Can you day hike into the Core Enchantments on a Colchuck Zone permit?

Yes. Day hiking in the Enchantments is permit-free, so a Colchuck Zone permit lets you camp at Colchuck Lake and push up Aasgard Pass into the Core lakes as a day trip, returning to your tent at night. That single fact is why the Colchuck Zone is the most useful backup in the whole permit system.

The odds case is strong too. The Core runs around 2% in the advance lottery, while the other zones run several times better. You are trading a night among the upper lakes for a dramatically higher chance of actually going.

The day-trip math is the hard part. From a camp at Colchuck Lake, Aasgard Pass climbs 2,200 feet in about a mile of loose rock with no maintained trail. Getting up, touring lakes like Inspiration and Perfection, and getting back down that scree before dark is a full, serious day. Start at first light and treat the descent with respect, since that is where tired legs cause problems.

Standard Enchantments rules apply in the Colchuck Zone: no dogs, no campfires, group size capped at 8, and mountain goats around camp that should never be fed. Colchuck Lake itself, sitting under Dragontail Peak, is hardly a consolation prize.