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Gitto,

The fact that exponents are written in smaller fonts, and in superscript means that bracketing can be dispensed with, where it would otherwise be required. So, you are right that the last across clue is T^(H + E - N). The 10th across clue is H(I!) (otherwise, it would be written (HI)!)

I meant to say earlier: not as hard as it looks but I still really enjoyed it. I don't always enjoy super-hard puzzles (especially when they're weekly with a deadline), but prefer something novel or interesting. In this case, I very much enjoyed the perceived complexity of the answers and entries compared with what was actually needed (and it was fun to apply the usual skills required here against a different set of results than normal, with their own particular properties). So a bit of a Trojan Horse, and the better for it.

Here's a thought experiment for those who've finished:

Suppose instead of the algebraic clue numbers, Piccadilly had just supplied normal clue numbers. Would the puzzle be solvable?

What if, in addition, there were enumerations?

Ah, a blank grid puzzle. I’ve not started this yet, so don’t know how helpful this might be, but here’s my usual reminder that I made a Listener-style grid tool which might help figuring out sequences of clue numbers and the like:

https://codepen.io/dcwarwick/full/wvdebxB

Don’t forget to set the grid size to 13x13 and turn on 180° symmetry.

After a few confidence-boosting weeks I must admit to feeling a bit defeated after staring at this thing for the better part of an hour and getting nowhere!

I have some relationships in terms of arranging some of the letters in a greater-than/less-than relationship, but it doesn't feel sufficient to make any real progress.

chrisremo - you can get most of the way with just the clue numbers. For instance (SH)^Y is the 20th down clue. It’s value cannot be less than 20. There are 47 clues in total and six after that on the down list so it’s maximum possible value is 41 (probably less).
There are a limited number of values for SH and Y.
Every number between 1 and a maximum of less than 47 need to be represented so if two consecutive down clues miss a number or numbers (say 23 followed by 25) then the missing number(s) (24) must be in the across clues.
Hope that gets you started.

Thanks Piccadilly. Looked daunting initially, but quickly fell out.
Neat device, not seen that before.

Thanks Smithsax at #20. I should have thought of that.

Gave up yesterday due to a false start. Started totally from scratch this morning and have finally finished this marvellous puzzle. I did have a major panic with the central row, but I finally saw common sense and the panic was over.

A novel take on the hidden locations device. Great fun. Thanks Piccadilly.