Q277: Unusually question here. Premise: I cast Awaken Construct on a 1 HD effigy, manifest True Mind Switch on it, and then play that as a character. My class HD erase pre-existing effigy HD, for all intensive purposes, via True Mind Switch, but does that mean if I later improve my effigy that those HD also wouldn't stack with my class HD? We are admittedly in murky waters here, because WotC really did there best to prevent playing a construct altogether, but nonetheless one can make interpretations from various sources. SRD references are best, but it is worth noting that I have a lenient DM, so a good argument is probably a successful argument.
Anger... rising...
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A 277: No. Your HD would not stack.
However, if the increased HD would somehow change any of this:
"
the type of your assumed body.
the Strength, Dexterity, and Constitution scores of your assumed body.
the natural armor, natural attacks, movement, and other simple physical characteristics of your assumed body.
the extraordinary special attacks and qualities of your assumed body, but you do not gain supernatural or spell-like abilities.
the possessions and equipment of your assumed body.
"
Then I think your character would benefit/hurt from the changes.
I do not think increasing your effigy's HD could grant mental ability score increases to you or increasing your HD could provide physical ability score increases to your effigy, considering you are still two separate beings.