Quandrix, the Proof

A spell's mana value is determined only by its mana cost. Ignore any alternative costs, additional costs, cost increases, or cost reductions.

If the card has {X} in its mana cost, you must choose 0 as the value of X when casting it without paying its mana cost.

The mana value of a split card is determined by the combined mana cost of its two halves. If cascade allows you to cast a split card, you may cast either half (as long as the resulting spell would have lesser mana value than the spell with cascade) but not both halves.

If a spell with cascade is countered, the cascade ability will still resolve normally.

Cascade triggers when you cast the spell, meaning that it resolves before that spell. If you end up casting the exiled card, it will go on the stack above the spell with cascade.

You exile the cards face up. All players will be able to see them.

Not only do you stop exiling cards if you exile a nonland card with lesser mana value than the spell with cascade, but the resulting spell you cast must also have lesser mana value.

When the cascade ability resolves, you must exile cards. The only optional part of the ability is whether or not you cast the last card exiled.

If you cast a card "without paying its mana cost," you can't choose to cast it for any alternative costs. You can, however, pay additional costs. If the card has any mandatory additional costs, you must pay those to cast the card.