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What is the closest that any cheapskate carriage comes to the very bottom of the extrusion? How much room is left on the extrusion under the carriage at that point?

Not sure what you mean by "very bottom of extrusion" foshon. Do you mean how close the cheapskate carriage comes to the top of the base plate? If so, that is 4.5" for me.

I'm thinking of mounting something on the side (non-belt area) of the extrusion. I just wanted to make sure that the carriage wouldn't hit it. Thanks!!

4,5" when the arms are parallel to the bed? (I'm not at my printer and can not measure now..)

The arms can't get parallel to the bed on a delta. This was measured with the print head between 2 towers as far out as it could go and measured at the other. It is not a precise measurement, ballpark.

Why not? I think this is what limits the outer position and the printable radius.
(To clarify: I'm talking about 2 of 6 arms, which are parallel.)

You can move to get the arms parallel to the table and there would be no gap, but this is not within the "normal" work envelope of the printer - 280mm diameter. At X=140, Y=Z=0, the height is 62.8mm/2.47" on my printer so my manual movement and measure was off a bit.

Thanks to clarify this.

So it is possible to bring them parallel and have an extended work envelope, but this is probably not recommended, because of the singular position.

Because of the geometry, I am not sure that its usable area out there or that the steppers could actually move to that location. I'll see what it can do when I have some time.