For spells with in their mana costs, use the value chosen for X to determine the spell's mana value.

The top card of your library isn't in your hand, so you can't cycle it, discard it, or activate any of its activated abilities.

If you cast a mutating creature spell, you must choose a target for it before the triggered ability from Vivien's last ability puts a creature onto the battlefield. The creature you find can't be the target of the mutating creature spell.

Normally, Vivien allows you to cast the top card of your library if it's a creature card, it's your main phase, and the stack is empty. If that creature card has flash, you'll be able to cast it any time you could cast an instant, even on an opponent's turn.

After activating Vivien's last loyalty ability, the delayed triggered ability it creates will put a creature card onto the battlefield before the creature spell that caused the ability to trigger resolves.

You'll still pay all costs for that spell, including additional costs. You may also pay alternative costs such as mutate costs.

If a card in a player's library has in its mana cost, X is considered to be 0.

Vivien lets you look at the top card of your library whenever you want (with one restriction; see below), even if you don't have priority. This action doesn't use the stack. Knowing what that card is becomes part of the information you have access to, just like you can look at the cards in your hand.

If the top card of your library changes while you're casting a spell, playing a land, or activating an ability, you can't look at the new top card until you finish doing so. This means that if you cast the top card of your library, you can't look at the next one until you're done paying for that spell.