My two penn'orth.......(just ruminating)....
If the nut won't shift by heaving with a box spanner and bar until the veins bulge it will sometimes loosen by putting a ring or open ended spanner on it an giving the spanner end a good thwack with a heavy hammer. The shock seems to do the trick. An impact driver/rattle gun is best buy would need a very deep socket. That thin nut seems to be a bit recessed into the sprocket so perhaps there isn't enough meat to get a ring or open ended on it securely.
If you can't determine exactly what the thread is (surely you can?) you could try it both ways as no harm will be done. Even whacking it the 'wrong' way will sometimes break the adhesion. Obviously you'll need to hold the g'box securely. Perhaps bodge up a bit of support for the g/box and clamp the chain in the vice jaws to hold it firm for a thwack or two?
P.S.
As Mick said, photos are confusing but it looks to me that the second thread of the right hand pair of the two threads is 'higher' (has slightly more thread showing above the nut) than the left hand ones where the second thread ring is touching the nut which makes me think it's a r/h thread.
Here, picture...........thousand words, so forth:
Sprocket nut snip detail.jpg
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