a rule that is safe for one is not automatically safe for the other.

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 two-nanometre table is that lamp through a triangular slit of half-width two nanometres, sampled every two. The factors are found as the fine table’s were — the line placed at every offset within reach, in steps of a twelfth of the smaller of its width and the slit, and the least share any entry or flanking pair reports of its true value taken — for the 1.2 nm Gaussian times exp(−t·n), now under the wider slit; with no notch they are the two-nanometre ceiling’s own factors exactly. The yellow notch’s optical depth comes from the truth or from the one-nanometre census’s fit read as −ln(1 − d), scaled to every line by strength, and a borrowed notch not called by the chosen rule is taken as none.

What this does not reach #

The same lamp twice. The borrowed notch comes from a fine table of exactly the lamp the coarse table holds. A notch read on one tube and carried to another of the same model would carry the two tubes’ differences too.

One slit ratio. The coarse slit is twice the fine one. A five-nanometre table, as many published spectra are, averages over four times the line’s width, and how its pair factor answers a shallow notch is not computed.

Strengths known. As before, the yellow’s notch is scaled to the other lines by their true strengths.

Still open: whether a five-nanometre table can borrow at all #

A five-nanometre slit is four times the line’s core, and its entries see a reversed line as little more than its area. If the pair factor’s rise with a shallow notch grows with the slit, a five-nanometre table’s ceiling for a borrowed notch may be lower than its unnotched ceiling at every depth the depth rule admits — and so, unlike the two-nanometre table’s, made less safe by the very notch that should protect it.

The calculation is the ceiling’s factors under slits of three, four and five nanometres at every optical depth, and each coarse table’s ceiling on the borrowed notch, called by the depth rule, against the true peak at every column depth. The prediction is that the pair factor’s rise grows with the slit, to a fifth at five nanometres, and that the five-nanometre ceiling on a borrowed notch still holds to five because the depth rule admits only notches deep enough to lower the factors again; if a called notch lowers a five-nanometre ceiling below the true peak, a coarse table should borrow the width and decline the notch. A wide table needs no notch to borrow makes that calculation.

A borrowed number obeys its new home #

The habit here is about carrying a measured parameter from the instrument that measured it to one that could not.

The notch was read correctly where it was read, and the coarse table’s arithmetic was correct for the notch it was given. The failure was in the passage: a number whose noise was harmless to the fine table’s ceiling meets an instrument whose response to that number runs the other way at small values.

The failure mode is to assume a parameter means the same thing to every calculation that uses it. A notch depth of 0.13 told the fine table to raise its ceiling slightly and the coarse table to lower its ceiling slightly, and only the second was dangerous. The test is to feed the borrowing calculation the parameter’s whole noisy distribution, not its clean value — here, every one of 440 readings — and to look for the reading that the new instrument cannot afford.