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If I connect the two white inputs (12 gauge and 10 gauge) together, wouldn't that put them in parallel? Providing a larger overall cross-sectional return path? Assuming that the commons are connected inside the power pillar.
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Yes it could be done and work well in most cases. The reason the electrical code does not allow it in houses or RVs is if the 30 amp circuity looses the common connection at the pedestal, you are depending on the GFCI at the post to detect the problem. This is because the 30 and 20 amp circuit breakers are only on the black wire (hot leg). In the best case the EMS will show a low voltage and shut the system down. There is a distinct plausibility the GFCI on the 20 amp plug will trip needlessly because it compares the hot leg with common current and trips if there is a difference. A power surge, as when an A/C cycles, could cause this difference in the common wire at the GFCI.