Magical Theorist

Sorry for the slow response. it's because from memory it seemed like a tough one. But on review I think the FAQ is pretty clear that the cost remains zero:

(2.16) “Minimum of 1”
If an effect allows a player to reduce a card’s cost, it cannot be lowered below one if the reducing effect has stipulation “to a minimum of 1.” If a players uses a combination of effects to reduce the cost of a card where one of the effects has the “minimum of 1” stipulation, and the other reducing effect does not, the “minimum of 1” takes precedence.

So the "minimum of 1", if present affects all cost-reducers. And things that other reducers may have brought to zero (even by themselves) will only reduce to 1 in its presence. But that is still a limitation on lowering.

Read the underlined part with a zero cost in mind and you should see that since the cost is already at or below 1, it cannot be lowered further.

There is nothing to suggest that these cards can raise the cost.

So, the "limit of 1", rather than creating a strict cost limit, is merely "infecting" all other cost reduction effects, with its restriction, and things that naturally cost 1 or less will essentially be unaffected.

Thanks, RP.  I'd definitely be surprised if someone made me drain a domain for the Mists of Lethe (for example), but my instincts are bad on this game and it wouldn't shock me if I was wrong again.  Still, the Magical Theorist wouldn't be very good at his job if there were actually times he worked against you!  

I remember the case you cited above.  Special Agent Clarkston and Lieutenant Wilson Stewart definitely don't play nice all the time.