A rod signal has no natural size

The rods’ route is priced by the lamp ended its account of where the model stops with a prediction it did not compute:

A real eye’s lens yellows with age, and an older lens cuts the short wavelengths before either the S cones or the rods see them, which should shrink the S shares under every lamp by a similar factor rather than change which lamps are extreme.

It is two claims. The second holds exactly. The first is wrong, and the way it is wrong is more interesting than a wrong sign, because there is no fact about the retina that decides it.

The same eye, two answers #

Between twenty and seventy-five the rod signal’s catch of a phosphor LED, divided by the S cones’ catch of the same lamp, goes from 10.5 to 21.2. The share the model’s departures actually use goes from 2.22 to 1.79. One doubles and the other falls by a fifth, on the same eye at the same ages.

• The difference is the S cone’s peak absorptance, which falls from 0.211 to 0.085 across the range — the lens keeps 86 per cent of the long-wavelength cone’s peak, 82 per cent of the middle’s and 40 per cent of the short’s.
• The model defines the rod signal as a tenth of each cone class’s own peak, so the shrinking S peak shrinks the signal entering the S channel by very nearly the factor the catch ratio grows by.
• The near-cancellation is not exact and it is not monotone: the phosphor LED’s share falls to 1.76 at fifty-five and rises to 1.79 at seventy-five.
• The ordering half of the prediction holds outright. The five lamps keep their order of S excess at every age — phosphor, three-emitter LED, tungsten, tube, daylight — without a single exchange.
• The practical advice reverses with the normalisation. Under the fixed-size reading an older eye makes the uncertain weight more expensive, not less.

Two readings of one sentence #

“A rod signal a tenth the size of each cone’s peak response” is how the rods are a fourth curve introduced rod intrusion, and it was a reasonable thing to write. A rod signal has to be given a size, the cones’ peak responses are the only sizes in the model, and a tenth of each is a modest intrusion.

It is also a statement with a hidden argument in it. A tenth of what the cone can do is not the same quantity as a fixed number of rod quanta, and the two come apart the moment anything changes a cone’s peak. Nothing in the census that introduced it changed one, because it was computed on a single median eye. Age changes one, by a factor of two and a half in the S cone alone.

The two families of curves go opposite ways. The solid curves — the rod signal at a fixed size — rise steeply: daylight from 4.6 to 11.0, the phosphor LED from 10.5 to 21.2. The dashed curves are nearly flat and slightly falling. Nothing about the retina differs between the two sets. Both are computed from the same aged lens, the same pigment templates and the same five lamps; only the definition of “how much rod signal” differs.

The two spectra say why the lens acts so unevenly, and they are the same pair the earlier essay used to explain why the LED is the expensive lamp. The S cones’ curve sits under 500 nanometres and the rods’ peaks at 507, which is only a little further along and is far enough. A lens’s absorption climbs steeply below about 500 and is nearly flat above it, so it takes most of what the S cones were catching and a modest share of what the rods were. That is a fact about where three curves sit on one axis, and it is the whole mechanism of the rising solid curves in the figure above.

It also says why the effect is largest under the LED. The LED puts four fifths of its light between 500 and 650 nanometres, under the rods’ curve and away from the S cones’, so the S cones were already catching little of it; taking most of the little they had is a proportionally larger change than taking the same share of daylight’s broad blue. The rising curves are steepest for the lamps whose light already sat where the rods are.

Which is right is a question about rod–cone coupling, and the model does not contain the answer. The physiology suggests the fixed-size reading is closer: rod signals reach the cone pathways through gap junctions and the rod bipolar circuit, and the strength of that coupling is a property of the retinal circuit rather than of how much light the cone in front of it happens to be catching. A cone whose pigment is starved by a yellow lens does not thereby receive a smaller rod signal; it receives the same signal against a smaller cone signal, which is the fixed-size curve exactly.

Where the cancellation comes from #

The near-flatness of the dashed curves is arithmetic rather than biology, and it is worth watching it happen.

The long- and middle-wavelength cones keep 86 and 82 per cent of their peaks at seventy-five. The short-wavelength cone keeps 40. That is the whole of the lens’s action: its absorption climbs steeply below about 500 nanometres, which is where the S pigment is sensitive and where the L and M pigments are not.

So the model’s rod signal into the S channel is scaled down by 0.40 between twenty and seventy-five. Over the same range the rod curve’s catch of a lamp, divided by the S cones’ catch, rises by a factor of about two — because the rods peak at 507 nanometres, comfortably clear of the steepest part of the lens’s absorption, and lose much less than the S cones do. Two point oh times nought point four is nought point eight, and the measured fall of the phosphor LED’s share is 2.22 to 1.79, which is 0.81.

The near-cancellation is therefore a coincidence of two curves’ positions, and it is close enough that its residual is not monotone. The share falls to fifty-five and rises after, because the lens’s absorption saturates at the wavelengths the S cone cares about while continuing to bite where the rods do. Nothing should be read into the shape of that residual; what it shows is that the quantity being plotted is a ratio of two things the model treats as unrelated.

The half of the prediction that holds #

The prediction’s second claim is about ordering, and it survives everything.

No lamp overtakes another except one pair, and that pair is the two whose values differ by under a tenth throughout. The phosphor LED is worth 3.05 at twenty and 2.60 at seventy-five and is the most expensive lamp at every age; daylight is worth 1.09 and 1.27 and is the cheapest at every age. The tungsten lamp and the fluorescent tube exchange places somewhere in the middle, and they are 1.89 against 2.05 at twenty and 1.77 against 1.74 at seventy-five.

That is the same transposition the earlier essay found when it ranked five lamps by the share and by the worth: those two lamps are close on every measure and their order is not stable. Calling it a crossing rather than a reordering is the honest description, and it is why the claim is stated as “no lamp changes which end of the table it is at” rather than as a rank correlation of one.

So a lamp’s structure decides whether the uncertain weight matters, and the observer’s age does not. Which lamps a mesopic match should worry about can be tabulated once, which is a useful thing to know and is what the earlier essay’s design argument was really resting on.

It is worth asking why the ordering is so much more robust than the level, because the answer is general. The level is a ratio between a rod quantity and a cone quantity, and the lens moves those two by different factors — so the level carries the lens. The ordering is a comparison of that ratio between two lamps read by one observer, and the lens’s factors are the observer’s, not the lamp’s, so they divide out. A quantity that survives a change of observer is one in which the observer appears identically on both sides, which is the same structure a gain is not an observer identified for a white point: what a common multiplier cannot reach is what a comparison can measure.

What this does to the earlier numbers #

Three consequences, and the first is the one that changes advice.

Under the fixed-size reading, an older observer under a white LED is in more trouble than a younger one, not less. The uncertain S weight enters against a smaller S-cone signal, so the same rod intrusion is a larger distortion of that channel. The earlier essay’s closing paragraph — that a warm LED dimmed in the evening should make the S weight more expensive still — is right for the lamp and, under this reading, right for the observer too, which doubles the effect in exactly the domestic case it named.

The three bars per lamp are the census everything here rests on, and they are the numbers this essay is putting a condition on. Under daylight the first two are 1.03 and 0.90 and the third is a third of either; under the phosphor LED they are 1.24, 0.45 and 0.96, and the S route alone is most of the first. Every one of those numbers is computed at the median eye’s age of thirty-two and under the scaled reading, and the comparison between lamps that they are for is untouched by anything here, because a common normalisation cancels out of a comparison between two lamps read by one observer.

What the figure cannot be read as is a statement about a person. A reader of sixty-five under a phosphor LED is not described by the middle bar at 0.45 unless the rod signal really does scale with their own S cones’ diminished peak, and the sentence in the earlier essay that invited exactly that reading — that an older lens should shrink the shares — is the one this essay is withdrawing.

Every rod-intrusion number computed so far is a number under the scaled reading. That is not an error: the scaled reading was stated plainly in the essay that introduced it and every census since has used it consistently, so the comparisons between lamps and between routes all stand. What does not stand is any reading of those numbers as a function of the observer, because the observer enters twice — once through the catches and once through the normalisation — and the two nearly cancel by accident.

And the matching experiment proposed alongside this one is unaffected, for a reason worth stating. The reference lamp must not move chooses a pair of lamps by the gap between their rod-to-S-cone catch ratios, and that gap is computed from the lamps and the curves rather than from the normalisation: rescaling the rod signal multiplies both lamps’ ratios by the same factor and leaves the ordering of the pairs alone. A design that depends only on a ratio of ratios is robust to exactly this, which is an accident of that design rather than foresight in it.

How the ages were computed #

The observer at each age is the site’s pigment-template eye with its lens age set and everything else at the median: macular pigment 0.35, optical density 0.30, peaks at the template’s medians. The cone absorptances follow from the pigment templates through the ocular media at that age, and the tristimulus matrix is the one fitted once on the median eye and used by every observer in the audit, so a difference between two rows is a difference in what the cones caught rather than in bookkeeping.

The rods are aged with the cones, which is the part it is easy to get wrong. They sit behind the same lens, and a first version of this aged only the cones: the S share then fell steeply with age while the rod catch stayed where it was, which is not an eye. The rod curve is the rhodopsin template through the ocular media at the same lens age, with no macular pigment, normalised to its own peak.

The raw ratio is the rod curve’s integral against a lamp divided by each cone class’s integral against the same lamp. The scaled share is that times the cone class’s own peak absorptance, which is the quantity the departures are built from.

Where this stops #

Lens age is one parameter and an old eye is several. Optical density falls a little with age, the macular pigment does not change much, and the pupil constricts — which reduces retinal illuminance and moves an observer further into the mesopic range, making the rod contribution larger by a route this model does not have at all.

The surface family is unchanged with age. The forty-two constructed surfaces are the same at every age, which is right — a surface does not know who is looking — and it means the spread of the departure across surfaces is being read through an observer whose S channel is much quieter at seventy-five. The ranking is not stable found that no departure has one size across surfaces; the spread here would need its own census before anything were said about which surfaces an older eye finds hardest.

The ocular media model is a template. It is a smooth function of age fitted to published lens-density data, and individual lenses vary by more than the age trend across a decade, so an observer’s own lens is the thing to measure rather than their birthday. The observer has no age is where that gap between a parameter and a person is set out.

And neither normalisation is measured. The fixed-size reading is more defensible on circuit grounds and it is still a stipulation: nothing here fixes the absolute strength of rod–cone coupling, and the two readings bracket a range rather than naming an answer.

Still open: whether a match can separate the two normalisations #

The two readings differ by a factor of two over a working lifetime, and that is a large enough difference to be measurable — which makes it a question rather than a permanent ambiguity.

The experiment is the asymmetric matching session described alongside this one, run on observers of widely separated ages under the pair of lamps that maximises the signal, with a weight fitted per observer. The two normalisations make opposite predictions about how the fitted weight varies with age: under the fixed-size reading an older observer’s matches move more per unit of weight and the fit should be tighter; under the scaled reading the movement should be very slightly smaller. The quantity to report is the fitted weight’s standard error against age, not the weight itself, and the two predictions differ in that quantity by about the factor of two the catch ratios differ by.

The design has an obvious hazard. An older observer differs from a younger one in every parameter at once, so a difference in fitted weight is not attributable to the lens without the lens being measured on each observer — which a densitometric measurement can do, and which turns the session’s covariate from an age into a number.

A normalisation is a model, and it is the one nobody writes down #

The habit is about the quantities a model is stated in.

The rod term was introduced as a fraction of a cone’s peak because a fraction is dimensionless and a peak was to hand. That is the ordinary way a term gets a scale, and it is invisible as a modelling choice precisely because it looks like bookkeeping — nobody argues about a denominator. It became load-bearing the moment a parameter was introduced that moves the denominator, and then it decided the sign of the result.

The move is to ask, before varying a parameter, which of the model’s quantities are defined relative to something that parameter changes. Here the answer is one: the rod signal’s size. Everything else in the observer model is an absorptance, and absorptances are what the lens acts on. Computing both readings costs a line and brackets the answer honestly.

The same trap is set anywhere a model gives a term a size by borrowing one. A field size is two changes is the neighbouring case: moving from a two-degree field to a ten-degree one changes the macular pigment and the optical density together, and a result attributed to “field size” is a result about a bundle. This is that one level further in, because the bundle is not two named parameters but one named parameter and one unnamed convention. Three curves for one space is the reminder that a set of cone fundamentals is a construction with choices in it, and a normalisation is the least visible of them.

The failure mode is a sensitivity analysis run over a parameter that the model’s own units depend on. The result is not wrong, it is conditional on a convention, and the convention is usually not in the sentence that states the result.