But in 1703 the Holy Office rejected this and stated that explicit belief in the Holy Trinity and Incarnation are necessary "by necessity of means", in other words, regardless of your sincerity, if you do not EXPLICITLY believe these core things, you cannot have supernatural faith.  That squarely rejects Rewarder God theory.

This has already been addressed.  Yet you repeat it as if it hasn't.

If I believed that people who do not have Catholic faith can be saved, then I'm going right back to the Conciliar Church and abjuring my schism.  There's no alternative.

You're in schism? How so?

Are the posters here (Lad, PV, etc) asserting that the Universal Ordinary Magisterium has been teaching heresy since the 16th century (vs Vatican II)? And if so, isn't that an assertion which translates into a defectible/defected Church?

2Vermont,
Here's my understanding of this question.
1.  Since the 1600s, the main instigators of changes to Traditionally-held views on EENS were theologians.  To some degree, it is part of their job to speculation and ask difficult questions.  On the other hand, if they are not corralled and kept quarantine from "normal avg Catholics" (who are not used to theological speculation) then such "novelties" can cause much scandal and confusion.
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2.  I'm not a historical expert on the Jesuits but it seems they were very involved in the push to water-down EENS (my guess is they were infiltrated) and they were eventually suppressed by one of the popes in the 1800s.  So, the infiltrators succeeded on 2 fronts - attacking EENS and weakening the Jesuits (who were, for the most part, great defenders of the Faith and anti-freemasonic).  The Jesuits were eventually un-suppressed but damage had been done to their reputation and also more infiltrators had joined.
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3.  The point is - theological opinion is not a "teaching" of the Church.  Theologians have no authority, no jurisdiction, no doctrinal weight.  Their job is to support the Vatican and do research for any of the pope's needs.  For example, before papal infallibility was defined at Vatican I, the pope had the best theologians and historians research the issue to make sure that the definition was clear, historically-accurate, etc.
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4.  Even Cardinal Dulles says that Pope Pius IX, when speaking on the invincibly ignorant, was following "current opinion" on the topic.  Pius IX was not "teaching" authoritatively, nor with any doctrinal weight.  He was using his personal capacity as a private theologian; he was not speaking in his formal, Apostolic Authority, head-of-the-church mode.
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5.  As has been shown, Pius IX's writings on the "invincibly ignorant" were interpreted liberally and incorrectly from his intentions.  He corrected his writings but such were ignored or not distributed as widely as the original.
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6.  Let's take a common example (below) and show how the Modernists have used their devilish trickery and cunning to attack dogma.  The below example will also show why catechisms can never be infallible - because doctrine requires precise terms and concise language.  Since the catechism is meant for children, or a child-like understanding of the Faith, the use of precision and concise language would hinder children from learning because it would require the use of more complex language and a larger vocabulary.
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7.  Let us remember a FACT of history, which Pope St Pius X lamented and which even freemasons have admitted - if Pope St Pius X had not become pope in 1903, the church was so liberalized and infiltrated then, that V2 would've happened at that time.  Let that sink in.  Pope St Pius X openly lamented that he was "all alone" and "surrounded by" modernists.  If he had not been elected, miraculously, then V2 would've happened 60 years earlier.
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8.  Let's also not forget that the pope right before St Pius X was Bl Pius IX, who was almost killed by freemasons, and was instead imprisoned in the Vatican.  The point is, freemasons were everywhere - all over Italy, Europe, and yes, even in the Vatican.  And they were able to sneak modernist language into docuмents like the catechism (see below) just like they did at V2.  All of the words/phrases I will highlight are general; they aren't precise theological words.  Therefore, these imprecise words allow Modernists to "re-define" them and thus, water down doctrine.
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From the Catechism of Pius X:

Q29. But if a man through no fault of his own is outside the Church, can he be saved?
A. If he is outside the Church through no fault of his, that is, if he is in good faith, and if he has received Baptism, or at least has the implicit desire of Baptism; and if, moreover, he sincerely seeks the truth and does God's will as best he can such a man is indeed separated from the body of the Church, but is united to the soul of the Church and consequently is on the way of salvation
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Ok, so let's look at the problematic phrases, which are open to interpretation (and which Modernists like Rahner, Dulles, etc) all used in their successful attempts to water down EENS to pave the way for V2's ultra-heresies.
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1.  Good Faith - what is this?  How is it defined?  How does one know who has it or not?  None one knows.  It's subjective, it's open-ended, it's open to sentiment and emotion.
2.  Implicit Desire - what exactly does this mean?  Is this St Alphonsus' implicit desire?  Or Rahner's "anonymous catholic" version?  It's not precise at all.  Dangerous theology.
3.  Sincerely seeks the truth - This seems pretty straightforward, right?  Except that only God knows who is sincere or not, since none of us can read hearts.  So, again, dangerous theology.
4.  God's will as best he can - Same as above; only God can know this.  Doctrine is black and white.  This is theological opinion and is dangerous for most people to think about.
5.  On the way of salvation - What does this even mean?  A liberal/modernist would define this as the person can "be saved in his current non-Catholic state".  An orthodox/traditional catholic would say that this means the non-Catholic is "progressing towards the full truth of the Church, which God will give to him, if he continues searching."
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Do you see the MAJOR DIFFERENCE between these 2 interpretations?  Do you see the major problem with imprecise language?  Don't you see how this was the exact same method used at V2?
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So, to answer the question...Did the Church actually "teach" error from the 1600s onwards?  No.  Theologians speculated and such opinions became widely known, which caused confusion.  And ambiguous and imprecise language was slowly introduced by masons and modernists into catechisms and other docuмents, just like at V2.  The only difference between then and now is that God gave the Church a saint in Pius X, who stemmed the tide of error, or else V2 would've happened in 1903.

Stanley, i'm done explaining things to you because you never agree or disagree with what I write.  You just continue to complain that you're being misinterpreted.   I want to know what you think, in your own words.  Please do not post articles or excerpts of catechisms, but speak plainly and tell us what you really think.

What is freely affirmed is freely denied.
And no, I've not said I'm being misinterpreted per se. I've said that Laddy, and to a lesser extent you, are attributing things to me that I didn't say, and I haven't said what I believe or don't believe.
The responses to posting excerpts from catechisms, encyclicals and so on has generated interesting responses. "That's not dogma", "that's from the expository part of Trent not the canons", "that's not infallible", and mental gymnastics to rephrase texts to make them say something different than their plain sense.

And according to you and Laddy, I'm a modernist Pelagian who follows St. Thomas.

LOL.

We can’t have a reasonable debate if you don’t explain yourself.  If you want to play childish games of hide-and-seek with your views, go ahead.  

And according to you and Laddy, I'm a modernist Pelagian who follows St. Thomas.

You're a self-avowed follower of St Thomas yet you also liberally interpret St Pius X’s catechism in a way that is contrary to St Thomas.  So, yes, you are a modernist because you hold both “a” and “not a” at the same time.  

We can’t have a reasonable debate if you don’t explain yourself.

Says the person who won't explain himself.

We should be able to have a reasonable discussion about doctrine without it being personal. But you and Laddy insist on making it personal with insults. THAT impedes reasonable discussion.

The fact is various church docuмents speak of invincible ignorance, baptism of desire, and implicit desire.

But none of them define very well (or at all) what these terms mean. What modernists do with that is not in the texts.

On the other hand:

Implicit desire just means they have a desire to get baptized but they haven’t openly told anyone (ie made it explicit).

That's also not in the texts.

You're a self-avowed follower of St Thomas yet you also liberally interpret St Pius X’s catechism in a way that is contrary to St Thomas.

The only "interpretation" of the St. Pius X catechism I did was to point out it teaches implicit BOD. Which it does, no? Is that "liberally interpret[ing]" the catechism?

I promise to do a better job of responding to your posts, Lad and PV, but, for now, are you saying that what is in these catechisms are actually teachings from theologians, not the Church?

Also, although I agree that catechisms are not infallible per se and that there can be error, I also believe that such an error must be immaterial to the faith.  These catechisms have been approved by popes to be used to teach the Faith.  How can there be something in them that is contrary to the Faith??

As for theologians teachings not being teachings of the Church I thought that there are a number of "theological notes" that determine whether they must be believed or not.

Are any of you familiar with the popular catechism "My Catholic Faith"? It has been around for a long time; the first printing was in the late 40's, I think, and has an imprimatur and nihil obstat. In many ways it is a good book for us to teach with and yet in the chapter on Baptism it suggests that according to some theologians there may be a fourth baptism called baptism by illumination.  Some of the faithful who lived around Powers Lake with Father Frederick Nelson (RIP) believed in this "baptism" and consoled parents who had lost a child in utero saying that if parents were to place their hands on the mother's belly and pray something like the Apostles' Creed the unborn would be baptized and saved. Now,don't ask me to explain something I don't understand and completely disagree with. I'm just relating what one of the believers tried to persuade me to believe.  But just look how far these theologians go and how much confusion they sow among the laity.