A narrow slit shrinks the error, not the bound

A declared width buys a factor of two gave a colour engine one extra number to bound its own error with. Two spectral tables, a lamp’s and a sample’s, each blurred through a five-nanometre slit and multiplied together, lose the covariance of the two factors inside every slit window, and Cauchy–Schwarz bounds a covariance by two variances. The lamp’s variance can be bounded if the lamp declares the width of its narrowest feature; the sample’s is estimated from its own table. Under a fluorescent tube, declaring the mercury lines’ true width of 1.2 nanometres brought the bound from 196 times the error to 86. It never failed at a true declaration. It was also nowhere near usable.

That essay found why. A narrow feature is not one narrow feature. Nothing in a declaration of 1.2 nanometres says the lamp has only one line inside a ten-nanometre window, and a comb of such lines approaches the two-point lamp that the loosest bound was already allowing for. So it proposed the repair from the other end: stop asking the lamp to be broad and ask the instrument to be narrow. A slit comparable to the declared width leaves no room for a comb. Its prediction was specific — that the looseness would fall once the declared width passed about half the slit, and that at a one-nanometre slit the declared-width bound would approach the bound computed from the true variances, “without anybody measuring a fine spectrum.”

The comb does go. The bound does not follow it.

The slit and the grid have to move together #

One thing has to be settled before any census is run, and it is the thing the slit is what makes it legal established for a single table. A triangular slit whose half-width equals the grid’s step makes a table that integrates the spectrum: the windows tile the wavelength axis, every nanometre of the lamp is counted once, and a line anywhere is seen by the two grid points either side of it. A slit narrower than the step does not. Its windows leave gaps, a laser line that falls between two grid points is not seen at all, and a census run that way reports numbers about a table that has thrown the lamp away.

So “a one-nanometre slit” here means a one-nanometre slit on a one-nanometre grid, and the census is run at four such arrangements: one, two, three and five nanometres, each slit on a grid of its own width. The five-nanometre one is the arrangement the two earlier essays used. Everything else is held: the same sixty-eight notched samples under the tube, notches five, eight, twelve and twenty nanometres wide centred every two nanometres across sixteen either side of the green mercury line, and the same sixty-eight under a three-laser projector; each lamp declaring the width its lines truly have, 1.2 nanometres for the tube and 0.2 for the lasers.

Smaller at every slit, looser at every slit #

Narrowing slit and grid from five nanometres to one takes the tube’s median error from 0.21 to 0.008, a factor of twenty-seven, and the declared-width bound from 15.6 to 2.3, a factor of seven. The bound therefore gets looser at every step — ×86, ×115, ×186, ×253 — while the true-variance bound stays between ×12 and ×15. At one nanometre no tube notch is certified within one colour difference, although every one of them is within a quarter.

• The prediction is inverted. The ratio the proposal expected to fall rises threefold.
• The bound is still a bound. It never falls below the error at any slit, on either lamp.
• What carries the looseness is the lamp’s half, and within it the peak: the tallest mercury line’s height, charged to every window of the spectrum.
• Under the projector nothing changes: its declared bound lies on its true-variance bound at every slit, as it did at five.
• The estimate from the tables alone, which failed on the line at five nanometres, fails on nothing at two or one. At a slit that narrow the declaration stops being needed for validity.

The tube’s curve runs the wrong way from left to right: tightest at five nanometres, loosest at one. The dotted line under it, the bound with both variances known, is flat — Cauchy–Schwarz itself is no looser at a narrow slit than at a wide one. So the growing gap between the two lines is all declaration. The projector’s two lines lie on top of each other at every slit and both rise a little towards the narrow end, for a reason that is the lasers’ and not the bound’s: a 0.2-nanometre line sits in one or two windows of a narrow grid and the notches mostly miss it, so the few errors that remain are sparse and the median is taken over near-zeros.

Why the ratio was the wrong thing to watch #

The ratio was the proposal’s measure, and it is a reasonable one when the error is roughly fixed and the bound is what moves. Here both move, and the error moves much further.

The dashed lines are the error the separation makes, and they fall steeply: under the tube from 0.21 at five nanometres to 0.08 at three, 0.028 at two and 0.008 at one. That is the result a second slit buys a quarter would have predicted if it had been allowed to narrow both slits and the grid. A narrow window holds little structure of either factor, and a covariance can only be as large as the structure it is the covariance of. The largest tube error in the census, 2.58 at five nanometres, is 0.21 at one.

The solid lines are the bound, and they fall too, but at a quarter of the rate on a logarithmic scale. The tube’s bound goes from 15.6 to 2.3; the projector’s from 47 to 4.1. Both stay above one colour difference at every slit. So the absolute statement a colour engine can make does improve, sevenfold, and it is still not a statement anyone could use as a tolerance. An engine at a one-nanometre slit can say its error is under 2.3 in the median case and under 3.7 in the worst. The error is under 0.21 in the worst.

The gap between the lines is what a user would pay for trusting the bound instead of the instrument. At five nanometres it is a factor of seventy-four between the median bound and the median error; at one nanometre, three hundred.

The comb goes, and the variance stays #

The prediction’s mechanism was the comb, and the comb really does disappear.

At five nanometres the lamp of largest variance consistent with the declaration is a comb, features spaced 3.9 nanometres apart, each reaching the declared peak. At two nanometres there is room for only one feature and the worst lamp is a single hump; at one nanometre it is still a single hump, pushed to the edge of the window where the triangular weighting lets its tail count against a nearly dark remainder. Nothing in the geometry prevents that: a feature 1.2 nanometres wide can sit anywhere, and a window one nanometre across is narrow enough that one feature and nothing is the whole of what it holds.

The variance those lamps carry is 1.11, 1.14 and 0.74 of a mean of one. The comb was the worst case at five nanometres, and the single hump nearly matches it at every narrower slit. So removing the comb removes almost nothing from the bound. The proposal pictured the declaration’s looseness as the extra variance of many features packed where the lamp really has one. It was the variance of any feature at all, placed where one could be.

That turns the question from the shape of the worst lamp to its height, and there the answer is sitting in the declaration.

Which half is loose #

A Cauchy–Schwarz bound on a covariance is a product of two standard deviations, the lamp’s and the sample’s, and a looseness can live in either. The census can separate them by mixing the true and estimated halves.

With the lamp’s true variance and the sample’s estimated from its table, the bound sits at ×13 to ×16 at every slit — within a fifth of the bound with both variances true. The sample’s estimate is doing its job: a notch is a smooth, broad feature in reflectance, a table’s local slope and curvature describe it well, and at a narrow slit they describe it better.

With the lamp declared and the sample’s true variance, the bound is ×93 at five nanometres and ×254 at one — within a few per cent of the declared-width bound itself. Knowing the sample perfectly buys nothing. Every bit of the looseness is in the lamp’s declared half.

And inside the lamp’s half, most of it is the peak. A declaration is three numbers: the table’s own value in the window, the width of the narrowest feature, and the lamp’s peak. The peak is the lamp’s highest point anywhere — for the tube, the top of the green mercury line — and the bound charges it to every window. In a window of phosphor continuum four hundred nanometres from any line, it still has to allow for a 1.2-nanometre feature reaching the height of that line. Replacing the lamp’s peak with each window’s own highest point, which only a fine spectrum can supply, takes the bound at one nanometre from ×253 to ×81, and at five from ×86 to ×35.

That is the diagnosis. The declaration’s weak number was never the width. It is the peak, which is one number for a whole spectrum and has to be true of the worst window in it. A narrow slit makes a narrow window, and a narrow window’s mean is small beside the tube’s tallest line wherever there is no line — so the ratio of peak to mean that the bound depends on grows as the slit narrows, and the allowed variance grows with it relative to the true one.

What is left after the local peak, a factor of five or six over the true-variance bound, is the width doing what a width can: it allows a feature where the lamp has none, and a window whose continuum is flat is charged a hump it does not contain.

The tables become enough on their own #

There is a second result in the census, and it matters more to an instrument maker than the first.

The tables cannot bound what they discarded found that Cauchy–Schwarz with both variances estimated from the blurred tables is not a bound: at five nanometres it falls below the error on six tube notches and twenty-two projector ones, on the line, where the error is largest. At three nanometres it fails on one tube notch and nine projector ones; at two and at one, on none under either lamp.

The reason is the one a grid is not a resolution gave for a single table. A table built through a slit narrower than the lamp’s features resolves them: the tube’s 1.2-nanometre lines appear in a one-nanometre table as the peaks they are, and the table’s local slope and curvature then describe the lamp’s structure inside each window instead of averaging it away. The estimate that failed because a table cannot carry what was narrower than its window stops failing when nothing is narrower than its window.

So at a narrow enough slit the declaration is not needed for validity. The estimate from the tables is valid on this census, and it is tighter than the declared-width bound by a factor of twenty: ×12 against ×253 under the tube at one nanometre. The one thing the declaration still adds is a guarantee that does not rest on a census — and that guarantee is the loose one.

Every notch, twice #

The two clouds show both results at once. Moving from five nanometres to one, the tube’s notches move left by more than a decade, because the error falls, and down by less than one, because the bound falls less. The one-nanometre cloud sits between 1.3 and 3.7 on the bound axis whatever its error, from under a ten-thousandth of a colour difference to a fifth of one. A bound that assigns 2 to a notch whose error is 0.0002 and 3 to one whose error is 0.2 is not measuring the notch; it is measuring the lamp’s peak against the window.

No point lies below the horizontal line at one colour difference, so no notch is certified. At five nanometres the cloud is flat in the same way, a decade higher. A bound with no correlation to what it bounds was the declared-width essay’s complaint at a single slit, where the rank correlation was 0.27. At one nanometre it is 0.05.

What the instrument settles and the bound does not #

The proposal ended on an alternative, and it is the one the census returns. “If the looseness stays near two orders of magnitude at every slit a real instrument has, then no arrangement of declarations produces a usable error bar, and the honest output of a bandpass calculation remains the arrangement of the measurement.” The looseness is two orders of magnitude at one nanometre, and more than at five.

What a narrow slit does is make the error small, and it does that by itself. At a one-nanometre slit on a one-nanometre grid, the largest error the separation made on any of sixty-eight notches under a mercury tube was 0.21 of a colour difference, and the median was a hundredth of that. That is not a bound; it is a census of cases, as every number in these essays has been. But it is a census of the error itself, and an instrument maker can state it: tables at one nanometre through a one-nanometre slit lose under a quarter of a colour difference to separation on notched samples under line lamps. A reader of those two tables knows more from that sentence than from any bound the tables can compute.

And the bound’s weak number has a name now. A second slit buys a quarter asked whether a pair of slits pays; one slit, two requirements found why one slit cannot serve two factors. Neither could say what a declaration had to contain to be useful, and this census says it: not the narrowest feature, which a narrow slit makes nearly free, but where the lamp is tall — its peak, window by window.

How the census was run #

Each notch is a smooth reflectance of 0.72 with a Gaussian notch of depth 0.66, under either the site’s fluorescent tube (a triphosphor continuum with four mercury lines 1.2 nanometres wide) or its three-laser projector (lines 0.2 nanometres wide). For each slit width w, the grid runs from 380 to 780 nanometres in steps of w, and each table entry is the lamp or the sample averaged through a triangular window of half-width w centred on that grid point, integrated on a tenth-of-a-nanometre sub-grid. The error is the CIEDE2000 difference between the colour computed from the product of the two tables and the colour computed with each window’s covariance added back.

Each bound is a per-window standard deviation product summed against the colour-matching functions into an XYZ box around the computed colour, turned into a colour difference as the largest CIEDE2000 at the box’s corners. The lamp’s declared variance is the largest a comb of Gaussian features of the declared width can have inside the window, with the table’s value as its kernel mean and the declared peak as its ceiling, searched over sixteen spacings and six offsets; the local-peak version replaces the declared peak with the lamp’s highest value inside the window. The sample’s estimated variance is read from the local slope and curvature of its table. A notch counts as failing when a bound falls short of its error by more than a millionth of a colour difference: two projector windows far from every line report errors of a ten-millionth from rounding against bounds of exactly nought.

What this leaves out #

The lamps are constructed. The tube and the projector are this collection’s analytic lamps, with line widths chosen to be typical rather than measured. A real low-pressure mercury line is narrower than 1.2 nanometres; a real laser’s width depends on its type. Neither changes the diagnosis, which is about the peak rather than the width.

The declared-variance maximiser is a search over a family, combs of Gaussians, and not a proof over every lamp a declaration admits. A declared width buys a factor of two said so, and it applies here: a lamp that is not a sum of Gaussian features could, in principle, have a larger variance at the same declaration.

The box-to-colour-difference step takes the largest CIEDE2000 at eight corners. CIEDE2000 is not convex, so the largest value inside a box need not be at a corner. On this census no bound fell short of its error, but that is a statement about these notches, not about the conversion.

And the samples are notches. A notch is the sample shape that made separation expensive in the first place. A sample with its own narrow structure — the line can be in the sample is where one was measured — would make the sample’s half loose as well, and the argument here that the sample’s estimate is harmless would need rerunning on it.

Still open: whether a narrow table can declare its own peak #

The looseness this census found is the lamp’s single peak charged to every window, and a fine spectrum would supply each window’s own. But a table built through a slit as narrow as the lamp’s features is already nearly a fine spectrum. A 1.2-nanometre line seen through a one-nanometre slit appears in the table at a known fraction of its height, because the blur of a feature no narrower than the declared width cannot flatten it by more than a computable factor.

The calculation is a local peak read from the table itself: in each window, the largest value any lamp with features no narrower than the declared width could have there, given the table’s values at that grid point and its neighbours. At a slit much wider than the lamp’s features that ceiling is useless, because a line hides inside a window and the table barely moves. At a slit comparable to them it should be close to the true local peak — and the local-peak bound was ×81 at one nanometre against the global peak’s ×253.

The prediction is that a bound built on it lands near that ×81 at one and two nanometres and near the global bound at five, so that the one declaration a lamp needs is its narrowest feature, and the peak is something a narrow table can be trusted to report about itself. If it lands near ×81, the declared-width bound becomes a factor of three tighter at no cost in declarations. If it falls short of that, the width alone cannot vouch for a table’s peaks and a lamp maker would have to state them.

A number is only as local as what it declares #

The habit is about noticing where a declared quantity applies, not just whether it is true.

Each of the three numbers the bound was given was true, and each was a statement about the whole lamp. Two of them, the width and the table’s value, apply to every window equally. The third, the peak, is a fact about one place in the spectrum that the bound had to assume could happen in every place. A narrower instrument makes windows smaller and more numerous, and a global number spread across more and smaller windows is a worse description of each. That is why narrowing the slit tightened everything but the bound.

The failure mode is to judge a declaration by whether it is true rather than by what it is true of. A true global maximum is a loose local constraint, and no instrument setting fixes that. What fixes it is declaring, or reading, the quantity at the scale where the bound uses it.