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Who wins?

Entropy Aegis I would think.

Imo, Monarch is certainly more powerful overall--but I can't imagine him causing any significant damage to the EA.

Idk, look at the regular Probes, how one did vs say Kyle Ion Green Lantern and easily won or how they was nuking cities all over the world and taking on the JLA.

Monarch post absorb captain atoms or prior to.

Monarch