Permit me to geek out a bit, I'm so excited!
The whole spot welding band thing had me going until Windy City suggested SS shim stock, which I have incoming, and a shear. That'll solve those problems.
But a bigger problem is stacking, securing, holding a circular pile of mesh segments so as to bind up into a proper biscuit. They move, are fiddly as hell, and don't want to behave.
I remembered I had a tiny device called a "sput welder", used to weld small wires to large aluminum pieces for anodizing. It's a capacitator discharge machine. Looks like this, just 8" or so across.
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What you do is let the machine charge up, then forcefully contact the work, usually with a wire in an alligator clamp. Instant very neat tiny welded wire. I played with it in different ways with all the test mesh pieces I'd accumulated.
By placing two mesh pieces on a ground plate, and aligning a rod along a channel, the machine is charged, and the positive rod is tapped onto the rod in contact with the mesh.
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There's a decent little POP, and it creates a hundred micro-welds where the mesh segments contact. Zero weld of the round rod. The welds are so small you can't even see them, but it feels like I dumped super glue in there.
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I found I could do 4 at a time using less than half the machine's power. So the plan is now: Cut a ring of copper, 2" to 4". Cut that into 2 semicircles, screw those to a base plate ground. Add a mesh segment or 3, sput weld, build it up in height until I have a half biscuit. Repeat, combine the two halves.
Thanks for letting me share. I demo'd for the wife... "Uh, that's nice, have fun."

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