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i believe it's a numerical puzzle this week, so the usual messages, wordings don't apply.

If you're helping with a clue, remember it's hints rather than answers that should be supplied. If you're supplying a 'gentle nudge' to the theme, aim to be helpful without being over-revealing.

OK, folks, time to dust off the old calculator! I always take a crack at these, but I rarely find that first door into the puzzle by myself. When you think the timing is right, I would so appreciate it if someone just gave the number of a clue that is a good jumping off point.

I don't even understand the first two sentences of the preamble. Is it too early to ask if my interpretation of "(2|3), (4|5) etc" is correct? If so, I'll probably give this one a miss.

Simond9x - as far as I can tell it means each letter represents one of two consecutive numbers in the formulae on the left and the other in the formulae on the right.

Bathmatters - hopefully not giving too much away to say factorials are your friend. They get very big very quickly.

Thanks SH. That's pretty much what I thought it must mean. I suppose it's the usual case of start solving and, hopefully, things emerge that make more sense of the preamble.

Giving quite a lot away really, Smellyharry. It’s pretty ridiculous to be asking for or giving Listener hints before the vast majority have even received the puzzle, in my view. You have three weeks before the solution appears, and it’s meant to be a challenge

I'm not a big fan of numericals, but this one has me gasping with amazement at the amount of thematic material. Every clue is relevant to the general subject matter (made clear in the preamble) and the final message is apt (and that's putting it mildly). In a word-based puzzle there are usually two things going on, and the setter's task is to keep two plates spinning at the same time. Here the level of complexity in the construction is at least one degree of difficulty more. I can't express my amazement strongly enough. It won't be Puzzle of the Year, of course, because it's "only" a numerical, but for my money the sheer elegance and complexity of the construction beats anything so far into a cocked hat. And as I say. I'm not a big fan of numericals.

Nice puzzle, slightly easier than the average numerical I'd say.

Quisling - apologies if you think that was too much. The first part was just clarifying the preamble and the second is a very basic generic rule for all numericals.

Still a hint, Smellyharry. I stand by my earlier comment. But there's no need to apologise to me. I'm only expressing my view, which may we be a minority one. The poster put the same request on at least one other site. Finding the way in is the hardest part of many numericals, so I think one needs to be cautious. As for clarifying the preamble, I have more sympathy, but the questioner conceded they thought it pretty much had to be that. Why did they not work on that assumption and see if it bore fruit?

To me the natural approach, and the only way to become a better solver, is to try as hard as possible to solve unaided, and only come here when one has reached a complete dead-end, not seek validation or help before even attempting it. Anyway, I've said my piece. I'll shut up.

"In across clues the letters consistently collapse into one state of their
pair"

My reading of this is that all letters in across are even or else all letters in across are odd. Is this correct?