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The authority of rulership for Adam is clearly over animals and the earth,

This is not a mystery. It’s plain and it’s clear. God himself speaks of the authority both have over the animals and all the earth. The authority of rulership for the woman is clearly over animals and the earth.
Adam identifies the animals before the woman is created, but he doesn’t name them, or give them names like “Tommy the cat”, “Leo the lion”, or “Sparky the dog.” The woman’s name was never “Woman”. Her name was Eve, not “woman”. There is a difference between naming a creature and identifying or recognizing one.
Adam had rule over the animals only because God himself gave this authority to both the human and the woman. But God gave both the human and the woman authority over the animals and the earth only after he created the woman. So did Adam have authority over the animals and the earth before the woman was created? Did God give Adam authority over the animals and the earth before the woman was created or only after?
God never once spoke to Adam giving him authority over either when Adam was alone. Adam also wasn’t given permission to eat food outside the garden when he was alone. When the human was alone, God only gave him trees within the garden to eat from. The human was only given permission to eat outside the garden after the woman was created and he was given this permission when the woman was with him, as God spoke to both giving them authority over the animals and the earth, and permission to eat fruit with seed from the enitre face of the earth. God had only given Adam trees of the garden when he spoke to him, before the woman was created, not trees on the entire face of the earth. So if Adam was given trees on the entire face of the earth, and rulershipp over the animals and all the earth, it was only when God SAID so, which was when he was with the woman.
If we cannot presume that the human had permission to eat from trees outside of the garden before the woman was created, then we also cannot presume he had authority over the animals (which he named) before the woman was created. This leaves us then without the presumption that he had authority over the woman too before he and the woman were given authority over ALL the animals and all the earth and given permission to eat from trees outside the garden.
If Adam identifying the animals as “elephant” or “tiger” doesn’t show he had any authority over them because he wasn’t even given authority over any while he was alone, but rather he was given this authority over ALL of them after the woman was created, then Adam identifying the woman as “woman” doesn’t show that he had authority over her either because it wasn’t given to him at all, and rather both were told to rule the animals and the earth.