Once you are in the advancement phase of characters, increasing the skill does not increase the specialisation.
And if you add a new specialisation, and have the skill, the specialisation adds the current (with advancements) value of the skill to the new specialisation.
Okay, these are damned difficult in Hero Lab, since Hero Lab is set up to re-calculate everything, every time the user changes something on the hero.
What you're going to have to rely on here is that every advancement on the advances tab has that same script to add the actual bonus from advancement that's in the CanAdvance component in advancement.core, and that all advancements run that script in the order they're listed on the Advances tab, from oldest to newest. That means that if, in that script, you can tell your thing that it started life as a specialization that was added during advancement, you can have it look up the current value of the skill it's based on. Since all the scripts are running in order, that value will only have the bonuses from any advancements added to that skill before the "New Specialization" advancement was added - not those after. Those specializations that were added during character creation will have a different script on their component that looks up the user-added value + bonuses from races, etc., but not from advancement, and adds that to the value of the specialization.
First, identifying whether a specialization was added during character creation or during advancement:
Means it was added as an advancement ( <> 0 means it was added during creation).
So, in a script on the specialization component, you can look up the non-advancement parts of the associated skill and add them as a bonus to the specialty.
Second, the advancement script. The first thing to do is to change the timing - you need it to come before the "Calc trtFinal" script, so that Calc trtFinal incorporates the amount this thing has been advanced, but since you also need to look up the user-added portion of the value (+ any bonus from races, etc. that apply to both the skill and specialty), you need your script to happen after the "Bound trtUser" script. So, Traits/1500 (or 2000 or 2500), if those other scripts are still using the timing from the Skeleton files.
Then, in that script, you'll need to check whether the thing you're currently working on is a specialization added during advancement mode:
Code:
if (tagis[component.SkillSpec] <> 0) then
if (origin.ishero = 0) then
field[spAssocSkl].value += field[trtUser].value + field[trtBonus].value + other values you need to sdd
(obviouly, I was guessing at the Id of your specialty component and the field you're using to store the value of the associated skill) (also, instead of "field[trtUser].value + field[trtBonus].value", you'll need to use whatever method you've set up to associate a specialty with its skill to transition to the skill, and add their values, rather than the trtUser and trtBonus from the specialty itself).
P.S. I like to use something other than trtBonus to store the bonus from advancement - I find it helpful to have the bonuses from races, etc. in trtBonus, and the bonuses from advancement in trtAdvance, or something like that, to make it easier to track down what came from what when I'm debugging.