A wide table needs no notch to borrow

A lamp’s spectrum is published as a table, each entry the lamp seen through a slit as wide as the table’s 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. A declared width buys a factor of two first made that promise, and the essays since have tested it on the hardest lamp the census holds: a tube whose every line is notched at its centre by a cooler column of the same atoms, so that each line’s peak is two shoulders with a dip between them.

A ceiling that knows the notch holds to the deepest column built the one-nanometre table’s ceiling for lines carrying the notch its own fit read, and it held at every column depth to five. The two-nanometre table cannot read a notch at all, and the notch fit declares the true width, and the true width runs out found its ceiling for an unnotched line failing from a column of three and a half. A coarse table can borrow a notch, but not a phantom gave it the fine table’s notch, scaled to every line by strength, and the borrowed notch held the two-nanometre ceiling at every depth, with a wider margin than the fine table’s own. One noisy reading’s phantom notch, read where there was no column, failed it, and borrowing only the notches deep enough for the depth rule to call them kept every reading safe.

Two nanometres is not the coarsest table a lamp is published in; five is common. The two-nanometre essay found a shallow notch lifting the table’s flanking-pair factor — the share of a line’s peak the pair of entries either side of it must together carry — by about a tenth, because two wide entries see a slightly flattened top better than a sharp one. The prediction was that this 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. The alternative was that a called notch lowers a five-nanometre ceiling below the true peak, in which case a coarse table should borrow the width and decline the notch.

Wide tables were never at risk #

Through slits of three, four and five nanometres, a shallow notch lifts the flanking pair’s factor by up to 15.2, 17.7 and 19.6 per cent, against 10.4 at two, and every table’s ceiling on the borrowed notch stays above the reversed lamp’s true peak at every column depth to five. But the same tables’ ceilings built for an unnotched line also stay above it, at every depth, by 4.7 per cent at least, where the two-nanometre table’s fails from three and a half. Borrowing the notch lowers every wide table’s margin at every column depth from a half to three — at five nanometres from as much as 26 per cent to under 7.

• The prediction holds on both counts. The pair’s rise reaches a fifth at five nanometres, and the borrowed ceiling holds.
• Its premise does not. A wide table did not need rescuing; only the two-nanometre table did.
• Borrowing is not free. On a wide table it spends margin the table had.

Drawn with no notch at all, the four tables’ ceilings part at once. The two-nanometre ceiling falls through the true peak at a column of three and a half; the three-, four- and five-nanometre ceilings rise with the first column, peak somewhere between one and a half and three, and fall gently, and none comes within 4.7 per cent of the peak at any depth. The five-nanometre table, the one the prediction worried about most, is the safest of the four at every column from three on.

That is not what a reader of the earlier essays would have guessed. Each essay found coarser tables harder: a two-nanometre table could not read the notch that a one-nanometre table could, and its ceiling failed where the finer one held. The natural extrapolation is that a five-nanometre table is worse again. It is better.

Why a wide entry is a safe entry #

A ceiling is built by dividing a table’s entries by a factor: the least share of a line’s true peak an entry can carry, however the line falls against the slit. That share falls steeply with the slit — from 0.273 at two nanometres to 0.199 at three, 0.155 at four and 0.125 at five, because a wider slit averages a 1.2-nanometre line over more empty spectrum. Dividing by a smaller number gives a higher ceiling, so a wide table’s ceiling sits, by construction, well above any line it could hold.

A notch matters to a ceiling only through what it does to those factors. At two nanometres the unnotched factor is large enough — the entry carries a quarter of the peak — that a notch which lowers the peak’s shoulders and spreads them can leave the entry carrying less than the factor assumed, and the ceiling falls short. At five nanometres the factor is already an eighth, small enough to cover anything a notch does to the line’s top; the slit sees a notched line and an unnotched one of the same area almost alike.

The pair’s rise, as predicted #

The prediction’s mechanism holds exactly. A shallow notch lifts the pair factor by 10.4 per cent at two nanometres, 15.2 at three, 17.7 at four and 19.6 at five, each rise peaking near half an optical depth and falling below the unnotched value by a depth of two. A notch half an optical depth deep flattens a line’s top without splitting it, and two wide entries either side of a flat top carry more of it between them than they would of a sharp one.

A higher factor means a lower ceiling, and a lower ceiling is a tighter bound — closer to the true peak, more useful, and less safe. That is how the borrowed notch costs the wide tables their margin: a column a half to three deep scales to notches near the depths where the factors rise most, the ceilings come down towards the peak, and the margin the wide table had without the notch is spent on precision it did not ask for.

On the borrowed notch #

Built on the borrowed notch, every table holds. The two-nanometre ceiling stands between 8.4 and 42.4 per cent above the true peak, the three-nanometre one between 8.6 and 37.3, the four-nanometre one between 7.9 and 31.0, and the five-nanometre one between 4.5 and 19.1 — the last within five per cent of the peak from a column of a half to two and a half. No called or uncalled notch the clean census reads brings any table below the peak.

The five-nanometre ceiling is the tightest of the four on the borrowed notch, where without a notch it was the loosest at every column from three. A five-nanometre table built for the fine table’s notch is the most precise bound the census has made of a reversed line’s peak from a coarse table — and it gets there by spending a safety margin the table already had, which is a trade a table maker should choose rather than fall into.

What borrowing buys and costs #

Subtracting the two ceilings shows where borrowing helps. At two nanometres it costs margin at shallow columns and adds it from a column of one and a half, and the added margin is what keeps the ceiling above the peak from three and a half on. At three to five nanometres it costs margin at every column from a half to three — up to 20 points at five nanometres — and gives a little back only at the deepest columns, where the column scales to notches so deep that the factors fall again.

So the rule the prediction offered as a fallback — borrow the width and decline the notch — is the right one for every table three nanometres wide or more, though not for the reason it gave. A called notch never brings a wide ceiling below the peak. It brings it closer, and the closeness is bought at the price of trusting a notch read by a different instrument.

The phantom notch, slit by slit #

The two-nanometre essay’s one failure was a phantom: at no column at all, with two thousandths of noise, the fine fit read a notch 0.133 deep, and the coarse ceiling built on it fell 0.7 per cent below the peak. Borrowed by every table, the same phantom fails the four-nanometre table too, by 0.05 per cent, and spares the three- and five-nanometre tables, whose lowest ceilings over all 440 readings stand at 1.039 and 1.010. With only the notches the depth rule calls, every reading leaves every table safe — the three-nanometre table’s worst at 1.059, the five-nanometre table’s at 1.012.

The four-nanometre failure is a reminder that a factor’s rise is not monotone in everything at once. A notch of 0.133 sits where the pair factor is rising at every slit, and whether that rise brings a given table below the peak depends on where that table’s entries happen to fall against the lines — four nanometres puts one of them where three and five do not. The depth rule removes the question by never borrowing a notch that shallow.

Where the run of slits has been before #

This is not the first time a wider slit has turned out kinder than expected in this collection. A narrow slit shrinks the error, not the bound found the same asymmetry from the other side: narrowing a table’s slit from five nanometres to one cut its colour error twenty-sevenfold and its declared bound only sevenfold, because the bound is built to cover the worst line the slit could hold and a narrow slit can hold sharper ones. A bound and the quantity it bounds respond to the slit differently, and the bound is the slower of the two.

The reversed-line essays met the same pattern in what a table can read. A reversed line is read as one line found a two-nanometre table reading a deeply reversed line as a single Gaussian about the width of its flat top — safely — where the one-nanometre table refused it. A notch in a line is never refused found the two-nanometre table never refusing a narrow notch at all, to a depth of five, because its slit cannot see the notch to object to it. Two nanometres hide the yellow’s notch from any fit then found that blindness costing the ceiling once the column was deep enough. Five nanometres is blinder still, and its ceiling, built with an entry’s share of the peak small enough to cover a notch it cannot see, is not hurt by the blindness at all.

What the five essays share is the distinction this one makes sharp: a table’s resolution and a table’s promise are separate things. A fine table sees a line’s shape and must therefore promise something about it; a coarse table sees only the line’s area, promises less, and is held to less. The two-nanometre table sits at the awkward point between — fine enough to make a promise that a notch can break, too coarse to see the notch that breaks it.

What a table maker should take from this #

At three nanometres and wider, build the ceiling on the line’s width and leave the notch out. It holds at every column depth the census reaches, by at least 4.7 per cent, and it does not depend on a second instrument’s reading.

At two nanometres, borrow the notch, and only when the depth rule calls it. That is the one table here the notch rescues, and the rule keeps the phantom out.

If a tight bound matters more than a safe one, a wide table can borrow too — the five-nanometre ceiling on a called notch is the closest bound in this census — but the margin it gives up is real, down to four and a half per cent.

How the tables were read #

The lamp is the census’s reversed tube: each of its lines 1.2 nanometres wide at its core, notched by a column of the same atoms at a third of its temperature, the green line’s column at an optical depth from nought to five and every other line’s in proportion to its strength. A table is the lamp through a triangular slit of half-width w, sampled every w nanometres and integrated at a tenth of a nanometre — the two-nanometre table exactly the earlier census’s. Each window’s ceiling is the larger of its neighbours divided by the single-entry factor, or the larger neighbouring pair divided by the pair factor, whichever is less, with the factors computed for the line nearest the window carrying the yellow’s notch scaled by strength: no notch, the true notch, or the notch the one-nanometre fit read, its depth d borrowed as an optical depth −ln(1 − d). Each ceiling is compared with the lamp’s true peak in its window. The noisy readings are the one-nanometre census’s twenty at each of eleven depths and two noise levels, borrowed always or only when deeper than the depth rule’s 0.2.

What this does not reach #

One tube. The census’s reversed tube has its lines where they are; a lamp whose lines sit closer together could put two lines under one wide entry, which the ceiling’s single-line factors do not cover.

One core width. Every factor is for a 1.2-nanometre line. A broader line fills a wide slit more, raises its factors, and could bring a five-nanometre ceiling down to where a notch matters again.

Noise in the coarse table. The noisy readings are the fine table’s; the coarse tables themselves are read clean.

Still open: whether a broad line brings the notch back #

A wide table is safe because its entries carry a small share of a narrow line’s peak; a line broad enough to fill the slit raises that share, and with it the chance that a notch matters. The census’s lines are 1.2 nanometres wide at their cores, and a high-pressure lamp’s lines can be several times that.

The calculation is the reversed tube with its line cores widened to 2.4 and 3.6 nanometres, read through slits of two to five, the unnotched ceiling and the ceiling on the true notch read against the true peak at every column depth. The prediction is that at a core of 3.6 nanometres the five-nanometre table’s unnotched ceiling fails from a column of about three, as the two-nanometre table’s did for the narrow core — that what matters is the ratio of slit to core, and a table needs the notch wherever that ratio falls under about two. If the five-nanometre ceiling still holds without a notch at every core, its safety comes from averaging over empty spectrum between lines, and the ratio is not the rule. A broad line brings the notch back to a wide table makes that calculation.

The coarse instrument was the safe one #

The habit here is about extrapolating a trend in an instrument’s resolution.

Every earlier essay in this run found the coarser table doing worse: it could not read the notch, its ceiling failed where the finer one held. The prediction followed the trend and asked only how much worse five nanometres would be. The trend held for what each table could see — a five-nanometre table sees less of a notch than any — and reversed for what each table could promise, because a coarse entry’s promise is weak enough to be safe.

The failure mode is to read a trend in an instrument’s resolving power as a trend in the reliability of what it reports. A coarser instrument resolves less and, for the same reason, often claims less, and a bound that claims less is harder to break. The test is to compute the claim at each resolution, not the resolution alone — here, the unnotched ceiling at every slit — before deciding which instrument needs help.