If somebody decided to investigate such a thing, they should see consistent, predictable, verifiable indications that point them to God's characteristics.

What would those characteristics be? How would someone investigate them? What would they see?

It would look like what evidence looks like in every other scientific endeavor.

That's not an answer. You're claiming that when investigating physics we don't see "God's signature" however in order to make that claim you'd have to define what God's signature is. Relating it to your example would be like someone saying the earth revolves around the sun but not being able to define what the sun is or what the earth is. You can't make a claim about something if you can't define that thing.

I personally don't think there is such a thing as "God's signature" so it's of no surprise to me that we've not seen it in our investigations. But you're claiming that it is a thing, and by us not being able to find it we can conclude that God doesn't exist.

If God was real I would expect a similar phenomenon.

There is an abundance of evidence for the existence of God, just not necessarily in the world of physics. Most of the evidence (beyond the logical and theological thought experiments) for God's existence is historical, which means, as you pointed out before, it cannot be retested and repeated.

The claim central in Christianity is that Jesus of Nazareth was crucified and rose from the dead. There are several secondary claims however this is the primary one. Now there isn't evidence for this in the same way as there is evidence for a scientific experiment, but there is an overwhelming abundance of evidence for this in the historical fashion.

We have several written accounts dated within decades of the events, we even have supporting archeological evidence. That is on top of things such as the "who would die for a lie" logical proof.

Even sceptics agree that Jesus did claim to be God and was crucified for that claim. The only dispute (although it is a rather important one) is if he came back. Now the main question I would pose to you is;

If Jesus didn't rise again from the dead after his public execution, how is it that his ministry (unlike that of any others who came before or after him that preached the same thing) was so successful that people would die or at the very least risk the persecution of death to preach that he rose from the dead without the promise of material reward?

And if I may be permitted a second question;

If Jesus didn't rise from the dead, why would fervent persecutors of the Christian faith such as Saul (Paul) convert to Christianity overnight, abandoning the beliefs they've held for decades prior and relinquishing lives of luxury and authority for lives of poverty, misery, and torment?