What do you think of the following when it comes to jurisdiction:

Papal mandate is a necessary condition but the instrumental cause is the sacrament.

In epistomology (i.e. major logic) we make a distinction between condition and cause.  Conditions are those that are necessary but do not actually cause the effect.  Example:  To make an oak tree we need sufficient conditions of humidity, soil, and warmth (sun) but these won't cause an oak tree to grow without an acorn which is the instrumental cause.  All these conditions won't make a pebble, a pumpkin seed, etc make an oak tree.
 
This opinion is based on historical argument.  Canon 26 of 4th Latern council admits the power of local to make other bishops for See's that have fallen vacant without prior Papal approval outside the Italian peninsula and this presumes their unity with the Holy See.

This permission was recinded by Pope Pius IX since communication by the late 1800's was possible. [But here is proof that even recently bishops had ordinary jurisdiction without expressed papal mandate; all the way up to Pius IX.  The change was ecclesiastical law not doctrine.]

In our day, we can't get prior papal approval, so the papal mandate is only implicit, since they profess union with Peter.  SSPX, SSPV, and some other independents do not since they act against the man they claim is pope or profess no moral certitude B16 is not pope.  

Sedevacantist bishops could be deemed Catholic but not with a formal diocese.  I would classify them as "bishops of noone"  See canon 309.  These are bishops of their congregations and priests but not of a diocese.

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Does anyone have a precise definition as to what exactly the "visibility" of the Church encompasses at a bare minimum say when it is hanging on the cross?

Is it:

1.  The Pope with the Bishops in union with him.
2.  The Four Marks (One, Holy, Catholic, Apostolic)
3.  Is it at least one Bishop (as a minimum).
4.  At least one Priest
5.  The valid Eucharist
6.  One or more Catholic lay-person with the faith?

I always thought it would be at least one Bishop.  Of course a true Catholic Bishop would be in union with the Papacy and he would be One, Holy, Catholic and Apostolic.

Just curious.

Not trying to start an unfriendly debate, though I do like a good debate.  

Most of us agree [or at least cannot honestly deny that the Doctors taught] that a public heretic cannot be Pope and that the Church must be visible.