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For checking the larger grid entries this site is particularly useful:

www.magicdragon.com/fig.html

Thanks. Id realized my mistake in the meantime (wrongly applying a condition that would be true if it were type g to my type h calculation.

Left half mostly done, might adjourn for the evening. Finding this a total slog though.

I believe I have finally finished. What a slog!

Am I correct?
There would be three possible fills except that in two cases there is a duplicate grid entry so that that value no longer complies with the rule that each entry is associated with exactly one property. E.g. 16d and 25a cannot both be the same because 16d gets property C and 25a gets property A - This is just an example. My duplicate occurs at different entries.

Which entries do you have duplicated buzzb ?

26 and 29

Please tell me the properties associated with 1ac and 9d

Bananabean:
In the set of numbers from 1 to 38, there are exactly two with property C (fourth power) - 1 and 16. So the grid entries at 1a and 16d have property C; they are both 4th powers. There are six numbers (1, 4, 9, 16, 25 and 36) with property A but 1 & 16 are already associated with C, so the four that are associated with A are 4, 9, 25, and 36, So 4d, 9d, 25d and 36d are all associated with A; i.e the entries are all squares.

It gets a lot harder after this...

One really needs to be very methodical to get through this puzzle. It is a slog but it's not quite as bad as it first appears, (though I haven't quite finished so may regret saying that).

I've created a spreadsheet with all the property letters listed against the grid numbers that satisfy them. It's then possible to identify all those numbers that appear against one property only. I think there are twelve such numbers. Once I'd used up all instances of a property I highlighted it as unusable for further entries.

Thanks buzzb but you have not mentioned 1d so I don't understand how you know that the four with property A are 4, 9, 25, and 36.
Can't 1d be a square?
Or could 1d have property C and 1a have property A?

There's no 1d.