let's say you cast force cage and made it permanent with permanency, either in an area of natural anti-magic, or in an area of mythic anti-magic field that allows permanency, regardless of how it was done, can you use a rod of cancellation in an anti-magic field to destroy the forecage? if it can cancel a Sphere of Annihilation that even a mage's disjunction has no effect on I assume it would.
Antimagic field spell would suppress the permanency effect for it's duration. If you were in an area of natural antimagic, you wouldn't be able to cast it in the first place.
Rod of Cancellation wouldn't work, I agree. I think artifacts are not affected by Antimagic Field spell, but I can't remember where I read that.
Rereading the antimgaic field spell, it specifies it doesn't work against artifacts. It's unclear if a dead magic or antimagic zone that is naturally occuring will affect artifacts or not.
So let's break this down:
You have a force cage with a permanent duration (let's assume this is allowed).
It is located inside an area of mythic antimagic field spell (natural antimagic or dead magic will prevent permanency for sure, unclear if force cage can be cast into a "natural" atimagic field but if dead magic I would say no). A regular casting of antimagic field spell will also prevent permanency so it would have to be mythic.
Then rod of cancellation doesn't work because of antimagic or dead magic.
Sphere of annihilation works because artifact (but unclear if it works in a dead magic zone or "natural" antimagic) field.
The main problems we have are we don't know if a naturally occurring antimagic field has the same exact functionality as the spell version. I have also seen dead magic areas described within the Pathfinder setting, but no explanation mechanical what that entails. It might a colloquial way of referring to natural antimagic field areas or it might be different.
Agreed, because the rod of cancellation won't function.
Regardless of antimagic field spell, "natural" antimagic field, or dead magic zone none of them would allow the rod of cancellation to function.
An AMF would suppress the Permanent Forcecage until the AMF ends, but it wouldn't end the Permanent Forcecage as if it was dispelled. As soon as the AMF ends, the Permanent Forcecage would return as normal. The same would happen if you took a Rod of Cancellation into an AMF; it would suppress the magical capabilities of the Rod until you take it back out of the AMF.
The force cage part wouldn't be affected, because it works like wall of force which isn't affected. But permanency would be...if that means the duration suddenly starts ticking till suppressed or is immediately suppressed is unclear.
Regardless, rod of cancellation isn't going to do anything.