Hi Neil,
thanks for a very intriguing essay. Calling on quantum mechanics will, I suppose, invite the usual criticism about how quantum mechanics is typically thought to play no role at the scales where we expect to find an explanation for goal-oriented behaviour. I think your take correctly deflects this criticism: while we may not find relevant quantum effects persisting at this level, this doesn't mean that quantum mechanics is explanatorily irrelevant. It may, on the contrary, serve to explain phenomena otherwise mysterious within a classical view of the world---a prominent example being the very stability and extendedness of ordinary matter, which is taken to be simply a primitive posit in classical mechanics, explained only later on by means of quantization and the Pauli exclusion principle.
So, seeing how it already helps dispelling the mystery of Descartes' res extensa's very extendedness, one should think it at least worth investigating whether it cannot do the same for his res cogitans!
On a different note, I'm not quite sure about your treatment of the Chinese Room, or your arithmetic variant: your AR is essentially something like the 'lookup table' machine, that can duplicate the performance of an intelligent agent merely by looking up inputs in a gigantic table, producing apparently intelligent outputs in return, thus 'fooling' any Turing tester. One usually supposes that, just as you claim there is no addition being performed within the AR, this lookup machine doesn't possess any intelligence---but I think a more accurate view is that it simply relies on 'stored' intelligence, since it took an intelligent being to draw up the table in the first place (otherwise, there are no criteria according to whether a stored reply is actually an 'intelligent' one).
Likewise, in order to draw up the results of the 'addition table' you propose, somebody (or something) had to draw up that table in the first place; and drawing it up could certainly be understood as 'performing addition'. So it's not that there is no addition in your AR, it's just that the addition has been time-shifted---the room relies on precomputation, as indeed computers often do (and people often used to do, see the logarithm tables of times past; which of course themselves had to be laboriously computed). Hence, a determined proponent of 'intelligence-is-as-intelligence-does' behaviourism might resist your argument there; but I think there are ample other arguments to show the hopelessness of the position.
Good luck in the contest!
Cheers,
Jochen