The paint's own light does not order the rooms

The painted census asks one question of thirty rooms: six box shapes, each with some of its faces painted and the rest grey, lit by the whole ceiling, and a satin finish laid on the grey faces. Does the finish make the room more colourful, and for how many of seventy-two paints? A finish adds colour only where it covers grey found it does in almost every room, and a grey finish mirrors the lamp onto the paint found a number that seemed to say which rooms: the send number, how much of the lamp’s light a grey face’s finish sends towards the paint, over what the matt surface would send. With one patch a face it ordered the rooms at 0.92.

Divided faces keep the census’s count and lose its rule re-solved every room with each face divided five or six times a side. The count survived — some paint gains in 28 rooms — and the rule did not: the send number, now matching the true box’s own, ordered the divided rooms at only 0.63, and three rooms whose finish sends less of the lamp’s light to the paint than matt gained for sixty-four paints or more. Those rooms were the low and flat boxes with their floors painted, and the essay read them this way: the lamp reaches a painted floor directly, and what a grey finish adds there is likely not the lamp’s light at all but the floor’s colour, reflected once already. The send number counts only the lamp’s light, and in those rooms it may be the wrong light to count.

The closing section turned that into a calculation. Weight the light arriving at each grey patch not by the lamp’s light but by the paint’s — the part of the arriving light that carries the paint’s colour — and ask again where the finish sends it. The prediction was that this coloured-light send number orders the twenty-four wall and floor rooms above 0.85 and brings the six painted-ceiling rooms into the same order, because it separates the lamp’s white from the paint’s colour. If it did no better than 0.63, the counts would be set by something no single geometric number of the empty room carries.

Neither number, nor its sign, orders the rooms #

Weighted by the paint’s own light, the send number ranks the divided census’s counts at −0.07 over the twenty-four wall and floor rooms and −0.02 over all thirty, where the lamp’s light ranked them at 0.63 and 0.51. In every one of the six rooms with its floor painted, the finish sends the floor’s light away from the floor — three to thirteen points less of it towards the paint than matt — and every one gains for all seventy-two paints. In the six rooms with the ceiling painted, the two numbers agree within half a point, because the painted ceiling is the lamp’s own face; those rooms gain for anywhere from none of the paints to all of them. The paint-lit number has converged. And even the room’s own response to the paint, read off the divided solve rather than the empty room, ranks the counts at only 0.71.

• The prediction fails outright. The paint’s light orders the rooms worse than the lamp’s, and not at all.
• Where the finish sends the paint’s light is not what makes a room colourful. The rooms that gain everywhere send it away.
• The painted-ceiling rooms cannot be reached this way. Their paint and their lamp are one face.
• No send number will order this census. The count is a threshold over seventy-two paints, and it has to be solved room by room.

Each room appears twice, at the same height. The smaller dots are the lamp-lit send number, the earlier essay’s: they lean upward to the right, loosely. The larger dots are the paint-lit number, and they do not lean at all. The six rooms at the top right of the lamp-lit dots — floor and side-and-floor rooms that send the lamp’s light towards the paint and gain everywhere — move to the far left when the paint’s light is counted.

Weighting the arrivals by the paint #

The send number is a property of each grey patch. Light arrives at the patch from every other patch in the room; the satin finish returns part of it as a lobe aimed by the direction it came from, and the rest the matt body scatters evenly. For each arrival, the number compares the share of the lobe’s return that lands on painted patches with the share the matt body would send there, and averages the difference over arrivals, each weighted by how much light arrives that way. The only thing that changes between the two numbers is that weight. The lamp-lit number weights each arrival by the light reaching the patch in a grey room under the ceiling lamp. The paint-lit number weights it by the part of that light the paint puts there: how much the arriving radiance grows when the painted faces’ reflectance rises, taken at the grey’s own reflectance by a small step either side.

That derivative is exactly the paint-coloured component of the arriving light. Light that reached the patch straight from the lamp, or only by way of grey faces, does not change when the paint changes, and carries none of the paint’s colour; light that has bounced off paint on the way does. So the paint-lit number says where the finish sends the paint’s light — back towards the paint, concentrating it, or away onto grey faces.

The computation is the divided census’s own directional solve, unchanged, on the same patches — five a side for the four lower boxes, six for the tall room and the slot, the division at which the tall room gains once its walls are divided found the census settled — with the arriving radiance taken at a painted reflectance a tenth above and below the grey’s. Alongside it the same run recomputes the lamp-lit number, which comes out equal to the divided census’s in every room to rounding, and its ordering of the rooms exactly the published 0.63.

The rooms that gain everywhere send the paint away #

The painted-floor rooms are where the prediction expected to win, and they are where it loses most clearly. In the low room with its floor painted the lamp-lit send number is −1.0 per cent and the paint-lit number −12.7; in the flat room, −4.7 and −13.5; in the wide room, +10.0 and −8.6. In all six floor rooms the finish sends the floor’s own light away from the floor by three to thirteen points, and in all six a grey finish colours every one of the seventy-two paints.

The geometry is plain once it is looked at. A painted floor’s light leaves upward, and it arrives at the grey walls and ceiling from below and at a slant. A satin lobe returns light roughly in the mirror direction — the aimed return the highlight is the lamp found on a single object — and the mirror of light coming up from the floor at a slant goes up and across — to the ceiling and the opposite wall, which are grey. The finish takes the floor’s colour and spreads it over the room’s grey faces. The matt body, by contrast, returns a good share of it straight back down.

That the rooms nonetheless gain everywhere says what the census is counting. A room’s colour, as the census reads it, is the sum of every face’s reflected light — grey faces included — against the lamp’s white. A grey face that carries more of the paint’s light is itself more coloured. So the finish can raise the room’s colour by sending the paint’s light towards the paint, concentrating it, or by sending it onto the grey faces, spreading it; the finish adds the room’s own colour found the second mechanism in one room, and here it is the mechanism in six. A send number measures one direction against the other, and the census rewards both.

The negative of the paint-lit number fares no better: it orders the wall and floor rooms at +0.07. Rooms that send the paint’s light to the grey faces do not reliably gain more than rooms that concentrate it on the paint; the side-walled cube sends it towards the paint by 5.3 points and gains for 32 paints, the side-walled slot by 9.7 and gains for all 72.

With the ceiling painted, lamp and paint are one face #

The prediction’s second hope was the six painted-ceiling rooms, which the lamp-lit number could not place: they all read negative on it, while five gained at every paint with one patch a face and three still do divided. The paint-lit number was meant to separate the lamp’s white from the paint’s colour arriving from the same face.

It cannot, and the reason is structural. The lamp is the whole ceiling. When the ceiling is painted, the paint’s light and the lamp’s light leave from the same patches, in the same directions, and arrive at every grey patch by the same paths; the part of the arriving light that changes with the paint’s reflectance is distributed exactly as the part that does not. The two numbers agree within half a point in all six rooms — the low room at −12.6 and −12.7 per cent, the tall room at −5.2 and −5.2 — while the rooms’ counts run from none to seventy-two. No weighting of the arrivals at the grey faces can tell these rooms apart, because at the grey faces they are the same rooms.

What does tell them apart is outside the send number altogether: the divided-faces essay found the painted-ceiling exception belonging to wide rooms, whose short grey walls see the ceiling overhead and return its colour across the floor, and failing in deep ones, whose tall walls send it down to a grey floor where it is diluted. That is a statement about where the light goes after the finish, not about where it comes from, and a send number is a statement about where it comes from.

The number has settled #

A number that fails to order the rooms might be failing because the faces are still too coarse to resolve it; the lamp-lit number itself was wrong by ten points with one patch a face. The paint-lit number at three patches a side lies within three points of its finest value in every room, and most lie on the line. Its failure is not a resolution effect. It is converged, and it is converged on a quantity that does not order the census.

Even the solved room orders loosely #

The last candidate is not a send number and not a property of the empty room. It is read off the divided solve itself, with the paint in place: how much faster a grey finish multiplies the room’s light at a high painted reflectance than at a low one, the log-slope of the finished-to-matt ratio between reflectances of 0.2 and 0.7. A paint’s colour is the contrast between the bands it reflects strongly and the bands it does not, and a finish that multiplies the room’s light more where the paint reflects more should raise that contrast. It is the most direct summary of the room’s behaviour a single number can give.

It ranks the thirty rooms at 0.71. Rooms whose finish favours the paint’s peaks do gain more often, but the seventeen rooms that gain for every paint are spread across most of its range, and four rooms with slopes between 0.0095 and 0.0107 gain for 72, 46, 39 and 35 paints. The count is a threshold over seventy-two paints of different widths and centres, most of which move by a per cent or two of chroma, and a threshold over a crowd sitting near it is decided by details no one number keeps. An open room hands over sooner and a probe at the wall prices the finish each found a number that ordered their own questions well; this census’s question is the count, and the count resists.

What this settles about the census #

The census has to be solved room by room. That was the divided-faces essay’s fallback, stated as the outcome if the coloured-light number failed, and it has. Two geometric numbers of the empty room — the lamp-lit send number and the paint-lit one — and one number of the solved room have now been tried, and none orders the counts better than 0.71.

The census’s central finding is untouched. A satin finish belongs on a room’s neutral surfaces: it colours some paint in 28 of 30 rooms and every paint in 17. What cannot be had is a rule that says which rooms from a number the room’s geometry supplies; the answer for a given room is its own solve, which the divided solver gives in minutes.

The earlier reading of the painted-floor rooms was half right. The finish there does add the paint’s light rather than the lamp’s; it adds it to the grey faces rather than to the paint, and the room’s colour counts both.

How the numbers were computed #

The rooms, paints, finish and solver are the divided census’s: six boxes painted five ways, seventy-two Gaussian paints, a satin lobe at roughness 0.2 on the grey faces, the lamp the whole ceiling, and every face divided into five by five patches for the low, flat, wide and cube rooms and six by six for the tall room and the slot, with the cone-averaged lobe coefficients and the directional transport of thirty unknowns instead of six extended to patches. The paint-lit weight is the central difference of the radiance arriving at each grey patch from each other patch, between solves with the painted faces’ reflectance at 0.6 and 0.4 and every grey face at 0.5, over the step; the lamp-lit weight is the radiance from a grey room at 0.5. Both numbers are area-weighted over the room’s grey patches. The response slope is read from the divided census’s interpolants of the room’s light against the painted reflectance, finished and matt. Rank correlations are Spearman’s over the rooms named.

What this leaves out #

The room’s colour is read as the census reads it. Summing every face’s reflected light is one choice; a viewer looking at the paint sees the paint’s face, and a census read on the painted faces alone would reward only concentration — the direction both send numbers measure.

Seventy-two paints, one finish. A census over fewer, more saturated paints, or a glossier finish, would move the counts away from the threshold where small effects decide them.

The derivative is taken at the grey’s reflectance. A paint’s peak bands sit well above it; the paint-lit weights there would differ in size but, the room’s transport being linear in each face’s reflectance up to interreflection, not in their distribution among directions by much.

Still open: the paint’s face alone #

The census reads a room’s colour as all of its light together, and both send numbers ask where the finish puts the paint’s light relative to the paint. A viewer standing in the room looks at the painted wall, and what colours that wall is only the light that arrives there.

The calculation is the divided census read on the painted faces alone: for each room and paint, the chroma of the painted faces’ reflected light with and without the grey faces’ finish, and the count of paints for which the finish raises it. The prediction is that on the painted faces the lamp-lit send number orders the counts well — above 0.85 — because concentration is then the only mechanism that counts, and that the painted-floor rooms, which spread the paint’s light onto grey faces, fall from seventy-two paints to few. If the painted faces’ counts are still disordered, the send number’s failure is about the finish’s lobe, not about what the census chose to count.

A number built for one mechanism measures only that one #

The habit is about checking what a summary number can see before asking it to rank.

The send number measures one thing: whether a finish sends light towards the paint or away from it. It was proposed because concentrating the paint’s light is one way a finish can add colour. The census counts a room’s colour in a way that rewards the other way too — spreading the paint’s light onto grey — and a number that sees only the first cannot rank a census that rewards both. Changing which light it counts, lamp’s or paint’s, changed nothing about that.

The failure mode is to refine a number’s inputs when its blindness is in its design. The test is to find a room the number calls wrong and ask whether any input would make it right; here the painted-floor rooms answered before the calculation was run.