On my theory, every one of the following statements are saying basically the same thing:

What if some were unfaithful? Does their faithlessness nullify the faithfulness of God? By no means!

But if our unrighteousness serves to show the righteousness of God, what shall we say? That God is unrighteous to inflict wrath on us? (I speak in a human way.) By no means! For then how could God judge the world? But if through my lie God’s truth abounds to his glory, why am I still being condemned as a sinner? And why not do evil that good may come?—as some people slanderously charge us with saying. Their condemnation is just.

through the law comes knowledge [as in direct experience] of sin.

For the law brings wrath, but where there is no law there is no transgression.

And the free gift is not like the result of that one man’s sin. For the judgment following one trespass brought condemnation, but the free gift following many trespasses brought justification.

Now the law came in to increase the trespass, but where sin increased, grace abounded all the more, so that, as sin reigned in death, grace also might reign through righteousness leading to eternal life through Jesus Christ our Lord. What shall we say then? Are we to continue in sin that grace may abound?

For apart from the law, sin lies dead. I was once alive apart from the law, but when the commandment came, sin came alive and I died. The very commandment that promised life proved to be death to me.

It was sin, producing death in me through what is good, in order that sin might be shown to be sin, and through the commandment might become sinful beyond measure. For we know that the law is spiritual, but I am of the flesh, sold under sin.

For God has done what the law, weakened by the flesh, could not do. By sending his own Son in the likeness of sinful flesh and for sin, he condemned sin in the flesh

What if God, desiring to show his wrath and to make known his power, has endured with much patience vessels of wrath prepared for destruction, in order to make known the riches of his glory for vessels of mercy, which he has prepared beforehand for glory— even us whom he has called, not from the Jews only but also from the Gentiles?

So I ask, did they stumble in order that they might fall? By no means! Rather through their trespass salvation has come to the Gentiles, so as to make Israel jealous. Now if their trespass means riches for the world, and if their failure means riches for the Gentiles, how much more will their full inclusion mean!

For if their rejection means the reconciliation of the world, what will their acceptance mean but life from the dead?

For just as you were at one time disobedient to God but now have received mercy because of their disobedience, so they too have now been disobedient in order that by the mercy shown to you they also may now receive mercy.

But the standard commentaries assign different topics. So:

What if some were unfaithful? Does their faithlessness nullify the faithfulness of God? By no means!

This is not about God being faithful to bring about the results that Israel had been commissioned to bring about (i.e. salvation to the world).  Rather, it simply means that God will continue to be righteous even when human beings aren’t.

But if our unrighteousness serves to show the righteousness of God, what shall we say? That God is unrighteous to inflict wrath on us? (I speak in a human way.) By no means! For then how could God judge the world? But if through my lie God’s truth abounds to his glory, why am I still being condemned as a sinner? And why not do evil that good may come?—as some people slanderously charge us with saying. Their condemnation is just.

God’s mercy is shown in the face of human sin.  “Doing evil that good may come” refers only to providing God an opportunity to forgive more.  It has nothing to do with actually using sin (especially Israel’s) to bring about world salvation.

through the law comes knowledge [as in direct experience] of sin.

We learn what sin is by reading the Law.  Why God gave the law and let it fail to do any real good is not answered or posited as a millennia long demonstration that human beings cannot save themselves.

For the law brings wrath, but where there is no law there is no transgression.

We learn right and wrong more plainly from the Law and yet still fail to obey it.  Why then even give the law?  Why increase wrath?  Either simply as a lesson against moralism or else simply to provide more sin to forgive.

And the free gift is not like the result of that one man’s sin. For the judgment following one trespass brought condemnation, but the free gift following many trespasses brought justification.

We are not to think that the parallelism teaches that as the one trespass led to judgment so the many treaspasses culminated in a free gift resulting in justification.  Rather, the judgement was executed on one trespass but the free gift of justification was executed on the guilt of many trespasses.

Now the law came in to increase the trespass, but where sin increased, grace abounded all the more, so that, as sin reigned in death, grace also might reign through righteousness leading to eternal life through Jesus Christ our Lord. What shall we say then? Are we to continue in sin that grace may abound?

Why did God want to increase the trespass?  Not because it was necessary to produce abounding grace, even though the question in 6.1 would lead one to think that the implication is exactly what hearers found in the statement.  No, he simply did it but then did more by grace.  Romans 6.1 is really asking if we should continue to sin to give God more opportunities to forgive us, not if we should sin to make grace abound further on the world the way that the trespass of rejecting Jesus resulted in abounding grace.

For apart from the law, sin lies dead. I was once alive apart from the law, but when the commandment came, sin came alive and I died. The very commandment that promised life proved to be death to me.

This is about a personal struggle with sin, not about how the Law in fact historically increased trespasses in Israel.

It was sin, producing death in me through what is good, in order that sin might be shown to be sin, and through the commandment might become sinful beyond measure. For we know that the law is spiritual, but I am of the flesh, sold under sin.

Again, this is about a personal struggle that happens to sound like the description of Israel’s and the world’s history in Romans 3-5.  Why God wants to aggravate sin in and individual, and how this description if accurate of 1) a regenerate person who cannot gain any victory despite Romans 6 and 8, or 2) an unregenerate person who nevertheless loves God’s law in his mind, is a matter of continuing controversy.

For God has done what the law, weakened by the flesh, could not do. By sending his own Son in the likeness of sinful flesh and for sin, he condemned sin in the flesh

This restatement of propitiation (Romans 3.21ff) could have been done at any time in world history and has nothing to do with the laws role in history of aggravating sin and increasing the trespass.

What if God, desiring to show his wrath and to make known his power, has endured with much patience vessels of wrath prepared for destruction, in order to make known the riches of his glory for vessels of mercy, which he has prepared beforehand for glory— even us whom he has called, not from the Jews only but also from the Gentiles?

The fact that Paul will later write that unbelieving Israel has only be partially hardened should not be permitted to interfere with this passage.  God desired to show his wrath, not in how he set forth Jesus as an propitiation, and then declared that event in his Gospel in all the nations, but in his desire to harden and punish Israel for unbelief.

So I ask, did they stumble in order that they might fall? By no means! Rather through their trespass salvation has come to the Gentiles, so as to make Israel jealous. Now if their trespass means riches for the world, and if their failure means riches for the Gentiles, how much more will their full inclusion mean!

This is narrative does not include the fact that Israel rejected and crucified Jesus but only that they rejected and persecuted the church later and thus jump-started Gentile evangelism sooner than the believing Jewish Church would otherwise have planned.  It has nothing to do with the mocking question in Romans 3 “let us do evil that good may come” and has nothing to do with the “the free gift following many trespasses” in Romans 5.  It is all entirely unrelated.

For if their rejection means the reconciliation of the world, what will their acceptance mean but life from the dead?

This is again, not about how Israel’s unbelief led to the propitiation necessary for world salvation, but only about how Gentiles got evangelized in the process of unbelieving Israel driving out believing Jews.

For just as you were at one time disobedient to God but now have received mercy because of their disobedience, so they too have now been disobedient in order that by the mercy shown to you they also may now receive mercy.

Same as above.

But is this really credible?  Is Paul dealing with a bunch of totally different topics that just happen to sound so similar?

Offering an elegant and simple solution is not absolute proof of veracity.  But I don’t see there is any way one can deny that my solution is much more simple and elegant than the ad hoc explanations that are put forward in the commentaries.