I meant that proving not all Christians always remain is not necessarily the same as proving some Christians ultimately fall away, because it's been conceded by some on your side that, sure, Christians don't always remain, that sometimes they walk after the flesh, but, they will argue, ultimately, Christ will uphold them and they will, ultimately, remain in Christ and walk after the spirit not after the flesh (as the argument goes: OSAS entails you will keep good works to the end). They don't concede instances of not remaining as necessarily proving possibility of ultimate not remaining.
I would never argue that. (Did I mention that I don't have a "side."

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I don't believe that one can be "in" one minute, "out" the next minute, and back "in" the next minute.
The doctrine you describe misconstrues Romans 8, thinking that when Paul mentions "walking after the flesh" he speaks about people who are living a "carnal" lifestyle. But Paul is not focused on sinful behavior in that context. Rather, he is focused on those who are overly obsessed with physical traits, such as being Jewish by birth or being circumcised.
The contrast is between our outward appearance and our inner selves. Outwardly, a Jew can be a male, a freeman, circumcised, a son of Jacob, a son of Abraham, someone who tithes, is devout, and keeps all the rules, among other things. If that person believes that outward appearance is the basis for God's favor, they are "walking according to the flesh." But if that person seeks God's favor through their faith, contrition, confession, their inward spiritual condition, and attitudes, they are "walking according to the spirit."
In Romans 2, Paul summarizes this principle, speaking about "true Jew." Notice how he compares and contrasts the inward man with the outward man.
Romans 2:28-29 For he is not a Jew who is one outwardly, nor is circumcision that which is outward in the flesh. But he is a Jew who is one inwardly; and circumcision is that which is of the heart, by the Spirit, not by the letter; and his praise is not from men, but from God.
To understand how Paul uses the phrase "according to the flesh," we can return to chapter one of his epistle.
Romans 1:3-4 concerning His Son, who was born of a descendant of David according to the flesh, 4 who was declared the Son of God with power by the resurrection from the dead, according to the Spirit of holiness, Jesus Christ our Lord,
Here, we see how Paul employs the phrase "according to the flesh," which answers the question, "Who is Jesus?" The answer is, "He was born of a descendant of David." He has the right to rule as the King of Israel because he is a direct descendant of King David.
In his second epistle to the Corinthians, Paul first acknowledges that we have known Christ according to the flesh, i.e as the son of David, then he says that we have dropped that distinction altogether and no longer know anyone that way.
2 Corinthians 5:16
Therefore from now on we recognize no one according to the flesh; even though we have known Christ according to the flesh, yet now we know Him in this way no longer.
Based on usage, we can rightfully conclude that Paul uses the term "according to the flesh" to indicate someone's identity. What is your name, job, family line, address, nation of origin, religion, and other such things? The answer to these questions identifies a person "according to the flesh." Jesus was a son of David according to the flesh; Abraham was Paul's forefather according to the flesh.
1. No, my concept of remaining comes from all of Scripture.
You were quoting the gospel of John was my point.
2. Likelihood is not addressed in that portion, only the possibility of remaining or not remaining is addressed.
The point is whether God saves or we save ourselves. If God saves us then all contingency is removed.
No, contingency is not even addressed in v1.
My point exactly. Your argument stops at verse one.
Yes, and Paul says we are to test ourselves to see if we are in the faith,
You might be thinking of Peter. Paul doesn't ask us to test whether we are in the faith. In Fact, Paul argues that God tests us to see if we will remain in the faith or not.
It's as clear as day : they were not remaining in Him
What isn't clear is whether they were given the Spirit of God and the Spirit of Christ.
We've gone over this, and, yes, every indication is he is a believer, because : a. if he were an unbeliever, he would already have been condemned (Paul wouldn't have said, "now that he has broken the rule 'each man is to be fully convinced in his own mind', he is condemned"), b. he is breaking the rule that is binding on believers (v5 "each man is to be fully convinced in his own mind... what ever does not proceed from faith is sin"--because "God's righteousness is revealed from faith to faith"), so he is condemned (another example is in 1 Co 8, where the believer who goes against what he believes is correct, partaking in idolatry, leads to his condemnation)--and all teachers agree with me, because it is obviously the case.
Where did you get the idea that Paul created a "binding rule?"