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The Garden of Eden story recounts how God created Adam and Eve, commanded them not to eat from the Tree of Knowledge of Good and Evil, and how they were expelled from the garden after they disobeyed Him, having been tempted by a serpent, and ate the fruit. As part of the Expulsion, cherubim and a flaming sword were stationed at the entrance to the garden, in order to prevent man from returning and eating from the Tree of Life.
A sword is stationed to guard Eden which you can draw it from Ultima Weapon in FF VIII. Lastly, this dude wield a sword.
In FF VI, Atma Weapon came with a package of a sword and a monster which is stated in the game itself stating.
In FFV II, Ultimate Weapon dropped a sword upon defeat. WEAPONS in FF VII are WEAPONS created by the planet. I dun think further details is needed since you played FF VII.
In FF VIII, Ultima Weapon wield a sword similar to Cloud's.
So there is no telling that Ultima Weapon is the sword itself since the sword is called Ultima Weapon too as based on the reference of Eden. As for Omega Weapon, I repeat myself:
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So you're suggesting that the party may have dispersed Utima Weapon instead of slaying it, where it reformed into Omega during time compression. It's possible, and Omega does first appear as a nebulous cloud of evil that materializes after you ring the bells. Both seem to be able to change from a noncoporeal to coporeal state.