The scientific model you have is one of assumptions.
It's not.
You like the tree, but the fossil record does not support the nested heirarchy.
Except that it does.
No species shows up in a layer where it shouldn't be.
We don't fine mammals mixed with trilobites for example.
The fossil record always has things suddenly appearing with not transitional fossils.
Fossilization is a rare process. Creating a fossil is really hard.
To assume you'll find such an abundance of fossils is like assuming that you can show me a photo of your face of every second of every day since you were born.
Reality simply doesn't work that way.
And the fact remains: no fossils are found in layers where they couldn't be.
You don't find mammals next to trilobites.
Your genetic understanding simply points out commonalities between creatures.
Nested hierarchies and genetic markers point to common ancestry.
This is how, btw, we can also tell your biological brother from some random person.
DNA allows us to establish levels of relatedness.
Courts do it all the time in paternity cases or to determine relatedness in inheritance cases etc.
It is an assumption that those commonalities prove common ancestry if all things.
It is not. It is based on an understanding on how reproduction works.
But all it really shows is common design.
No. Again: nested hierarchies. Not mere "commonalities".
It is only an assumption that it shows common ancestry and evolution
It is not.
It is a demonstrable fact that an evolutionary process would result in a nested hierarchy and ONLY in a nested hierarchy.
And it's exactly such a pattern that we observe in life.
You still have no evidence that the evolutionary process ever really occurred.
Except for the fact that every single aspect of the evolutionary process is demonstrable and observable AND that all independent lines of evidence (comparative anatomy, comparative genetics, fossil record, geographich distribution of species, etc) ALL converge on that same answer and that same hierarchy.
At best the only thing you might be able to state is that mammals descended from mammals and cold blooded creatures from cold blooded creatures and fish from fish. That's it.
And that all those also share ancestors.
But again the fossil record shows no such thing.
The fossil record is completely consisted with such a thing.
We don't find mammals next to trilobites.
We also never find mammals with feathers or reptiles with inner ear bones.
Because nested hierarchy.
And since you cannot observe such a thing happening
We constantly observe it happening. In the wild, in the lab, in virusses, in bacteria, in agriculture, in breeding programs,...
and you cannot test such a belief all you have left are assumptions.
Evolution is
extremely testable as it makes an inumerable amount of testable predictions. Which is one of the reasons why it is such solid and established science.
Considering the ridiculous amount of testable predictions in evolution... IF evolution is false, then demonstrating it as false should be trivial.
Let me ask you this. What is the common ancestor of vertebrates and invertebrates?
Determining that 2 species (or individuals) share an ancestor is one thing. Identifying that ancestor is another.
Let's dumb it down so that you can think clearly about it, because you have to much emotional baggage attached to your evolution denial. Let's just stick to humans for now...
Consider this hypothetical:
Suppose there is this tribe somewhere in the middle of nowhere of 1000 individuals, 200 of which are newborn babies, less then 6 months old.
For some reason, the tribe is attacked and all 800 non-babies are massacred and burned until there is nothing left.
The other 200 are sold into adoption around the world. Their place of origin not communicated and lost in the pages of history for all of time.
Let's say 2 babies are sent to New York. They are brother and sister but nobody knows that. Each gets adopted by seperate families.
Years later, both their DNA ends up in a research project for some reason.
==> now, it can be determined, based on their DNA, that they are siblings. Yet, their parents are completely unknown.
So there you have it.............
Determining common ancestry is one thing.
Identifying that common ancestor, is another.
The second, does not exclude the first.
So as for your question "What is the common ancestor of vertebrates and invertebrates?"...
The answer is simply: I don't know, but that doesn't stop us from being able to determine that they both share ancestry.