Is there a purist method to join/lock the edges of an origami magic ball so it's still squishable?
For certain corrugations which are "rolled" in order to get an object with rotational symmetry, you can overlap the two edges of the paper and have the model keep its shape. However, I doubt this would work with the magic ball since it's an action model. Its point is to be squishable and just arranging the layers one over the other prevents this in two ways:


Squishing introduces a lot of tension which just arranging two corrugated edges above each other can not withstand.
Overlapping multiple layers introduces some asymmetry: if you overlap just a single row of the corrugation, you create a stripe which has two layers of paper while other places have just one. This will cause any deformation to not be laid out equally over the whole model, and this will not only look bad but also locally increase the tension which makes the model come apart. Using more rows than just one in order to make a stronger lock only increases this effect. Perhaps if you use a sheet twice as wide as the original in order to get exactly two layers of paper over the whole surface, it might be enough to withstand some squishing, but the layers would come apart along the free edge anyway.


Basically, if there is a long free edge, this technique might work for static models, but not for a squishy ball. So, the only way forward would be to get rid of the long free edge. Imagine we were not folding a corrugation but just forming a cylinder out of the sheet and wanted it to stay in one piece. You might try to make pleats around the perimeter of the cylinder so that there are many short spans of free edge instead of one long span. The wall of your cylinder would then look something like this:

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Joining two edges when the walls have such pleats would probably work. Now imagine treating this pleated paper as if it were a plain sheet and folding the corrugation on that.

I haven't tried this in practice but it might work.I am, however, afraid, that in practice these extra layers would change the behavior of the structure, and in practice you would still get noticeable artifacts.