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I agreed with Buddy. There is another thematic clue in the undeciphered grid body. This points to only one solution that I can find.

Ok, I accept that if the answer from that work is taken as the only answer to 55, 56 then my solution is wrong and I am sorry for dragging out the debate. I thought that any 1, 2 with A=P was accepted in which case there is one which meets 13 and 19 also.
Luckily I don't submit so no harm done.

Well that was a challenge! I definitely needed the hint here about using the grid diagonal. Thanks to Wikipedia for listing the 1a 7a instances, only one of which fitted the diagonal length and presumed 55a 66a hint.
Did anyone discover a way to compute the cipher? Preamble WONDERFUL COMPUTABLE … seems to indicate there is an algorithm.

I think “wonderful computable algebraic formula” likely refers to the formula to find the area.

I doubt there is any way to compute the cipher especially since there are so many ambiguous unchecked cells.

As recommended earlier, you should use the fact that the resulting crossing entries are words or phrases or abbreviations (38a is problematic). In many cases there is only one possible letter to choose (you can assume that no letter decodes to itself) or just a couple. 19a is particularly good for this. That will establish a set of decodings that you can use for the other three edges.

Start with the perimeter letters which occur checked more than once.

Many thanks Ozzy. I take it that there is no key which permits deciphering but simply random substitutions. And yes, the right-hand column was the give-away to completing the deciphering.

Many thanks Quisling, your comment confirming the two entries you refer to.

Really struggling with this one for whatever reason—I have one initial clue left to solve, 13dn, which I can’t seem to figure out. I also have no clue how to tackle the cipher; it seems from responses in this thread that it’s not very difficult, which guess means I’m missing something obvious because I can’t figure out how to proceed with so many unchecked letters. Any help welcome!

(Of course I immediately just solved 13dn. Still no clue how to proceed with endgame after a few days of staring.)