Rashmi, Eternities Crafter

If you cast a card "without paying its mana cost," you can't choose to cast it for any alternative costs, such as emerge costs. You can, however, pay additional costs. If the card has any mandatory additional costs, such as that of Cathartic Reunion, those must be paid to cast the card.

For spells with X generic mana in their mana costs, use the value chosen for X to determine the spell's mana value. For cards in your library with X generic mana in their mana costs, X is considered to be 0.

If Rashmi's ability resolves and the spell that caused it to trigger has been countered, use that spell's mana value as it last existed on the stack to determine whether or not you may cast the top card of your library.