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As soon as I posted I saw it. One error was all it took... Great puzzle. Thank you Oyler

I have a full grid, both before and after instruction, and am now doubting myself after trying to check something. This is a wonderful construction, because at first sight the last line of the preamble implies the solution is not unique (when clearly it is).

I believe I understand how that trick is pulled off, and to confirm it to myself I've been looking for a particular connection between two entries in the grid AFTER applying the instruction - which I'm just not seeing. (I'm being intentionally vague of course and hoping somebody understands what I'm talking about.)

Could anybody confirm which two entries are related in the way I think they must be?

When I had about half the digits I deduced the message, and worked backwards from the other letters to resolve some ambiguities I couldn't otherwise figure out. That still left me with a problem or two but they were resolved by considering the post-instruction grid state. That's obviously not the correct solve path and I fear I've made a mistake which led to the ambiguities not being resolvable in the first place. My guess it was a website I used to show me 4 digit primes which I now know was bizarrely incomplete, so I suspect I resolved one entry too soon and that led to an ambiguity elsewhere. (The site listed some but not all of the four digit numbers, with a tick next to the primes. Why some (but not all) non-primes were listed I don't understand, but later I found an actual prime was not listed at all! Should have such with OEIS like usual.)

To smst. If you go to crossnumbersquarterly.com there is a link to a set of our tables. It lists all of the primes less than 10000 in a rather nice way that allows you to see things more clearly.

smst, I may not be following you correctly, but there is an ambiguity in the pre-instruction solution, which is resolved by following the instruction. Does that help?

Sorry, still stuck at an earlier stage. I keep getting that 4dn is less than 8dn (which is itself quite high). Has anyone else hit this problem?

I don't think it does unfortunately, acornz, but that is good to know. (I had an ambiguity in the bottom half which could be resolved after the instruction.) I didn't explain myself clearly enough, but that's because I'm trying to be vague.

The preamble tells us that there are at least two solutions which can satisfy the clues, and that one (the final grid, "B") can be arrived at by applying the instruction to another (the initial solution, "A"). Somehow the solver must have arrived at A in order to receive the instruction which makes B. So there must be something else in the solve path other than just the clues which at some crucial early point sends the solver down the path to A instead of to B (or some other C or D or whatever).

The only way I can see for that to happen is a restriction outside of the clues themselves, which A must satisfy but B does not. I can't see evidence of that in my B, so I imagine something is wrong with my A (but not much because I got a message and it seems likely to be correct).

If that's still too vague I'll have to remember to come back here after the closing date!

Thanks oyler. I'm going to start again and will use that site instead. I'll might even let myself use a computer this time (perhaps this confusion is what I get for attempting pencil and paper only for a change!).

smst - I may have had similar challenges as found the ambiguity challenging as I had the wrong NW corner for a while (initially) and only realised because the message made no sense.

Seems a bit odd that you have to reverse engineer from the likely message but perhaps I am
missing something.

Very clever composition mind to have two solutions that work.

Yes, Dylan! I have the same frustration. I’ve checked my working but am obviously missing something.

I'm still struggling with the initial grid fill--it's been a very slow and incremental solve so far. I have fewer confirmed digits than I'd like, a fair few binary options (many of which seem as though they will be impossible to disambiguate purely mathematically), and several more cells with several possible options.

My progress has slowed and while I'm sure there are some more deductions I can make, I'm starting to wonder if at this point I'm meant to try and infer the message. I have a plausible first word, but beyond that the possibilities become so wide that it just doesn't feel like the intended solve path is to try and guess at them.

Do I likely need to keep pecking away at the individual clues? Or should I be looking for some entirely new angle of attack?