'Positive doubt' is a canonical term applied after a Rite or the administration of a sacrament has been found, by the correct authorities and in accordance to specific guidelines , to be lacking in some essential element to render it possibly invalid. 

Correct.  Many Trad clerics (including +Tissier from the sspx and Fr Scott) and many others who were non-sspx, have reached the same conclusion, since the 1970s.  This is not some new development.

Now you are correct when you state that very significant changes have been made to the episcopal rite of consecration, but has it been established as invalid? No.

These very significant changes = positive doubt.

Positive doubt = possibly valid.

Has a 'positive doubt' been established? Certainly there are serious reasons to cause one to have doubts - not knowing one way or another - but has a 'Positive doubt' been established?

Positive doubt = serious reasons to cause doubt.  Yes, these serious reasons establish doubt.

You make mention of the Anglican Rite that Pope Leo XIII declared to be "absolutely invalid, null, and void" but this has no bearing on the Catholic New Rite as it have not been declared upon.

The problems with the anglican rite are also present in the new rite.  This is part of the "serious reasons" why there are doubts.

In fact, as we have established, even his Lordship Williamson believed the New Rite to be valid. Why was that?

He's in the minority opinion.  But even he conditionally consecrated +Vigano, so he obviously had some doubts.

We have also established that His Grace Archbishop Lefebvre only re-ordained when each individual case was investigated on its own merits; not simply because they were ordained in the New Rite. Why was that?

In the time when the new rites came out, til Lefebvre's death (60s-early 90s), there were still TRUE RITE bishops who were ordaining.  These ordinations would've been valid.  But now...all the TRUE RITE bishops (pre-V2) are dead.  You can no longer compare +ABL's time to ours.  

Now along comes Mr. Ladislaus who declares that the SSPX are all modernists because they do not treat the New Rite of Ordinations as invalid as he wants them to.

  Positive doubt does not declare anything invalid.  You continue to misinterpret the argument.  Are you doing this on purpose?

If no 'positive doubt' has been established in the New Rite,

There are many, many reasons to establish positive doubt.  Go look up what Bishop Tissier said many times, and also go read the article posted from Fr Scott.  All these reasons still exist.

if it has not been declared as invalid,

Only the Church can declare it invalid.  This is stupid argument and has no place here.  All that is required is positive doubt, which is, AS CANON LAW SAYS, is to TREAT IT as invalid.  

The bar is pretty low for positive doubt, because The Church, through Canon Law, does NOT WANT THE FAITHFUL TO TAKE RISKS WITH SACRAMENTS.  Going to positively doubtful sacraments IS A RISK TO YOUR SALVATION.

then I as a Catholic must be content with this ruling trusting that the SSPX will make the right decision about each individual case as it has always done. 

1.  The new-sspx isn't infallible.
1b.  The new-sspx isn't the Church; they can't make "rulings".
2.  The new-sspx has changed their view on the new rites, in opposition to Bishop Tissier and Fr Scott (and others).
3.  The new-sspx has a similar view towards to the new rites as Indult communities and no longer agrees with the 'common opinion' of Traditional communities.

If you are content with placing your salvation in the hands of an organization which isn't infallible and which is no longer Traditional, then that's your decision.  But don't act like you weren't warned.  And don't act like your decision is based on canon law.  And don't act like your "contentment" to follow the new-sspx will absolve you from the moral consequences of going to doubtful sacraments. 

YOU have the job of saving YOUR soul.  The new-sspx can't save it for you.  I hope you will reconsider the above and pray about the huge, moral problems which positive doubtful sacraments cause.  Your duty to God outweighs your loyalty to the new-sspx.

'Positive doubt' is a canonical term applied after a Rite or the administration of a sacrament has been found, by the correct authorities and in accordance to specific guidelines , to be lacking in some essential element to render it possibly invalid. Now you are correct when you state that very significant changes have been made to the episcopal rite of consecration, but has it been established as invalid? No. Has a 'positive doubt' been established? Certainly there are serious reasons to cause one to have doubts - not knowing one way or another - but has a 'Positive doubt' been established?

You make mention of the Anglican Rite that Pope Leo XIII declared to be "absolutely invalid, null, and void" but this has no bearing on the Catholic New Rite as it have not been declared upon. In fact, as we have established, even his Lordship Williamson believed the New Rite to be valid. Why was that? We have also established that His Grace Archbishop Lefebvre only re-ordained when each individual case was investigated on its own merits; not simply because they were ordained in the New Rite. Why was that?

Now along comes Mr. Ladislaus who declares that the SSPX are all modernists because they do not treat the New Rite of Ordinations as invalid as he wants them to. And this is the crux of the matter isn't it? I am a traditional Catholic of the Roman Catholic Church. I am not a member of your personal church. If no 'positive doubt' has been established in the New Rite, if it has not been declared as invalid, then I as a Catholic must be content with this ruling trusting that the SSPX will make the right decision about each individual case as it has always done. It is the same of the New Rite Mass. You have declared it invalid. You have no right or authority to do that. You can think it if you want but you cannot impose that on other Catholics. Out of principle, I would never attend a New Rite Mass, but neither do I condemn a fellow Catholic for going to one. Bishop Williamson was of this mindset too. As I said, if it gives your personal peace of mind to formulate this list of transparency, then go for it, do your best - make sure you get those child abusers on there too. But don't use it as tool to bash the SSPX with. That's your own personal beef and has no place here.

Are they paying you to do this?

Monsignor Byrnes is in Georgia as of a few years ago.

"All that is required is positive doubt, which is, AS CANON LAW SAYS, is to TREAT IT as invalid."  Please may I have the full reference from Canon Law to ascertain the context.  

I can't find that in there, must be made up by some of the resident armchair theologians. But what is in there is this:
Canon 732 
§ 1. The Sacraments of baptism, confirmation, and orders, which imprint a character, cannot be repeated.
§ 2. But if a prudent doubt exists about whether really and validly these [Sacraments] were conferred, they are to be conferred again under condition.

Conditional ordinations are not required by law to be public.  

Canon 1007
Whenever ordination is to be repeated or any of the rites supplied, whether absolutely or under condition, this can be done outside the [usual] times and secretly.

Yes but....a positive doubt REQUIRES one to avoid the mass/sacrament AS IF IT WERE INVALID, so says canon law.

Absolutely wrong.  Positive doubts are based on evidence.  Positive doubts exist WHILE WE WAIT for the Church to make a ruling.
Go do what you want.  You're not a Trad, that's for sure.  Traditionalists have had positive doubts for 50 years.
Ok, until we receive a concrete decision from ORTHODOX rome (not new-rome), then the positive doubt remains.
The ORTHODOX and TRUE Church hasn't ruled on the matter, so positive doubt remains.
This sentence proves that you still don't understand what 'positive doubt' is.
Don't have this info handy at the moment.

Correct on every point.  No, it does not require ROME to determine positive doubt.  If someone saw the video of Bishop Neal Webster attempting to consecrate Father Pfeiffer, there's positive doubt right there.  If I found that some priest was using Welch's grape juice for Mass, I could go beyond positive doubt to even asserting straight-out invalidity.  Once the Church has defined what the matter and form are, any deviation from those suffice to establish positive doubt.  There are some cases where it might be questioned whether the doubt be positive or negative, and in those cases the matter might be referred to the Holy Office ... once that's orthodox and dependable, as Wojtyla actually declared one of the schismatic Rites to be valid despite its lacking an actual consecration.

BUT, even then, positive doubt suffices to impose on us the moral obligation to avoid a Sacrament, but a much lesser threshold than positive doubt can suffice to justify the conditional "re"-administration of the Sacrament.  You can't do it willy-nilly for no reason, but you can for any reasonable consideration.

So, for instance, with conditional confirmation.  Since there was a tendency for bishops in the Conciliar Church to mess with the essential form and at one point to even substitute olive oil with palm oil, that suffices to conditionally Confirm anyone coming over from the Conciliar Church.  I do not have to establish a postiive doubt for a specific case.  Sure, let me go do an investigation, costing thousands of dollars, to determine who did the confirmation, then try to interview him if he's alive, or find other witnesses, to establish whether or not that particular bishop may have messed with the form on that occasion and whether that diocese was using palm oil at the time, etc.  That's just ridiculous.  Of course, the SSPX claim to be "investigating" intention, which Pope Leo XIII explicitly states that the Church does not and cannot know, since it belongs propertly to the internal forum -- yet SSPX can somehow find out due to some amazing new investigative techniques.

Given the Novus Ordo propensity to dabble with Sacraments, I would say that the general state of the Novus Orod would suffice to conditionally confer any Sacraments confected within the Conciliar Church, even Baptism.

But of course, these are specific concrete cases ... and the status of whether a postiive doubt exists in a RITE would create positive doubt across the board.

So, for instance, before Pope Leo XIII made a ruling on Anglican Orders, there were probably already some theologians out there who held they were valid, while others that they were invalid, or at least doubtful.  That dispute would have sufficed to create conditions more than sufficient to conditionally ordain all Anglican priests converting to the Catholic faith, and even to REQUIRE it, since that would establish positive doubt, since some reasonable theologians could make weighty arguments asserting its invalidity.

Now, after the teaching of Pope Leo XIII, he declared them straight-out invalid so that no conditional form was necessary, and absolute ordination required.

So there are levels here ...

1) prudent doubt -> conditional PERMITTED
2) positive doubt -> conditional REQUIRED
3) positive invalidity -> absolute REQUIRED

In other words, even IF one claimed the threshold of positive doubt was not met (which is absurd on its face), prudent doubt would justify a conditional administration.

So why WOULDN'T the SSPX do it?  Quite simple.  They don't want to offend the Modernists.  Period.  So intead of ofending Modernists, they'll risk subjecting the faithful to possibly invalid Sacraments and require them to have their consciences troubled by the possibility.  So they throw the peace of soul and possible salvation of souls under the bus, sacrificing it on the altar of having some kind of regularization with the Modernists.  Period.  End of Story.

They deceive by falsely exaggerating the threthold require to administer conditional ordination, conflating the question of how re-administration of a character Sacrament would be a sacrilege (and ignoring the fact that these are CONDITIONAL administrations).  Those conditionals would be sinful if there was no prudent reason, but something trivial, or where someone did it for no reasons, "just in case".

Monsignor Byrnes is in Georgia as of a few years ago.

Thanks you.  I'll update his listing.  That's the kind of thing I was hoping to keep track of on this site, if possible.  In addition to getting their pictures, to track where they migth be.