The room settles after the eye does

Somebody walks in from the snow and switches on the light. Four things in that sentence are still moving, and every model on this site computes exactly one of them.

Each of those numbers was computed against a target that was not moving — the pools, the pigment and the junction alike. The adaptation pools relax toward a fixed drive; the pigment recovers toward a fixed light level; the lamp warms toward a fixed junction temperature. Put them in one room and none of them has a fixed target, which is not a harder version of the same problem — a pool relaxing toward a moving drive is not a step response, and the closed-form two-exponential the adaptation essays use is the wrong object for it.

The claim #

The slowest clock in a room is not in the observer, and most of what it leaves behind cannot be adapted away by an observer of any speed.

• A room takes 295 seconds to settle to within a unit, against 130 for the same room with the lamp already warm. The lamp more than doubles it.
• A minute in — the point at which the eye is conventionally said to have adapted — the surface is still ΔE00 6.1 from where it will end up. With the observer settled from the first instant it is 5.2 of that. The residual is the lamp.
• And 76 per cent of what is left survives an observer who adapts perfectly and instantly. A warming lamp changes the shape of its spectrum, and a change of shape is not a gain in any basis.
• The pigment’s clock is worth almost nothing here, which is the same finding from the other side: a bleach is a diagonal gain, the pools are driven through it, and they track it out at whatever speed it runs.

What is moving #

The lamp is a phosphor-converted white LED built from its parts — a pump line, a converter absorbing along a Beer–Lambert path and re-emitting broad, and a red nitride band. Its junction temperature rises first-order with a time constant of 150 seconds, which puts nine tenths of its colour change at 405 seconds. Three data-sheet slopes act at once as it warms: the pump peak moves to longer wavelengths, the die loses radiant efficiency, and the converter loses quantum yield while its own emission moves. The net is a lamp that is dimmer and cooler in colour temperature than the one that was switched on.

The eye arrives adapted to daylight and carrying a bleach from it. The two neural pools are integrated forward at a twentieth of a second against a drive recomputed from the lamp; the pigment gain multiplies that drive rather than sitting beside it, which is the arrangement that makes a steady bleach cancel exactly and is not an approximation of convenience.

The one approximation made for cost is that the lamp is sampled on a one-second grid and interpolated. Its time constant is 150 seconds, so a grid a hundred and fifty times finer than the thing it samples ought to be free, and assertTheGridDoesNotDecideIt requires that halving both the grid and the integration step moves nothing.

The decomposition #

Four traces, and the third is the one that matters.

The room as it is starts at ΔE00 27.4 and is at 6.1 after a minute.

With the lamp frozen at its final spectrum, the eye alone reaches a unit in 130 seconds. That is the number every adaptation experiment measures, because an adaptation experiment is run under a lamp that has been on since before anybody arrived.

With the observer settled from the first instant and only the lamp moving, ΔE00 5.2 of the 6.1 remains. Nine tenths of what is left after a minute is not the eye.

And with an observer who tracks the lamp perfectly, with no lag at all, 4.7 remains — 76 per cent. That is the part no adaptation of any speed could have removed.

The three partial answers do not add to the whole and are not expected to: 3.1 plus 5.2 is 8.3 against a room that measures 6.1, which is 36 per cent out. A gain that is still moving multiplies a stimulus that is still moving, and the product of two departures is not the sum of them.

Adaptation cancels gains, and only gains #

The interesting division in that list is not between the observer’s clocks and the lamp’s. It is between the quantities that are gains and the quantities that are not.

A photopigment bleach reduces each cone’s sensitivity by a factor. That is a diagonal matrix in the cone basis. Chromatic adaptation applies a diagonal matrix in the same basis, and it is driven by the signal after the pigment has scaled it — so whatever the pigment does, the pools see the result and undo it. The pigment can run at any speed at all and stay invisible, which is why removing it from the computation entirely changes the room’s residual at a minute from 6.08 to 6.12.

A warming lamp does something structurally different. Its pump peak moves three nanometres, its converter’s emission moves with it, and the ratio between the two changes — so the shape of the spectrum is different, not merely its overall level or its balance between three cone classes. A surface lit by it returns a different set of ratios. There is no diagonal matrix that undoes a change of shape, and there is nothing an observer can do about it however fast they do it.

That is the rule this room produces, and it sorts the four clocks in a way their speeds do not:

A state variable that acts as a gain is invisible at any speed. A state variable that changes the shape of a spectrum is visible at every speed.

It explains why nobody has ever needed to model photopigment kinetics to predict a colour, and why every lamp specification in existence carries a warm-up clause.

The eye is chasing something that is still moving #

The adaptation gains at any moment are the gains appropriate to a lamp the lamp has already stopped being. That sounds like a small correction and it is measurable: the same eye in the same room is further from settled when the lamp is warming than when it is not, by up to half a unit at the worst point.

But the lag is the smaller of the two effects. The larger one is the irremovable part, and the reason is worth stating in the language of the earlier essays. Chromatic adaptation is a hypothesis about what changed — it assumes the change was a light, and a light is a multiplier. That hypothesis is right about most of a warming lamp, which is why the eye follows it so well; it is wrong about the part where the converter’s balance shifts, and that part is what is left.

What a specification would have to say #

A lamp is specified by a chromaticity, a colour rendering index and a colour temperature, all measured after a stated stabilisation period — typically thirty minutes, sometimes an hour. An adaptation experiment is specified by a viewing condition and a settling time, typically a minute.

Both documents are internally correct. Between them there is a period of several minutes in which neither applies: the lamp is not the lamp that was measured, and the observer is not the observer the viewing condition assumes. Every colour judgement made in that window is made under conditions no document describes.

The size of it is the number this essay computes. A minute in, a surface is six units from where it will be; three minutes in, still one; and a unit is the tolerance a paint or a print is accepted against. A viewing booth switched on for a quick check is a different booth from the one in the specification, and the difference is larger than the thing being checked.

Two more readings of the same room say what happens over the first five minutes rather than the first fifteen, which is the interval anybody actually waits.

Who found it, and when #

The eye’s side of this has been measured since the 1950s and the standard number for “settled” — a minute, sometimes two — comes from experiments in which the light was already on. That is not an oversight; it is the only way to isolate the observer, and isolating the observer was the point.

The lamp’s side arrived with solid-state lighting and belongs to engineering rather than to vision. A filament reaches thermal equilibrium in a fraction of a second because it is the hot part; a light-emitting diode is a junction bonded to a heatsink, and what takes minutes to settle is the heatsink. Every data sheet carries a stabilisation clause and every photometric standard specifies a warm-up before measurement, precisely because the number measured at switch-on is not the number the lamp will hold.

The two literatures do not cite each other and have no reason to. What is new here is not either clock but the observation that they overlap, and that in the overlap the larger term belongs to the lamp.

The four traces, in one number each #

A compact way to hold the result. Take a saturated blue surface, a minute after the light goes on:

The last row is the one worth staring at. Removing photopigment kinetics from the model entirely — a mechanism with a two-minute time constant, a fifty per cent bleach on arrival and a measurable tint of its own — changes the answer by four hundredths of a unit.

The two departures are orthogonal #

The decomposition is presented with a caution: the partial answers do not add, 3.10 plus 5.17 is 8.27 against a room measuring 6.08, and the reason given is that a moving gain multiplies a moving stimulus. That is the right mechanism and it leaves the impression that no composition rule is available. One is, and it is exact to within a per cent.

The two departures add in quadrature. √(5.17² + 3.10²) = 6.03, against the room’s measured 6.08 — 99.1 per cent of it. Solving instead for the angle the two would have to subtend for the composition to be exact gives 88.9 degrees.

So the eye’s lag and the lamp’s transient are very nearly perpendicular contributions to the same distance, and the failure of the linear sum is the ordinary failure of adding the legs of a right triangle rather than anything special to gains and stimuli. That is a more useful statement than they do not add, because a quadrature rule can be used: given either partial and the whole, the other follows, and a reader can check the table against itself.

It also says something about the two effects. Orthogonality here means the eye’s residual and the lamp’s residual point in different directions in colour space, which is what the essay’s own mechanism predicts — one is a gain error along the cone axes and the other is a change of spectral shape, and a change of shape is by construction the part no gain reaches. The 88.9 degrees is the “no diagonal matrix undoes a change of shape” claim, measured.

Two figures that need a decimal more #

Two of the essay’s roundings run in the flattering direction, and both are worth correcting because the essay’s argument does not need them.

“Nine tenths of what is left after a minute is not the eye” is 85 per cent, not 90: 5.17 of 6.08. Eighty-five per cent still carries the point, and the gap matters because the same paragraph’s other fraction — 76 per cent irremovable, 4.65 of 6.08 — is exact. The removable part of the lamp’s residual is the difference, 0.52 units, which is also the half a unit at the worst point the lag is described as costing two sections later. Those are the same number arriving twice, and saying so ties the two paragraphs together.

And “nine tenths of its colour change at 405 seconds” does not follow from a 150-second time constant. A first-order transient reaches nine tenths at 150 × ln 10 = 345 seconds; 405 seconds is 2.70 time constants and is 93.3 per cent. The sentence joins the two with which puts, and the arithmetic does not put it there.

The resolution is almost certainly in the essay’s own description rather than in an error. The junction temperature rises first-order; the colour change is three data-sheet slopes acting on a spectrum and then a correlated colour temperature computed from it, and none of that is linear in temperature. So the colour lags the temperature, and 405 against 345 is that lag. What the sentence should not do is present the second number as a consequence of the first, since it is a consequence of the first plus a nonlinearity, and the size of the lag — sixty seconds, or forty per cent of a time constant — is itself a result about how a white LED’s colour tracks its heatsink.

The pigment is slightly better than invisible #

The last row of the summary table is quoted as the striking one and it is quoted only for its size. Its sign is worth a sentence too.

Switching the photopigment off entirely moves the room’s residual at a minute from 6.08 to 6.12 — 0.7 per cent, and upward. Removing a mechanism makes the observer very slightly worse off, which is not what invisible would give: an exactly cancelling gain would leave the number unchanged to the last digit.

The cancellation is therefore near-exact rather than exact, and the small residue helps. That is what the arrangement predicts: the pools are driven through the pigment gain, so a bleach that is still recovering delivers a drive that is slightly closer to the settled one than the unbleached drive would be — the eye arrives from snow already partly turned down, and turning down is most of what walking into a dim room requires. Four hundredths of a unit is not an effect anybody should build on, and it is the right sign, which is a better check on the arrangement than the magnitude alone.

Reading the same decomposition two minutes in rather than one says which of the three clocks is still running at the point somebody starts working.

The other direction: walking out #

The scenario runs one way round because that is the one with four clocks in it, and the reverse is worth a sentence because it is not symmetric.

Walking out of a lit room into daylight puts the observer under a source that has been stable for four and a half billion years. Three of the four clocks are still running — the two pools and the pigment — and the fourth has been removed, so the settling is the eye’s alone and the published minute is the right number. The pigment matters more in that direction than in this one, because the bleach is being acquired rather than shed and it is acquired far faster than it recovers.

That asymmetry is a property of the room rather than of the observer, and it is the sharpest form of this essay’s point: how long a colour judgement takes to become stable is decided by whatever in the scene is slowest, and in an artificially lit room that is not the person.

What was computed, and how #

The two pools are integrated forward rather than solved. The step is a twentieth of a second, which is a twentieth of the fast pool’s time constant, and the drive at each step is the lamp’s white in the cone basis multiplied by the pigment gain.

Everything is reported as a distance from the settled state — the same room, the same eye, an hour later — because that is the quantity each of the three source models is separately about and it is the one a reader can check by waiting.

The scenario’s two illuminances come from the census the bleaching essay already carries: three hundred thousand trolands outside, which is snow, and a thousand in the room. Five minutes outdoors before coming in.

The surface is a saturated blue, which is the class of stimulus this effect is largest on for the same reason it is largest in every observer-metamerism result here: a warming lamp’s spectrum moves most where its pump line is, and a blue surface is returning most of its light from exactly there.

Half an hour is longer than anybody waits and is the honest end of the trace, because it is where the curve has actually stopped moving.

Where it stops #

The lamp is one lamp. A filament has no converter and no phosphor, so its warm-up is a temperature and nothing else, and it stays exactly on the Planckian locus throughout. A fluorescent tube warms in a different way again — mercury vapour pressure rather than a junction — with a longer clock and a different direction. The 76 per cent is this device’s number, not the class’s.

The observer has two pools and a pigment and nothing else. There is no rod contribution, which for somebody walking in from snow into a lit room is defensible and would not be for a darker room. There is no local adaptation: the whole field is treated as one drive, so nothing here is about what happens at an edge.

And the room has no geometry. The lamp is a spectrum with a level, not a source with an angle, so a surface at the far side of the room and one directly underneath are given the same light. Adding that back is the difference between a room and a lamp, and it is a separate argument.

Where the ladder goes next #

The rule that adaptation cancels gains and only gains is a claim about the form of a change rather than its size, and it should be checkable against everything else on this site that changes a spectrum. A shadow, a bounce off a coloured wall, an interference filter and a dimmer are four such changes, and each of them is a gain or is not.

The obvious next computation is that census: take every spectral change the site models, decompose it into the part a diagonal matrix can represent and the part it cannot, and report the ratio. It would predict which of them constancy handles well without needing a single new mechanism — and it would be a prediction, because nothing about the arithmetic knows which changes people find easy.