I will say at the outset that I'm not in either Byers' or Gore's league in math skill. Still, when I read Byers' article I was uncomfortable; he lost me at some point, and no amount of study could bring me back to the trail. Note, too, that, in the end, both authors make similar changes in the nut and saddle locations, so the actual effect is similar. As Gore points out, in many cases it's better to simply do something
that tends in the right direction, as the result will be 'closer' than it would have been, even if it's not 'perfect'. In this case, perfection is a will-'o-the-wisp that will constantly elude you, so in practice it might not make much difference which system you follow. Still, it's usually better to do things for the right reasons, if only because you're less likely to run into nasty surprises.
In 'Contemporary Acoustic Guitar ; Design and Build' Vol 1, 'Design', on page 1-65, Gore writes:
"Two writers on the subject, Byers and French, assume a solution by Morse approximating a clamped boundary condition, anticipating significant inharmonicity. (snip [equation given]) However, this does not match our experience, which is that inharmonicity is rarely an issue, implying that strings as normally found on a guitar are better described by a pinned end condition." He goes on to describe a test which confirms this.
'Inharmonicity' is the condition where each higher partial of the string is relatively sharper than it 'should' be, due to string stiffness. This is certainly real: the question is whether it causes problems with intonation. The theory here is that the ear does not (as tuners do) home in on the strongest partial, and determine pitch from that. Rather they tend to use a sort of weighted average of a number of strong partials to determine the pitch. The closer to 'harmonic' those are the more secure the sense of pitch. In theory, partials that are shifted significantly upward in pitch will move that sense of pitch sharp.
A string with a 'pinned' end is free to rotate vertically around the fixed point of the saddle top, while 'clamped' end is not, and must bend in order for the string to vibrate. This bending in effect shortens the string a bit, and raises the pitch. You do
see this on wound strings that break over the saddle with a sharp bend. The windings pack together on the bottom of the curve, and the string actually rises a bit in front of the saddle, which can be clearly seen. Even with plain strings we're talking a matter of degree here.
In 'The Physics of Musical Instruments'. Fletcher and Rossing give the equation for determining the change in tension of a displaced string. It is:
T = T0 + ((E*A/L0) * delta L), where
T = final tension
T0 = original tension
E = Young's modulus (a measure of how much force it takes to stretch something by a given amount)
A = the cross sectional area of the string (or the core in a wound string)
L0 = the original length
delta L = the change in length
What this says is that for a given string and displacement the rise in tension will be proportional to the cross section area of the of the string and the Young's modulus. Materials like steel that have a high E value don't stretch much when you push them aside, and the tension rises a lot, so they need a lot of compensation. Nylon stretches more, all else equal, so you see less tension change and pitch rise for a given displacement assuming the same string diameter, and need less compensation. However, nylon strings, because they're lighter, have to be thicker to carry a given tension, and the cross section area is greater, which reduces the advantage a bit. This is, of course, particularly true of a plain G string, which is why it needs the most compensation.
I used the term 'red herring' because, of course, the G string is also the stiffest string, by quite a bit. For a given material the stiffness of a round string goes as the fourth power of the diameter. That makes the G string about 4 times as stiff as the high E if the same material is used. That's a larger difference than in the cross section, which is about double for the G string. Intuitively an explanation that relies on stiffness rather than area seems more likely.
Because of it's high stiffness a plain G string is, in fact, much more inharmonic than the high E. This is one reason, I think, for the somewhat 'fuzzy' sense of pitch it can give. This gets worse as you fret the string higher, since the stiffness has even more effect on the inharmonicity for a shorter string. The whole issue would be moot if Classical players could get, and would use, wound G strings that held up reasonably well. It's rare for a steel string guitar to be set up with a plain G string, except for Blues playing, where they like to 'bend' the pitch a lot. Steel Gs use metal windings, which hold up reasonably well. On nylon they use plastic, which wears fast, but sounds pretty nice while it lasts. The other issue is, of course, 'zip', which is a matter of left hand technique.