You can't really say that the breakeven point is X number of days, as it depends more on the number of trips and the length of each trip.
For example, the DVC Gold pass is $699 (all prices excluding taxes). A 10 day park hopper (the equivalent ticket to an AP) is $588 (May 9, 2020 start date), and a 1 day ticket for the day time is $195. So in this case, 11 days of tickets is more expensive than the Gold AP. BUT - a 3day Park Hopper during Christmas week is $506 and a 1day hopper in May is $195 for a total of $701 for a total of 4 days.
So as you can see, you can't really say that breakeven is X days. You can say 2 trips provided each of those trips is 3 or more days long.
Since there's too many potential situations; I'll try to keep it simple.
Apples to apples, a DVC AP allows you to go whenever and to park hop. So we'll compare it to a flex/park hop ticket. As such, there's only 3 potentials
1) 11 Consecutive Days to Breakeven
If we're talking consecutive days, ten days is the max for a ticket. It just happens that gate price on a 10 day, park hopper flex ticket is roughly $670. Considering the Gold AP is $699 currently, you would breakeven on the 11th day. And since there isn't an 11 day ticket, you'd have to buy a second ticket.
2) 4 Single Days to Breakeven
If we're talking single days, it would cost roughly $230 at the gate for a flex park hop which means 4 days/trips is breakeven.
3) 2 Trips to Breakeven
Every thing else in between is simple as gate prices for a 2 day flex hop ticket is roughly $350 and $660 for a 9 day flex hop ticket. Any 2 combinations of those will put you over $699.