Das Urzeitwesen des Anbruchs

If the target of The Dawning Archaic's last ability has {X} in its mana cost, you must choose 0 as the value of X when casting it without paying its mana cost.

To determine the total cost of a spell, start with the mana cost or alternative cost you're paying, add any cost increases, then apply any cost reductions (such as that of The Dawning Archaic's first ability). The mana value of the spell is determined by only its mana cost, no matter what the total cost to cast that spell was.

You cast the spell while The Dawning Archaic's last ability is resolving and still on the stack. You can't wait to cast it later in the turn. Timing restrictions based on the card's types are ignored. If you cast a spell "without paying its mana cost," you can't choose to cast it for any alternative costs. You can, however, pay additional costs. If the spell has any mandatory additional costs, you must pay those to cast the spell.