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Assuming my suggestion of 6 words produced by the numeric values is correct, does the 4th word in this vary for the digits of each number?

Gosh, I'm making heavy weather with this. So far have identified the 61. My way in was via 25 and 26 down and I have a small number of possibles for 3d and 26a, but I can't see a logical next step. Surely I don't need to brute force this? Any hint (however oblique) as to how I might think about my next step would be greatly appreciated!

Piffleworthy I think you need both logic and brute force!

Piffleworthy, I don't think brute force is needed, but you do need to narrow down some options. Since you have 26d you know the first digit of 26a (which = O). There are four options. One of those options is more likely for the first expression in 26a (though the others are not theoretically impossible). This will give you a likely value for UM.

It pays to list the numbers in numerical order and fill in the letters as you go along. At some point you will see coherent bits of the message appearing, which helps to place other letters approximately.

Gitto, I have yet to complete one of the sequences, but for the sequence starting at 14d, and one other sequence, the 4th word of the message is consistent and unchanging for the digits of all numbers in the sequence. It is different for each sequence.

Hi I am in the same boat as piffleworthy earlier ! Assuming my numbers for 26Ac and 26 Down are correct ( the former a little higher than the latter), my problem is to make 27 Down gel with 31 Across. Is the second digit of 31 an even number, for example? Any hints, nudges will be much appreciated !!

Thanks Murky, now got all 3 sequences and in doing so managed to confirm that my grid is 100% correct. My problem was assuming the instruction was additive to the start point which led me down many many dead ends, trying consistent and inconsistent variations of the fourth word to each relevant number. A truly remarkable construction and finding 3 sequences that conclude in the same way is extraordinary.

primeprospect, I don't know how many letters you have managed to match to a number, but 31A was a late solution for me. I would concentrate on the advice from murky@24 and then look at all the different variations of SUM clues.

I add my praise to Elap for the elegance of this puzzle. The fact that the letters formed the clues AND the hint seemed sublime to me. I wouldn’t know where to begin! I felt joy I when I sorted the letters as instructed and found the hint. There was a short time applying the hint and finding the parameters. Having the grid completely agree with the sequences was reassuring. A great puzzle that I throughly enjoyed.

Thank you Elap, another elegant construction. No need for coding- pencil, paper and calculator perfectly adequate.