The slot's loss survives every finer division

A gloss finish on a wall is a mirror with a blur, and in a room it hands some of the light arriving at the wall back in directions a matt wall would not. The painted census asks what that does to the colour of a painted room. Six box shapes, each painted five ways — side walls, four walls, side walls and floor, floor alone, ceiling alone — are lit by their whole ceiling, the rest of each room grey, and a satin finish is laid on the grey faces. A finish adds colour only where it covers grey found that such a finish usually makes the room more colourful, and the essays after it tried to say why.

The most recent of them changed what was read. The paint itself gains all or nothing read every room on its painted faces alone — the light a viewer looking at the paint receives — and found every room’s verdict all-or-nothing: in twenty-eight rooms the finish makes all seventy-two paints more colourful, and in two it makes none. The two are the tall narrow slot, one unit wide, two and a half tall and only 0.35 deep, with its floor painted and with its ceiling painted. There the paint loses about a twentieth of a per cent of its chroma.

That number is small enough to distrust. The census solves each room with every face divided into strips — five a side for the four lower boxes, six for the tall room and the slot — and the tall room gains once its walls are divided had already shown a sign changing with the division: one patch a face said the tall room gained for no paint, six strips said fifteen. The prediction was that the slot’s sign holds at seven and eight strips a side, since the tall room’s gain is itself positive and small and the slot sees even less of itself; and that if it did not hold, the slot’s loss was the division’s.

The sign holds, and the coarse census had it wrong #

Solved from one strip a side to eight, the slot’s paint gains colour from the grey faces’ finish at one, two and three strips — 0.28 per cent undivided, a hundredth of a per cent at two, a thousandth at three — and loses from four strips on, for every one of the seventy-two paints, with its floor painted and with its ceiling painted alike. The loss deepens at every finer division: −0.031, −0.044, −0.052, −0.053 and −0.055 per cent from four strips to eight with the floor painted, −0.032 to −0.059 with the ceiling. Run out to no strip width along the square of the strip width, it settles near −0.058 per cent for the floor and −0.065 for the ceiling, six and ten per cent beyond its value at eight strips. The floor and the ceiling stay twins to within half a hundredth of a percentage point at every division. And the paints that lose most are the bright and broad ones, the bright losing more than twice what the dark do.

• The prediction holds. The slot’s loss is not the division’s; it grows as the division refines.
• The coarse solves were wrong in sign. A census at one to three strips a side would have put the slot among the gaining rooms.
• The six-strip census was already right, and within about a sixth of the converged loss.

The two lines sit almost on top of each other, the floor’s and the ceiling’s, and run from a gain at the left to a loss at the right. The undivided solve’s gain of 0.28 per cent is off the top of the scale. At two strips the gain is a hundredth of a per cent and at three a thousandth or two; between three and four the line crosses zero and keeps falling, steeply at first and then more slowly. From six strips on, each extra strip adds about a thousandth of a per cent to the loss, and the curve is visibly bending towards a limit.

What the division was getting wrong #

The slot is the box in which division matters most, and the reason is its shape. Its two big faces, the end walls, are one unit by two and a half and face each other across 0.35; its floor and ceiling are one unit by 0.35, and the ceiling is the lamp. A satin finish on the grey faces sends part of the light arriving at each grey face back in a lobe around the mirror direction, and in a box this narrow, where every face sees the others at a grazing slant, where that lobe points depends sharply on where on the face the light arrives.

One patch a face assumes every point of a face sees the room the way its centre does. For the slot’s tall end walls that is plainly false: the upper part of each end wall sees the lamp at a steep angle, the lower part almost edge-on. The tall room gains once its walls are divided traced the tall room’s sign change to exactly this — its end wall’s upper patches sending the lamp’s light across to the painted walls and its lower patches sending it to the floor, something one patch a face averages away. The slot’s end walls are as tall and much closer together, and their patches see the painted floor and ceiling at slants that change even faster with height.

The census does not take the slot’s crossing apart; it measures where it falls. What it shows is that the answer stops changing sign once each end wall is cut into four bands of height, about sixty centimetres each in a room two and a half metres tall, and that from there each finer cut moves the answer the same way by less. The coarse solves were not imprecise versions of the right answer; they were answers about a room whose end walls see the lamp at one slant, which no real room’s do.

Where the loss settles #

A discretisation error that falls with the square of the patch size makes a straight line when plotted against 1/n², and the slot’s losses at six, seven and eight strips lie close to one. Run out to no width, the floor’s line reaches −0.058 per cent and the ceiling’s −0.065. The line through seven and eight alone gives −0.059 and −0.066, within a thousandth of a per cent of the three-point line, which is the check that the curve is in its asymptotic range and the extrapolation means something. Five strips sit a little off the line, as a coarser division should. The census’s own six-strip value, −0.052 and −0.054, is within about a sixth of the limit.

The boundary belongs to the quadrature found the directional solver’s reach in roughness set by how it samples the lobe, and priced finer sampling as a purchase. Division of the faces is the same kind of purchase. Eight strips a side is 384 patches and about 123,000 unknowns, one radiance for every ordered pair of patches that see each other, and it took the solver about forty minutes for each painted arrangement; six strips is 216 patches. What eight strips bought is the knowledge that six were already close.

Every paint crosses together #

The crossing is not a mean drifting through zero while some paints stay on each side. At two and three strips every one of the seventy-two paints gains; at four, every one loses. The spread between the paint that changes most and the one that changes least grows with the loss: at eight strips with the floor painted, from −0.020 per cent to −0.171. The all-or-nothing verdict the previous essay found on the painted faces survives the refinement exactly — the slot is a room in which the finish costs every paint, not most of them.

Which paints lose most #

The loss belongs to the paint’s own light, and a paint that returns more of it loses more of it. At eight strips the paints with a peak reflectance of 0.85 lose 0.075 per cent of their chroma on average, those at 0.6 lose 0.056, and those at 0.3 lose 0.033 — more than twice as much at the bright end. Among band widths the broadest, a hundred nanometres, loses most at every brightness, 0.123 per cent for the bright broad paints. That is the same ordering the previous essay found in the gaining rooms, where a brighter paint always gained more than a darker one of the same band: the finish redistributes the paint’s own light, and there is more of it to redistribute from a bright broad paint. In the slot the redistribution runs the other way.

Why the floor and the ceiling agree #

The slot’s floor and ceiling are not the same face. The ceiling is the lamp, and painting it paints the source of all the room’s light; the floor is the face opposite, which receives the lamp’s light only after it has crossed the whole slot. The previous essay found the two gaining the same colour to a hundredth of a point in every box while the light the finish added to them differed by up to eighty-two points, and read it as the paint’s return to itself deciding the change. The slot keeps the twinship through every division: the floor’s and the ceiling’s mean changes agree within half a hundredth of a percentage point from one strip to eight, and cross zero at the same division.

The bars are the twinship measured directly. At one and two strips the floor and the ceiling agree to the last digit shown; from three strips on the floor’s change sits a little above the ceiling’s — gaining slightly more at three, losing slightly less from four — and the gap grows with the loss, to about four thousandths of a percentage point at eight strips against losses of five to six hundredths. The difference is under a twelfth of the loss at its largest, and it never changes sign: whatever separates the two faces is a small correction to one shared effect, not a second effect of its own.

That is a strong hint about what the loss is. Whatever the finish does to the paint’s colour here, it does through the paint’s own light making its way back to the paint across the grey faces, and that path is the same length from the floor as from the ceiling. The lamp’s light, which reaches the two faces by very different routes, would not give the same answer twice — which is also why the paint’s own light does not order the rooms found no number of the empty room predicting how much a paint gains. A gloss finish takes colour out of the whole room kept a ledger of every flux between a room’s faces and found the finish moving colour between them rather than only removing it; in the slot the ledger’s net for the painted face comes out a little negative, and does so identically at either end of the box.

Trusting the solver at eight strips #

A finer division is worth nothing if the solver behind it has changed. The directional transport was built to reproduce the radiosity solution it replaced — thirty unknowns instead of six checked that a matt room comes out to ten parts in a quadrillion — and dividing the faces adds patches without changing the transport’s equations. The six-strip slot here reproduces the painted-face census’s six-strip numbers paint for paint, to the last digit, which is the check that the seven- and eight-strip solves are the same calculation carried further rather than a different one.

What the division does change is the cone averaging: each patch’s lobe is averaged over the cone the receiving patch subtends, and smaller patches subtend smaller cones, so the lobe is sampled more finely at every step. That is the error a straight line in 1/n² describes, and the three finest divisions fall on one.

What the slot is, and what it is not #

The slot is the extreme of the census’s boxes — the narrowest, with the lamp’s face smallest against the walls — and the one room where a grey finish takes colour from the paint. That verdict is now settled by division. It is a small effect, a sixteenth of a per cent of chroma at most, and no viewer would see it as a change in the paint; what it marks is the boundary between rooms where a grey finish adds colour to the paint and rooms where it subtracts, and the slot sits just across it.

Divided faces keep the census’s count and lose its rule found that re-solving every room on divided faces kept the census’s count of gaining rooms while changing which rooms they were. This is the same result one box at a time: dividing the slot’s faces left the census’s verdict on it unchanged at the division the census used, and showed that a coarser census would have got it backwards.

How the slot was solved #

The slot is a box one unit wide, two and a half tall and 0.35 deep, its whole ceiling emitting, its floor or its ceiling painted and the other faces grey at a reflectance of a half, with a satin finish — a microfacet lobe of roughness 0.2 on a dielectric of index 1.5 — on the grey faces. Every face is divided into n strips along each of its sides; the unknowns are the radiances leaving each patch towards each patch it sees, the lobe averaged over the cones the two patches subtend, and the transport solved by Gauss–Seidel to a residual of 10⁻¹¹. At each division the painted face’s reflected exitance is computed matt and with the grey faces finished at seven Chebyshev reflectances spanning the paints’ range and interpolated between them. Each of the seventy-two paints — six hues, four band widths, three peak reflectances — gives a spectrum of painted reflectance; the painted face’s light under D65 is its exitance band by band, and its chroma is read in CIELAB against the lamp’s white. The change is the chroma with the finish over the chroma without, less one. At six strips the census’s own painted-face reading is reproduced paint for paint. The extrapolation is the least-squares line through the mean changes at six, seven and eight strips against 1/n², evaluated at nought.

What the census does not reach #

Uniform strips. Every face is divided into the same number of strips along each side, so the slot’s long end walls get strips two and a half times taller than wide; a division proportioned to the gap, as the tall-room essay proposed, needs more patches than the solver can hold.

One roughness. A glossier finish sends a narrower lobe whose direction depends still more on where it starts, and the slot’s sign could depend on roughness as it depended on division.

One box. The slot and the tall room differ only in depth, and one gains while the other loses. Somewhere between them the sign turns.

Still open: the depth at which the finish stops costing the paint #

The slot loses and the tall room, the same box a full unit deep, gains. Between 0.35 and 1 there is a depth at which a grey finish neither adds nor removes colour from a painted floor, and it says how enclosed a room has to be before a glossy grey surround starts to cost the colour of the surface it surrounds.

The calculation is this box solved at depths from 0.35 to 1 — 0.45, 0.55, 0.7 and 0.85 between — with its floor and with its ceiling painted, at four strips a side, where the slot’s sign is already the converged one, and the painted face’s mean change read at each. The prediction is that the sign turns between 0.45 and 0.55, nearer the slot than the cube-like end, because the slot’s loss is small and the tall room’s gain is large, and that the floor and ceiling turn together. If the sign turns only near the tall room, the slot’s loss is not a property of narrow rooms but of the slot’s one particular shape.

A small number is a question, not an answer #

The habit here is about what to do with a result near zero.

A loss of a twentieth of a per cent is exactly the kind of number a discretisation can produce, and the census had already shown a sign moving with division in the tall room. The tempting response is to call the slot’s sign uncertain and move on; the useful one is to buy the division that settles it. Here the purchase cost two more solves and settled more than the sign: it showed the sign wrong at coarse divisions, right from four strips, converging, and a sixth larger than the census reported.

The failure mode is to treat a near-zero result as noise without asking what would make it signal. The test is to change the thing the result depends on — here, the division — and see whether the number moves towards zero, away from it, or settles. A number that grows as the method improves is not noise.