Examples: Could it remove a cubic inch of material from the center of a bowling ball?
Could it remove and replace a single electrical component from the circuit board (which would require a complexly defined volume)?
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Precision transports do seem to be possible, but are vanishingly rare and presumably prone to a high failure rate. The best examples I can think of are the Ferengi transporting Mrs and Miss Troi out of their clothes in TNG: Ménage à Troi
And the "fetal transport" performed by the EMH in VOY: Deadlock
Both require closely adjacent (or connected) volumes to be transported from within larger objects.
Klingon transporters can, at least.
In Star Trek IV: The Voyage Home, Scotty beams an exact volume of water along with
two humpback whales into the cargo bay of a Klingon ship.
We know he beamed a volume to an exact set of dimensions because otherwise the water, which can be seen through the transparent aluminium wall, would splash around to conform to the new container and would disturb the
I believe the transporter was capable of precision transport.
In the TNG episode Rascals, Captain Picard and three others are physiologically reduced to the age of 12 during a transporter mishap. Doctor Crusher eventually ascertains that their de-ageing is the result of missing viroxic sequences in their cellular RVN.
The episode is resolved with Dr. Crusher programming their original sequences into the transporter, which then manages to reconstruct them as adults.
I'm not entirely sure how the transporter does this, but it is implied in the episode that it could insert the missing sequences back into position. I'm slightly hesitant to accept that theory - such a control of transporter technology would have resulted in advances in transporter assisted medicine, engineering, construction, etc. (We have seen hints of this)
TNG era transporters also had a biofilter, by means of which a transporter could remove biological contaminants of humanoids as they were being beamed aboard.
The above are two instances of transporter precision; one deals with beaming matter into place, one deals with beaming matter out.
So, I believe that the transporter was fully capable of removing a cubic inch from a bowling ball, or a component from an electrical circuit.