A variation on AotH and cave entrance.
In this case, if an AotH surrounds the mundane (only) entrance to a regionne, does the effect of the AotH extend into the regio?
I would say no. To have the Aegis cover the regio, I would want it cast within the regio. This goes double if the regio is bigger than the basic area of the Aegis.
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I would say it varies depending on the regio. some it will, some it will not. Magic isn't nice and predictable like science.
We've always treated regio levels as completely separate for Aegis purposes. The Aegis only exists on the level it was cast.
But I wouldn't have that as an absolute rule either, more of a default so long as it doesn't cause weirdness or become narratively inconvenient. I can see there being situations where it makes sense for the opposite to be true. In particular, if the regio level is smaller than a boundary (so contained within a single structure, or even just a single room) I'd probably allow an Aegis to propagate down, personally.
So for the cave I'd say yes if the cave is a standard structure or smaller, no if larger, with the possibility to overrule either if it helps the story.
Regiones definitely need their own Aegis. Regiones are, for all intents and purposes, separate spaces entirely from our world; it's just their entry conditions that are accessible within the physical space. If you cast an Aegis that covers a train station (not medieval but whatever this is an allegory) you don't extend protection over the other city the train tracks lead to.
On the other hand if you put an aegis in the first floor of a building the second floor may be protected. Not only does nobody know for certain what the N-space geometry of regios looks like, the very question is beyond the ability of people in a medieval paradigm to ask. As such I would expect what knowledge exists to be comprised of accumulated lore rather than models.