Before measuring a wire resistance in the laser cutter, I checked the resistance of the two test leads on the Aneng AN8009 meter (“Check your zero!”) to show an unsteady reading around dozen ohms.
Poking around inside showed the internal fuse apparently making poor contact with its holder, as poking it changed the random values:

Two tiny drops of Caig DeoxIT stabilized the reading around 1 Ω across several different combinations of test probes, so I declared victory. There is surely an offset calibration buried in the firmware, but it’s no longer a trimpot available to service technicians.
The ceramic fuse has an internal resistance of about an ohm, but swapping it for a replacement fuse with 0.2 Ω resistance didn’t materially change the results. It’s worth noting those glass fuses are slightly longer than they should be, surely due to their leads, and required slightly bending the fuseholder clips.
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