1) An auxiliary bishop has no office at all;

He's still a bishop.  The spiritual powers (in a crisis), are WAY more important than the human office.  That's why an excommunicated priest can hear confessions in an emergency, because his spiritual powers persist even when he's considered cut off from the Church.

2) An auxiliary bishop has no formal apostolicity (i.e., he has only material apostolicity = episcopal succession, but not formal apostolicity, which adds to material apostolicity jurisdiction/office).

You have it backwards.  His spiritual powers are the formal apostolicity; his human/jurisidictional office is the material side of things.  His spiritual powers can never be taken away (the Divine/Apostolic nature); his human/jurisdictional office depends on the current rules of the time.  The whole idea of an auxiliary bishop is a modern concept anyways.

If that weren't so, every schismatic bishop would be part of the ecclesia docens (and part of the visible Church), which is crazy.

I don't know about this, but if they were valid, they are still successors of the Apostles and can hear confession in an emergency. 

All of the Trad Bishops are successors of the Apostles and are part of the visible church, whether novus ordo-ites want to admit it or not.

You can't simplify things jesy because they are more nuanced than you like, or these are the types of results you get.

You missed my point (which is normal).  In peaceful times, human/church rules apply.  But you gave the hypothetical example of a crisis where only 1 auxiliary bishop was left on earth.  In that case, human/church rules no longer apply and due to the emergency, you go to basic principles, i.e. sacramental/Divine rules.

Non-sequitur.

It's evident from the context that you have no idea what this Latin term even means.

He's still a bishop.  The spiritual powers (in a crisis), are WAY more important than the human office.  That's why an excommunicated priest can hear confessions in an emergency, because his spiritual powers persist even when he's considered cut off from the Church.
You have it backwards.  His spiritual powers are the formal apostolicity; his human/jurisidictional office is the material side of things.  His spiritual powers can never be taken away (the Divine/Apostolic nature); his human/jurisdictional office depends on the current rules of the time.  The whole idea of an auxiliary bishop is a modern concept anyways.
I don't know about this, but if they were valid, they are still successors of the Apostles and can hear confession in an emergency. 

All of the Trad Bishops are successors of the Apostles and are part of the visible church, whether novus ordo-ites want to admit it or not.
You missed my point (which is normal).  In peaceful times, human/church rules apply.  But you gave the hypothetical example of a crisis where only 1 auxiliary bishop was left on earth.  In that case, human/church rules no longer apply and due to the emergency, you go to basic principles, i.e. sacramental/Divine rules.

I agree with you completely.  There's a notion of authority inherent in the notion of the episcopacy.  In fact, the Greek term means "overseer", the idea being that they have oversight over priests, and the essential form of the ordination refers to the principle or "chief" power of the priesthood, meaning the the recipient would be a chief over other priests.  It's almost of the essence of the episcopacy to have the authority, but it remains latent until it's activated by the reception of ordinary jurisidiction.

1) Non-sequitur.  The question is whether the total absense of any bishops teaching or governing (i.e., with formal apostolicity) is compatible with the Christ's constitution of the Church.  Clearly, it is not.

If one bishop is alive, auxiliary or diocesan, the Church would (arguably) still exist.  An auxiliary bishop has teaching/governing authority inherent in his sacramental powers.

2) I have no idea what you are talking about here, and I suspect that neither do you.  Material apostolicity = episcopal succession.  Formal apostolicity = the addition of jurisdiction/office.  Nobody disputes this (not even sedes).  Visit the CE under "apostolicity."

3) Schismatics are not successors of the Apostles, despite material apostolicity.  See aforementioned article in the CE. 

Sorry if I screwed up the terms but my point is, the above distinctions between material/formal apostolicity go out the window in a crisis.  I don't want to discuss schismatics because it muddies the water.

4) Not even in a state of necessity, does a schismatic bishop gain office/jurisdiction/formal apostolicity.  All he gains is the validity of his sacraments (which come not from his power of order, but from the request of the faithful, and are supplied not by any latent powers of order, but by the Church).

I don't want to discuss schismatics because it muddies the water.

I agree with you completely.  There's a notion of authority inherent in the notion of the episcopacy.  In fact, the Greek term means "overseer", the idea being that they have oversight over priests, and the essential form of the ordination refers to the principle or "chief" power of the priesthood, meaning the the recipient would be a chief over other priests.  It's almost of the essence of the episcopacy to have the authority, but it remains latent until it's activated by the reception of ordinary jurisidiction.

Me too.  Even priests participate in the hierarchy of jurisdiction.  When they receive a celebret from a bishop they have ordinary jurisdiction in the internal forum for that territory.  When they hear your confession and assign a penance, that penance is binding on you.  So an auxiliary bishop has at least that much ordinary jurisdiction even if technically he is not an ordinary (and neither is the priest).  Even a priest who doesn't have a celebret still participates in the hierarchy of jurisdiction when he hears a confession in another territory due to common error (i.e. he didn't know he wasn't authorized to hear the confession).  So their are different levels of participation from minimum (the priest without a celebret) to maximum (universal and ordinary of the pope).  But they are all still part of the hierarchy both of orders and of jurisdiction.  Even the cleric without any minor orders still is a member of the hierarchy.

1) Non-sequitur.  The question is whether the total absense of any bishops teaching or governing (i.e., with formal apostolicity) is compatible with the Christ's constitution of the Church.  Clearly, it is not.

2) I have no idea what you are talking about here, and I suspect that neither do you.  Material apostolicity = episcopal succession.  Formal apostolicity = the addition of jurisdiction/office.  Nobody disputes this (not even sedes).  Visit the CE under "apostolicity."

3) Schismatics are not successors of the Apostles, despite material apostolicity.  See aforementioned article in the CE. 

4) Not even in a state of necessity, does a schismatic bishop gain office/jurisdiction/formal apostolicity.  All he gains is the validity of his sacraments (which come not from his power of order, but from the request of the faithful, and are supplied not by any latent powers of order, but by the Church).

For #2 above, we are supposed to believe that the pagan-idol-worshipping-adultery-abortion-sodomy-promoting apostate Bergoglio has apostolicity of doctrine?  Read your own reference material.  They point out that apostolicity of doctrine is guaranteed by apostolicity of mission.  So you would have us believe that Bergoglio is teaching the same doctrine that Christ taught to the apostles?  You need to get your head checked.  Also, it's not even clear that Bergoglio is a priest, never mind a bishop.  So how exactly could he be sent if he doesn't profess the faith, doesn't preach the faith, and doesn't even have valid holy orders?  You are arguing technical points of theology while your "pope" is worshipping pagan idols.  It makes you look like a fool.

What in the wide world of sports is "apostolicity of doctrine?????????????"

I think you made up a new term, in order to attack it.

If you're not careful, you'll outmaneuver yourself.

Read the CE article that you referenced.

SJ with an immature response.  I’m shocked!  (sarcasm alert)