OK: I see we're going be arguing the same points here as we are in another thread.
There have been several attempts made over the years to calculate the compensation needed at the nut and saddle. These go back at least to the 80s that I'm aware of. There seem to be two major reasons why strings go sharp when they're fretted: string stiffness and tension change. Byers' work addressed the first as the 'major' cause. Gore argued that Byers' model was incorrect, and did some experiments and mathematical evaluations that demonstrate that. He then went on to set up a model that allowed for calculating the correction based on tension change. I have not seen Byers' rebuttal, if any, to Gore's assertions.
I will say that I never was convinced by Byers' model, but found Gore's to be more congenial.
So far as I know, neither author denies that both causes can come into play; what's in dispute is the proper model of string vibration, and the resulting balance of the different factors.
"'carbon' strings stretch differently than nylon,"
and prawnhead replied:
"Some strings labeled as "carbon" will be stiffer than some strings labeled as "nylon"."
'Stiffness' is a section property, not a material characteristic. When we say that a material is 'stiffer' we need to specify that it's for a string of the same diameter. If this is the case, if a 'carbon' string at a given diameter is indeed stiffer than a 'nylon' one, it's because the 'carbon' material ha a higher Young's modulus. When either string is displaced the tension will increase somewhat. Given the same diameter and displacement the one with the higher Young's modulus will see a larger tension increase. Assuming the initial tension was similar, the one with the higher Young's modulus will thus give a larger pitch rise when fretted. This is not directly related to the stiffness, in the sense that it was not resistance to bending itself that caused the pitch rise: both stiffness and pitch rise are rather functions of the Young's modulus.
One of the advantages of 'carbon' strings is that they tend to be denser than 'nylon', so a thinner string can be used to achieve a given pitch and tension. Since the stiffness of a plain string goes as the fourth power of the diameter and the Young's modulus, a thinner 'carbon' string with a higher Young's modulus could actually be less stiff than a nylon string for the same tension and pitch. Even if the stiffness does not affect the intonation directly, a more flexible string will tend to sound better. It will tend to have a more 'true' overtone series and possibly lower losses, both of which are also issues with plain 'nylon' G strings.
Again, I'm not as qualified as I'd like to be to argue the math behind all of this. I'm simply putting forth the case as I understand it.