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Now, the fallen world.
1. I have never really thought this question through in its potential complexity. I know the Augustine/Wesley schtick, of course, the one we read in all the theology textbooks and Wesleyans used to preach. God created the world perfect. Adam sinned. Sin came on all the world, both the creation and humans. We now have a sin nature that makes us sin. Christ's death atones for the sin acts we do, the Spirit removes our sin nature so that we do not have to sin any more.
This is a mixture of Paul, Augustine, and Wesleyans. What Paul taught, I believe, is that the creation has been under the power of Sin since Adam. With regard to the world, it became subject to corruption and decay. With regard to humanity, the weakness of our flesh makes us susceptible to the power of Sin. The Spirit empowers us to live righteously and fulfill the law of love.
But is this the greatest common denominator of Scripture? It is the model that won out in the Western church, the Catholic side. It is the model that has dominated Protestantism because Luther's base camp was in Paul. But Paul is only one voice in Scripture and the West is not the whole church.
2. Adam plays no significant role in the theology of the OT itself. That is to say, Genesis 4 through Malachi do not use Adam to make any argument at all. He is generally absent from those pages. Here you have to remember that most of us do not read the Bible for what it actually says. The question of Adam's significance for the OT is not debatable--he simply is not given a thought from Genesis 4 on. But we might read him into the pages because of the glasses we wear when we read the OT.
Similarly, Adam plays no clear role in the NT outside of Romans 5, 1 Corinthians 15, and possibly Hebrews 2. Again, most people--including pastors--don't read the Bible. They read into the Bible. We have no evidence that Adam featured at all in the theology of Matthew or Mark or John or James. Our default reading of the Bible is through theological glasses, and we see things that weren't originally there.
Did Jude believe in Adam's Fall? We don't know. He quotes 1 Enoch, and the Fall in 1 Enoch came not with Adam but when angels sleep with human women in Genesis 6, had giant offspring, from whose dead bodies the spirits of demons originated.
I'm digressing simply to point out the extent to which our organization of Scripture is dependent, not on Scripture itself, but on the traditions of organization we have inherited growing up in the church.
3. What most Scripture does assume is that we are separated from God and in need of atonement and redemption. Even here, however, the assumptions of Leviticus are one strand within the OT and one that stands in some tension with the prophetic strand. Jeremiah 7, Micah 6, Isaiah 1 all deny the importance of sacrifice over and against a more fundamental obedience to God in relation to how you treat other people. In the NT, Hebrews makes explicit the end of the sacrificial system.
The proximity to God in Scripture still varies. In the OT, Israel stands closer to a right standing with God than the pagans do. The NT sets us on a trajectory in which all people stand equally close to such a right standing. It is not that humanity has come closer to God but God has come closer to humanity in Christ.
So we can at least say that the common denominator of Scripture is that the default state of humanity is one of separation from God. It is a state of sinfulness in need of reconciliation and redemption.
4. But what is this sinfulness? In the OT Law, the need for sacrifice primarily has to do with "ceremonial impurity." Theoretically, "high handed" sins resulted in death, and since most of the OT does not have a conception of an afterlife, death was the ultimate purging.
I don't think there is any evidence that the standard of sin was absolute perfection. This perspective comes from a couple obscure verses in Paul, lifted to inappropriate centrality. The standard of sin, in the NT, becomes motive in relation to love. Jesus points in this direction in the Sermon on the Mount. John's writings support this perspective. Paul's writings support this perspective.
The Greatest Common Denominator of Scripture might thus view the understanding of the Law so prevalent in Christianity today as an immature one that has not fully grown into the NT. Sin has always been, for God, any act of will that wrongs another, including God. One can wrong another person both intentionally and unintentionally, but it is the intentional ones that most concern God.