Can you cast a card exiled with Sunforger by paying its mana cost instead of using Sunforger's effect?

Sunforger's activated ability lets you unattach it, search your library for a white or red instant with mana value 4 or less, exile it, and then cast that card this turn without paying its mana cost. The card is exiled specifically as part of this ability's resolution, and the only casting permission granted is the 'without paying its mana cost' clause.

Under CR 601.1 and 116.1, you may only cast a spell when you have explicit permission to do so. An exiled card has no general rule allowing it to be cast simply because it is in exile — you need a specific effect granting that permission (CR 400.1, 406.3). Sunforger grants only the 'without paying its mana cost' permission, not an additional option to pay the mana cost instead.

Because there is only one casting permission attached to the exile (casting without paying its mana cost), you must either use that permission during the turn or not cast the card at all. You cannot substitute an alternative payment — CR 702.2c (which covers 'without paying its mana cost') does not allow you to choose to pay the cost instead; the alternative is the only avenue available.

Concrete example: You activate Sunforger and exile Lightning Helix. You may cast Lightning Helix for free that turn. You cannot say 'I'd rather pay {R}{W} for it normally' — the only permission in existence is the free cast, so it's free-cast or nothing.

If for some reason you choose not to cast the card that turn, it simply remains in exile with no way to cast it later, since no further permission is granted by Sunforger or any general rule.