Nothing, I would think
Either he used a specific counter-free-transfiguration spell, in which case it wouldn't affect anything that was not free transfigured, or he uses Finite and only puts enough magic into it to overwhelm anything a first-year student could produce.
I suppose he could have put in just enough juice to counter one spell from a first-year but not enough to counter two, but that is relying on a second layer of Dumbledore error.
That's even assuming that Finite or Finite Incantatem can affect artifacts, which we don't know to be true. Have we seen any signs of such?
enough magic into it to overwhelm anything a first-year student could produce
That wouldn’t be quite enough; Harry could have got someone else to help, though indeed it is unlikely.
assuming that Finite or Finite Incantatem can affect artifacts, which we don't know to be true
You’re quite right about this. Presumably there’s some sort of “stabilizing” element for artifacts and spells that are meant to be (semi)permanent, precisely to avoid accidental Finite. (Not necessarily brute-force resistance spells, it could be just a safety thing to make sure you really want that cancelled.)
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