Sorry about the misleading info on the stepper motor - I found the info on the internet - but now I can't go back to it - so Stirling is probably a lot nearer the mark than I.
As to your other question - there isn't really any way to test the axis to find if it is loosing steps. I can write a program to check that the direction of A is correct in relation to X and Y movement which was your other question, but, in the absence of any feedback as to which way the knife is actuall pointing, then any check merely relies on the DRO's of the system - and since these keep accurate position (even is the axis misses steps) that is not much good - it would always say it is correct.
I have not had any experience of tangential control so cannot comment on absolute stop - v - continuous
I personally would get a bigger motor, or gear down the one you have got. The Gecko drive - if it is like mine - will take a substatially larger motor. Mine are eight wire, 2.5amp per segment, at 7.5v and give 220 Ncm - and they work well and at £23 are cheap. You could probably run them with your 56 v supply.
I cannot get my head ropund your speed figures, however - you are quoting 5000 m (metres)/min - do you mean mm (millimeters). If you also have this acceleration and speed on your knife - IS IT NECESSARY. The knife, at best will only be moving a short time and only moving a few degrees, I would seriously cut down the speed of the knife rotation - you don't need it, and this is always one of the main reasons for missing steps. I have managed to coax 40 ins per minute out of my motors, with Geckos - but I have cut this to 20 ins per minute in deference to accuracy over speed.
The time you would loose by cutting your knife rotational speed by half would be minimal, but might just give you the accuracy you are looking for.