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Have a full grid but, having seen the contributions on this thread, denied that the next stage is too difficult. Even so I'd be grateful if someone could help me parse 4 down - what is the extra letter?
Thanks

The extra letter in 4d is S. The "energy" which has to be lost is a 2-letter word.

Thanks Will37 - see it now

After filling the grid ……………………

Note by Admin - Post amended as I am told it gives sway the entire end game

Extraordinarily clever piece of construction. Drxx, it’s not mathematics, it’s not even sums.

You mathematicians are a strange bunch - don't you ever get bored with logic, calculations, 'proofs'... (or whatever false flag you want your maths to sail under)? I certainly do.
If someone else had offered a timely hint to the endgame, I'd have kept my own offering - and my opinion - to myself. But please don't try to convert me (the missionary zeal in this thread would put a sect of religious fundamentalists to shame).
Believe me, there are others like me who hanker back to a time when Listener puzzles were complex and ideas based - we're just not very well represented online.
Enjoy the love-in amongst yourselves, and please accept my not wanting to be part of it (you'll probably have a few more just like this before the year's out - maybe drop a timely hint in future?).

Drxx and all Forum members.

If you have any concerns about Listener posts giving away too much information, please contact me direct so that I can amend the relevant post.

I do not read all the posts, just glance in now and again, but I do read all my emails and take action accordingly, as I have in relation to post 34.

Good for you DrXX

By the way the purists are talking about arithmetic, really

This one seems to have divided us. It wasn’t so hard to have a hypothesis or two about the endgame, but to be fair to the critics, if you didn’t have decent spreadsheet skills, the mechanics of trying those hypotheses out would have been a bit of a chore ( especially if your first hypothesis was wrong).

My biggest worry this week was that I might have a grid entry wrong, which could make the endgame impossible. However, I did have a wrong letter and the endgame actually made it obvious.

To be fair, the maths involved was pretty basic. The challenge wasn’t the arithmetic. For me, a decent challenge and fun.

Thank you, woodlouse - they mean much the same to me but I'm no expert. My preferred term would have been 'sudoku' but, for some reason, that was vetoed the last time around (although the word seems entirely fitting to me).

Gregwatsonuk - your empathy is much appreciated, thank you (but the suggested aids are anathema to me).

Norah, I never worry about too much being given away - too little has always been my gripe.