Why does the hair light need a magenta gel on a green screen shoot?
Because the screen is lighting your hair at the same time your hair light is, and both of them put green into the same pixels - but only one of those two lights is yours to change. A minus-green (magenta) gel subtracts green from yours, at the place where the screen is adding it. Measured on this keyer with the screen 0.8 m behind a blonde head and the hair light three times the key: the screen leaves +5.09 levels of green on the arc the hair light covers, and a 1/8 gel takes that to +0.81 - an 84% cut, for 0.12 of a stop of light. It does nothing for the other 61% of the outline that the hair light never reaches; that part is despill's job, and despill only removes about half of it.
Your hair is lit twice, and only one of those lights is yours
On a green screen set the hair is lit twice: once by the hair light you hung behind the subject, and once by the screen itself, which is a 3.0 x 2.4 m lamp sitting less than a metre behind the head. In the model I ran - a 0.11 m sphere for the head, the screen lit to a flat 70 IRE, one key light from the camera side and one hair light from behind and above - the screen supplies 14.2% of the green irradiance landing on the arc the hair light covers when that light is three times the key, and 33.2% when it is only as strong as the key. Both contributions land on the same pixels.
Only one of those two is yours to change. You cannot dim the screen without wrecking the key, and you cannot move it without changing the shot. You can, however, change the colour of your own light for the price of a gel. That is the whole reason the advice exists: a minus-green gel subtracts green from the one source you control, on exactly the pixels where the screen is adding it.
What the screen actually leaves on the hair
I rendered that head and ran it through this site's keyer at its defaults - similarity 42, edge softness 14, spill 35, despill 10, key sampled from the screen's own colour - and measured what is left against the same head photographed with no screen behind it. On the arc the hair light reaches, dark brown hair keeps +4.84 levels of green at 0.4 m, +2.54 at 0.8 m and +0.74 at 1.5 m. On the rest of the outline, which the hair light never touches, it keeps +10.49, +6.33 and +2.03 - roughly two and a half times as much.
Blonde hair, which reflects several times more light than dark brown, carries roughly twice as much again: +5.09 levels on the lit arc at 0.8 m and +12.36 on the rest. Note that these are the levels left after the default spill suppression has already taken its cut. They are small enough to be invisible on a set monitor, which is why the problem usually surfaces later, once the composite is built and the hair edge no longer sits in the new background.
What a minus-green gel actually does to the light
Gel fractions scale optical density, not transmission: a 1/8 gel is the full-strength filter raised to the power of one eighth. Modelling a full minus-green at 42% green transmission, 1/8 passes 89.7% of green and 92.2% of the light overall - 0.12 of a stop. A quarter costs 0.23 of a stop, a half 0.46, a full gel 0.87. So 1/8 is the largest dose you can add without touching your exposure.
The consequence that matters is that the gel is a percentage, not an amount. It removes about a tenth of whatever green your hair light is putting on the hair, so the same 1/8 removes more when that light is hot and less when it is dim - while the screen's contribution is fixed by distance and exposure. One number therefore cannot be correct for every setup, which is the reason this article has tables instead of a single recommendation.
The dose, measured
Blonde hair, screen 0.8 m, hair light three times the key: no gel leaves +5.09 levels, 1/16 leaves +2.92, 1/8 leaves +0.81, 1/4 leaves -3.25, 1/2 leaves -10.64 and a full gel -22.73. Negative numbers are a magenta cast, which is the failure mode of over-gelling. At that distance and that hair, 1/8 removes 84% of what the screen added, which is a good description of why it became the standard piece of advice.
Change any input and the right dose moves. Dark brown hair under the same light needs slightly less: 1/8 overshoots to -0.84, and the zero crossing sits between 1/16 and 1/8. Move the subject to 0.4 m and blonde hair needs roughly a quarter. Push the screen back to 1.5 m and blonde hair needs about 1/32. The rule behind all of it: the dose follows the ratio of screen green to hair-light green at the hair itself.
Where the gel cannot reach
In this geometry the hair light covers 39% of the head's outline - 1209 of the 3064 rim pixels. The other 61%, the sides and the underside, receives screen bounce and no hair light at all, and on that part the gel does nothing whatsoever: +6.33 levels at 0.8 m without a gel, +6.33 with 1/8, to two decimal places. The same holds at 0.4 m, where the unlit arc sits at +10.49 either way.
So the gel is not a general cure for spill. It is a cure for the arc you can see lit, and it is worth checking which arc that is before you buy into it. The rest is either a lighting problem - a second kicker, or more distance between the subject and the screen - or a job for despill, which only acts where green has risen above the pixel's own red and blue. Here it fires on 62.2% of the unlit arc at 0.8 m and 95.8% at 0.4 m, against 3.4% of the lit arc.
Why the sliders cannot stand in for it
This tool's spill suppression is a green-channel limiter. It acts only where green exceeds both red and blue, and it can never pull green below that floor. On the lit arc your hair light keeps red comfortably above green, so at the defaults just 3.4% of those pixels qualify at all. Pushing spill and despill both to 100 moves that arc from +2.58 levels to +2.48 - a 4% improvement, which is to say none.
The same sliders on the unlit arc do real work: +8.32 levels with them off, +6.33 at the defaults, +3.54 at maximum. Even there, about half of what the screen deposited survives every setting, because the floor is the hair's own red channel and the contamination sits underneath it. The two fixes are not substitutes. The gel works where your light is; despill works where it is not.
What it costs you
In exposure, very little: 0.12 of a stop for 1/8, rising to 0.23 for a quarter, 0.46 for a half and 0.87 for a full gel. That is the practical reason 1/8 is the dose people repeat - it is the most correction you can buy before you have to reopen your stop.
There is also the screen to think about, because a hair light hung behind the subject also throws light onto it. Measured, the screen absorbs up to about half of its own illumination from the hair light before anything shows: at 50% spill the keyed residue is still 0.00%. At 100% it starts leaving 1.33% behind, and at twice its own illumination the screen stops keying altogether - 97% of it survives. The gelled version is marginally safer at every step, 1.33% against 0.52%, because minus-green relieves the green channel, which is the one that clips first.
Related questions
Does it have to be 1/8, or will any magenta gel do?
It has to be sized, and 1/8 is only right for one part of the range. Measured with the screen 0.8 m behind a blonde head and the hair light three times the key, 1/8 leaves +0.81 levels of green, 1/16 leaves +2.92 and 1/4 overshoots into a magenta cast at -3.25. Dark brown hair under the same light needs less: 1/8 already overshoots to -0.84. At 0.4 m blonde hair needs about a quarter; at 1.5 m it needs about 1/32. Start at 1/8, because it costs 0.12 of a stop, then compare the rim against a frame shot with the screen struck.
Do I still need the spill sliders if I gel the hair light?
Yes, for the part of the outline the hair light never reaches. In this model the hair light covers 39% of the rim and the gel does nothing measurable on the other 61%, which is also where the contamination is two and a half times worse. There the sliders earn their keep: on the unlit arc at 0.8 m, spill suppression off leaves +8.32 levels, the defaults leave +6.33 and both at maximum leave +3.54. On the arc your hair light does cover, the same sliders manage +2.58 to +2.48, so they are not doing the gel's job.
Will the gelled hair light ruin my green screen?
It takes more spill than most people expect. The screen tolerates a hair light contributing up to roughly half of what its own lights contribute before the key notices - the residue is still 0.00% at 50%. At 100% it starts leaving 1.33% behind, and at 200% it fails outright, with 97% of the screen surviving the key. The gelled light is slightly kinder to the screen than an ungelled one at every level, because minus-green takes the load off the green channel, which is the first to clip.
Does this matter at all with dark hair?
Less, but it does not disappear. Dark brown hair keeps +2.54 levels on the lit arc at 0.8 m against +5.09 for blonde under the same light, because there is less albedo for the screen's light to work with - so the right dose is smaller, somewhere between 1/16 and 1/8 rather than a flat 1/8. What never pays on any hair colour is stacking gels: a full minus-green left -19.75 levels of magenta on the lit arc and still cost 0.87 of a stop.