Okay. I will try again.
Axiom is not the right word. Hypothesis is better. Let’s put it this way:
A Hypothesis Not an Axiom
H1: We hypothesize that God does not ever direct in the course of evolution.
H2: We hypothesize that God does direct evolution towards outcomes He desires.
That is not enough information to analyze H2. So, let’s articulate some specific mechanisms of guidance. If any of these are undetectable in DNA, than we know that God’s guidance would be undetectable.
H2.1: God alters the environment, changing selective pressures to bring about outcomes he desires.
H2.1 is obviously undetectable in DNA. For example, we know that the extinction of dinosaurs 65 million years ago, likely by an asteroid, enables the rise of mammals and of us. We have no way of knowing if the divine purpose of said asteroid was to enable our rise. Perhaps it was. H2.1 would not be detectable in our DNA, and can profoundly influence the course of evolution.
H2.2: God ensures the survival of specific individuals, carrying perhaps currently deleterious but long-term important mutations, so as to bring about outcomes he desires.
H2.2 is obviously undetectable in DNA. Perhaps God preserved some key mammal linages from extinction when said asteroid hit, to ensure that we also would eventually arise. H2.2 would not be detectable in our DNA, and can profoundly influence the course of evolution.
H2.3 God causes mutations that could have happened by other means, but He inserts these into genomes to ensure evolution produces outcomes he desires.
H2.3 is undetectable in DNA, but perhaps not obviously.
Let us start with the well grounded claim (in theology) that God desires animals with the “human-condition” to arise: which we will define here as written language, civilization, and a full theory of mind. Consider humans, chimpanzees, gorillas, bonobos, and orangutans, which all had the same starting point. However, neither chimpanzees, orangutans, bonobos or gorillas arrive at the the human condition.
We know that differences in these lineages are almost entirely explained by neutral evolution (let’s say, about 30 million bases of difference). Most differences between us are just neutral. Yet, we got to the “human condition” and they did not. We also know that some of these mutations caused us to be very different. Which mutations and how many? We are getting some answers, but from neutral theory we expect there to be on the order of just 1000s of mutations (let’s say 2 thousand bases of differences).
Perhaps God ensured that the right mutations would arise in our lineage so that the “human condition” could arise. Perhaps this could have happened on its own, but the chances were 1/100 or 1/1000, and God just made sure it would happen. What would have to do to ensure this? Not much. Let’s say He just made sure the right 100 mutations took place. There is no way we would find these mutations as anomalous against 30 million bases of neutral evolution, and 2,000 mutations of natural evolution. Therefore, H2.3 would not be detectable in our DNA, and can profoundly influence the course of evolution.
H2.4: God causes clusters of mutations that would not otherwise be likely occur simultaneously, to enable evolution to overcome difficult to cross “barriers,” so as to produce outcomes he desires.
H2.4 is probably least obvious.
What is the chances of starting from that point and becoming “human”? Perhaps we are tempted to say 1/5, but that is not quite right. We just do not have enough data to tell. The reality is that our best estimate is somewhere between 0 and 1/5, because we have no idea if our rise was a fluke or not. After all, there is just one example of animals with the “human-condition.” Perhaps the chances are 1 out of a million, or a billion.
Perhaps there is a specific set of 10 mutations that have to take place simultaneously in the same individual (or even group of individuals). H2.4 would not be detectable in our DNA, and can profoundly influence the course of evolution.
I am not at all saying that this is actually the case. Rather, we have no direct view of the precise genomic history of our lineage going back millions of years. We do not know the precise fitness landscape along these paths. Perhaps there is a barrier there we cannot expect to see. Before you call this fantastical, there are secular scientists who have proposed just this type of barrier. The existence of such a barrier would explain why humans alone are the only animals in all the earth’s history that appear to have arrived at the “human condition.”
Once again, I’m not making a strong claim that such a barrier exists, but just saying that there is no way of knowing from evidence one way or another. We do not know and we cannot know.
What this Means for ID
It appears that we have identified four types of guidance that God could use in evolution. Not one of these four guidance mechanisms is detectable in DNA, and all four would have a profound effect on the course of evolution. I am sure we could imagine more mechanisms too. It would be hard to find one that would be detectable
Based on what we know of biology, especially neutral theory, we do not expect God’s guidance to be detectable in DNA… I hope that @EricMH, @pnelson, @bjmiller, @AJRoberts, @Winston_Ewert, and @Agauger are reading along. They would see that it actually would take intentional effort for God to reveal Himself in evolution. There is no good theological reason to think that God would try to reveal His existence in DNA. That is just is not how the God of the Bible appears to work.
Given this analysis, the failure of ID to find strong evidence of guidance in evolution is not evidence against guidance. Despite their intuitions, we just do not expect nay evidence.
What This Means for Sufficiency
In the same way, we can’t say that evolutionary science demonstrates that “we don’t need God” to explain the course of evolution. That is not true at all. We just do not know. We have no control experiment that shows the evolution of earth with and without God. Instead, we know that evolution is profoundly shaped by contingency. God might just be one of those contingency. Maybe without His involvement, evolution would have taken a very different course. Or maybe not. It is also possible that God did not ever intervene, or that he does not exist.
Even if I’m not correct about one of these cases, you have to show how all are detectable to demonstrate that God’s guidance should be detectable. The case for undetectability is distinct in each case.
Science is Silent on God
Al this goes to say that science is silent on God’s guidance. @gbrooks9 has a totally scientifically consistent understanding of reality, even if it is ultimately a theological position because it invokes God. Science is neutral on God. It does not tell us if He guided evolution. Instead, it tells us there is no evidence one way or another.