The slope limit was exact all along

The object-colour solid holds every colour a surface can have under a light, and its boundary is reached only by ideal pigments that switch between reflecting everything and nothing at an instant. The limits assume a pigment that switches instantly asked what a real colorant gives up by not switching instantly, and a run of essays has compared three models of a real one against the ideal under nine lamps — a comparison the lamp outweighs the pigment model found the lamp dominating. A slope limit bounds how fast reflectance may change across the spectrum; bands in nanometres build a reflectance from absorption bands no narrower than forty nanometres; bands in energy fix each band’s width in energy instead, the model written in energy, the bands spare the green set beside the other two. A direction through the solid counts as lost when a model reaches less than ninety-nine per cent as far as the ideal.

Those counts rest on an asymmetry. The band models are found by searching, and a harder search pulls the band pigments away from the slope limit found that searching them from 594 starts instead of a handful lowered their counts by a sixth. The slope limit is solved by a dynamic programme that walks the spectrum band by band, choosing a reflectance from 161 levels and never changing it by more than the slope allows. That is exact — but exact on its grid, and the essay’s closing section pointed at the gap: a reflectance restricted to 161 levels might fall short of what a continuous one reaches, so the slope limit’s count could fall on a finer grid, the same correction from the other side.

The prediction was that it falls by under a twentieth. The slope limit is a limit on the rate of change, and a staircase on 161 levels already follows any ramp the slope allows to within a 160th of the full range. If the counts fell by as much as a sixth, the census’s comparison of models would have been fair after all, both sides flattered equally, and the band models’ lost sixth would have been the grid’s.

The count does not move on any grid #

Solved again on 321 and 641 levels — each band’s allowed change scaled to 40 and 80 levels, so that the transition stays at 40 nm — every census direction under every one of the nine lamps reaches exactly what it reached on 161, to the last digit the arithmetic carries. Every lamp loses the same directions, 463 over the nine. Solved on a grid of only nine levels — nought, an eighth, a quarter and so on to one — with a band allowed to change by one level, every reach is again the same. Every optimal reflectance is a staircase whose steps are whole eighths wherever the lamp gives light. So the census’s count is exact for every reflectance with that slope, not only for those on its grid; and the slope limit’s lead over the band model searched from 594 starts, 1.38 times as many lost directions, is a lower bound.

• The prediction holds in the strongest form available. It asked for a fall under a twentieth; the fall is nothing.
• The grid was never doing any work. A dynamic programme on nine levels gives the same answers as one on 641.
• The asymmetry between the models is now one-sided and known. Every remaining uncertainty in the comparison belongs to the band models, and resolving it can only widen the slope limit’s lead.

Every slope-limit bar carries the same label because every grid gave every lamp the same count. Thirty-one directions under three narrow emitters, 89 under daylight, 77 under tungsten, 83 under the white LED, 33 under the triphosphor tube and as many with its lines doubled, 14 under the tube’s coating, 21 under clear mercury and 82 under the halophosphate tube — the census’s own numbers, which the search essay already quoted, now with seven grids behind each. The band model’s bars are the searched counts, shorter under every lamp. The tube’s coating and clear mercury lose fewest of all, as a tube spares a pigment through its phosphors and a mercury lamp’s few losses are deep found for each.

Why the grid cannot matter #

The census asks, for each direction, how large a weighted sum of a reflectance’s values can be: the reflectance in each of 81 five-nanometre bands, times that band’s weight for the direction under the lamp, summed. The reflectance must lie between nought and one in every band, and must not change by more than an eighth between neighbouring bands — a full swing from nought to one takes eight bands, forty nanometres. That is a linear sum maximised over a set bounded by flat walls, and such a maximum always sits at a corner of the set.

At a corner every band is pinned by a wall: either it sits at nought or one, or it sits exactly an eighth from a neighbour that is itself pinned. Follow the pins and every band’s value is a whole number of eighths up from nought or down from one — and because one is itself eight eighths, those are the same nine values. No corner of the set has a band between two eighths, so no grid finer than the eighths can offer the maximum anything it lacked. The census’s 161 levels hold every eighth, as levels 0, 20, 40 … 160, and the twenty-fold surplus between them was never used.

Under daylight the direction the slope limit reaches least well asks for a narrow peak in the green — weights positive only from 495 to 540 nm. The ideal pigment reflects fully across those ten bands and nothing elsewhere. The slow pigment climbs eight steps of an eighth, holds at one for three bands, and falls eight steps, reaching 76.1 per cent as far as the ideal; the 641-level solution lies exactly on the nine-level one. Moved to the other lamps, the staircase changes shape — two peaks under the three emitters and the tubes, a plateau under clear mercury — and keeps its steps. Under the mercury lamp a stretch of the spectrum gets no light at all, the weights there are nought, and any reflectance serves; there the two solutions may differ, and both are optimal.

This is also why the prediction’s reasoning, though its conclusion held, pointed the wrong way. It expected a finer staircase to follow a ramp more closely. But the best slow pigment is not trying to follow a ramp; it is trying to get from nought to one, or back, as fast as allowed, and the fastest way is the steepest ramp, which on a five-nanometre grid is exactly a staircase of eighths. The slope limit’s best reflectance never wants a value between steps.

Seven grids, one answer #

The argument predicts more than that the finer grids change nothing: it predicts that any grid holding the eighths gives the same answer, however coarse. Nine, 17, 33 and 65 levels each give 463 lost directions over the nine lamps, as 161, 321 and 641 do, and every direction’s reach agrees with 161’s to the last digit. The line is flat across a seventy-fold range of grids, and its only requirement is that a forty-nanometre transition be a whole number of levels per band.

The band models’ searched counts, 335 and 311, sit below it. They come from a search, and a search reports the best reach it found — never more than the true best — so each band model’s true count of lost directions is at most its searched count. The slope line cannot move and the band lines can only fall.

The same holds at every transition width the slope census has used, because the argument never used the number eight. A 10 nm or 20 nm transition is exact on five levels, a 40 nm one on nine, 80 nm on seventeen, 160 nm on thirty-three, direction for direction under daylight. What changes with the width is how many directions are lost — two of 154 at 10 nm, 89 at 40, 114 at 160 — which is the census’s original finding, now with no grid behind it to doubt.

The lead is a floor #

With the slope side exact, the comparison between models has one direction of uncertainty. The search essay found the slope limit losing 1.17 times as many directions as the band model on the census’s search and 1.38 times as many on the harder one. Every further improvement in searching the band model can only lower its count, so 1.38 is the least the true ratio can be. Under each lamp separately the same holds: the ratio ranges from ×1.08 under the tube’s coating to ×1.74 under the triphosphor tube, and every one is a floor.

The energy model, whose bands are fixed in energy rather than wavelength, stands the same way: its searched count of 311 is an upper bound, and the slope limit’s lead over it, 1.49 times, is likewise a floor. What the census can now say without hedging is that a pigment limited only in how fast its reflectance changes, at forty nanometres a full swing, loses at least a third more of the solid’s directions than a pigment built from forty-nanometre absorption bands — a statement about every pigment of each kind, not about the pigments a search happened to find.

That is the reverse of a common intuition. A slope limit sounds like the more permissive model: it allows any shape at all, provided the shape changes slowly, where the band models allow only sums of a few bands. But a forty-nanometre absorption band falls from its peak to half in twenty nanometres and keeps falling; its flanks are steeper near the middle than an even ramp of the same overall width — the steepness a sharp edge is bought with depth found a band buying by going deeper — and the census’s directions reward steep edges. The slope limit forbids any edge steeper than its ramp anywhere, and the bands do not.

When the step does not divide the range #

The eighths are a convenience of the census’s numbers, not a condition of the argument. Follow the pins again for a slope whose step does not go into one a whole number of times — a band allowed to change by 0.15, say, which is a transition of 33⅓ nanometres. A band pinned from nought takes the values 0.15, 0.30, 0.45 and so on; a band pinned from one takes 0.85, 0.70, 0.55. The two families no longer coincide, so the corners use two interleaved sets of values instead of one evenly spaced grid — fourteen values in all, nought and one included — and an evenly spaced grid holding both needs a level every 0.05, twenty-one levels. A step of 0.13 would need a level every hundredth.

The exactness survives that, because it never depended on the levels being even. What the corner argument says is that the best reflectance takes finitely many values, every one of them a whole number of steps from nought or from one, and a programme that offers exactly those values finds the maximum. What fails for an awkward step is the convenience of an even grid: a grid of 161 levels holds 0.15 and 0.85 only because a 160th divides them, and a step of 1/7 is held by no grid of 161 levels at all. For that slope the census’s grid would have been an approximation, and the remedy is not more levels but the right ones.

That is why the transition widths in the figure above were solved on the fewest levels holding their steps rather than on one fine grid for all of them: the number of levels a width needs is a property of its step, and every width the census uses happens to have a step that divides one.

What a census count is worth #

The slope census has been quoted for a run of essays as the number of directions a slow pigment cannot reach, with a footnote that the number was exact on a grid. The footnote can go. For a reflectance sampled every five nanometres and changing by at most an eighth per band, the count is exact, and every direction it lists is a direction no such reflectance reaches within one per cent — not one the dynamic programme missed. A colorant chemist given a slope budget can treat the list as a proof of impossibility.

It is exact on the five-nanometre wavelength grid, which is a different grid. The reflectance is sampled at 81 wavelengths, and the slope is a change per band. A reflectance sampled at one nanometre with the same slope per nanometre could place its staircase’s steps between the census’s wavelengths and reach a little further where the weights change fast — notably under line lamps, whose lines the five-nanometre grid smears across a band. That is a question about the spectrum’s sampling, not the reflectance’s, and the argument above does not touch it.

How the grids were solved #

The directions, lamps, observer and ideal are the census’s: 148 to 157 directions per lamp through the object-colour solid, CIE 1931 2° colour-matching functions at five nanometres from 380 to 780, and each lamp’s ideal reach from the positive part of the direction’s weights. A direction’s weights are the lamp’s power times the direction’s mixture of the three colour-matching functions in each band. The slope-limited reach is the dynamic programme’s best weighted sum on a grid of levels, with each band’s level differing from its neighbour’s by at most the transition’s step: 20 levels a band on 161, 40 on 321, 80 on 641, one on nine, two on seventeen, four on 33, eight on 65. The reach is divided by the ideal’s, and a direction is lost below 0.99. The band models’ counts are the search essay’s, from 594 starts per lost direction.

For the transition widths, each width’s step on 161 levels is 160 × 5 nm divided by the width, and the fewest levels holding it are the step’s own denominator — five for 10 and 20 nm, nine for 40, seventeen for 80, thirty-three for 160 — never fewer than five, the least the programme takes.

What this does not reach #

The wavelength grid. Every reflectance here is sampled at five nanometres. The argument makes the slope limit exact on that sampling and says nothing about a finer one.

The band models’ true counts. They are bounded from above by the search and not from below. How far the band model’s searched reach falls short of its best is the census’s one remaining unknown.

Other slopes. The census’s slope limit is the same at every wavelength. A colorant’s real limit may be steeper in some regions than others; a varying step keeps the corner argument only where each band’s step divides the range.

Still open: whether the line lamps’ losses live in the wavelength grid #

The five-nanometre grid treats a mercury line six nanometres wide as most of one band, and a slope-limited pigment can only change by an eighth across it. Under clear mercury the census’s deepest losses ask for a cliff between the green line at 546 nm and the yellow at 578, and a pair of lines asks for a cliff, not a gap found fifteen directions still short by more than a per cent however deep the band model’s cliff was cut.

The calculation is the mercury lamp’s census on a one-nanometre grid — the colour-matching functions interpolated, the lamp’s lines at their census width of six nanometres and narrowed to three and to one and a half — with the slope limit solved exactly at the same 40 nm transition, now five levels a nanometre, and the ideal recomputed. The prediction is that on the one-nanometre grid the slope limit’s mercury count falls by about a third at the census’s line width, because a cliff can then fall between the lines rather than across them, and that it falls further as the lines narrow. If the count does not move, the mercury lamp’s cost to a slow pigment belongs to the lamp, not to the census’s sampling of it. A slow pigment splits mercury’s lines only if its swing fits between them makes that calculation.

Exactness is a property of the answer, not the method #

The habit here is about where an approximation actually sits.

The census described its slope limit as exact on a grid of levels and its band models as searched, and a later essay reasonably asked whether the grid was flattering one side as the search had flattered the other. The way to settle that is not to keep refining the grid and watching the numbers, which can only show that they stop moving; it is to ask what the answer looks like. Here the answer is a staircase of eighths, and a staircase of eighths fits on any grid that holds them.

The failure mode is to treat a discretisation as an approximation without asking whether the true optimum already lies on it. The test is to find the structure of the solution — here a linear objective’s corner — and check what grid that structure needs. When it needs fewer levels than were used, the method was exact all along, and the uncertainty in the comparison lives entirely on the other side.