Of course, the conclusions drawn from math can be falsified, or at least we hope to be able to falsify most of them. I'm not sure why you say this —is it to show the difference between math (which we can trust) and the 'special pleading' of invoking first cause? The comparison is between math (axiomatic) and causation (axiomatic).
Mathematics is the logical study based on a series of axioms. These axioms can be used as a model of the real world and so maths is useful as a logical method of studying aspects of the world that can be approximated mathematically.
What is special pleading —whatever can't be proven? But anyway, the law of causation does not say "everything has a cause"; it says, "all effects are caused." First Cause is not an effect. But I don't mean to exasperate you; I expect you meant to say that just because all effects are caused does not imply that there is anything that is not an effect. But that to me is logically vapid. It seems the kind of thinking that science has always shunned. You seem to me to be, like most atheists and others, to go to whatever degree is necessary to avoid the obvious here.
This strikes me as rephrasing causation to remove the obvious contradiction but in fact simply makes the axiom false. Specifically that sources of causation need not be caused themselves.
Nevertheless, I agree that proof of first cause would be nice. It would be nice to see the circle closed, such as is done with mathematical 'proving' of an equation by coming at it from another direction. The problem is, science and philosophy uses causation, math and other logical formats and laws as verification of theory all the time. Few of the notions involved in quantum theory can be verified physically. And much of what is verified physically can only be interpreted mathematically and by sequence of causation and other logical reasoning.
Extreme edge science like quantum theory might be harder to demonstrate... but the mathematical analysis is based on data from the physical world, not simply conjecture.
So why the shunning and even scorn involved in the question of First Cause? Is it really simply because there isn't any point in pursuing proof?
Two reasons in my opinion.
One it seems to be an attempt to at an explanation without data and possible without the possibility of data.
Secondly I think people have a negative association due to it being commonly used without independent justification to simply insert the religious preferences of whoever is using it as a justification.
Then maybe, and so it seems to me proper, we need a good definition of species. Obviously, it is not every variation that is a separate species. But I see no problem with all arachnids descending from a pair of spiders, or all canines from a pair of dogs, in the few thousand years it has been.
It is actually difficult to define the edge cases.
One common method is by ability and inclination to reproduce, but that is not practical with extinct species.
It's interesting that you are comfortable with all arachnids being one set... but that represents more variety in genetics and structure than all primates or even all mammals.
The problem is that species is fundamentally a human label for a biological concept that doesn't have hard barriers.
Many people are happy with accepting that all cats or all canines being one group... but if you accept canines, why not bears as well?
This pattern can be demonstrated genetically to apply across all life and the fossil record support this diversification.
I don't know what 'non-coding DNA' is, as, I assume, opposed to 'coding DNA' and why it is relevant to mutation rates.
Non coding DNA doesn't directly "code" for the proteins that make up our bodies.
This means that mutations of this area are vastly less significant to the animals and can be passed on indefinitely.
There is a phenomenon apparently common to modern science that assumes that things are now as they always have been, for example that radioactive decay rates have always been the same, and operate by the same principles.
It is not simply an assumption, it is also supported by evidence.
As with all the physical evidence for events over deep time a sufficiently powerful and deceptive creator could have left false evidence, but it seems unjustified to assume it.
Wait, what? Where are they? Ok, yes, that's cheap. But I really don't get your point. What species? Do you think the Ark theory, sans TOE, would imply that all diversity as we have now, plus extinct varieties, would be represented on the ark? Why should so many species have representation on that boat, when only one pair in a species might do the job, that diversifies into several of what we call species (again, I'm not sure where to define 'species', here), without calling it TOE?
Dinosaurs, mega fauna mammals, synapsids, non human hominids and all manner of radically different plants and environments.
All these things leave evidence in the same places.
In my state there is evidence for giant salamanders, glaciers, continent sized river valleys, hibernating dinosaurs, volcanoes, saber toothed marsupials and giant birds.
These things can't all exist in the same place at the same time.
This is a little humorous, here, you arguing against the migration of DNA and I for it! But at least I have the excuse that I don't know what I'm talking about!
There's nothing wrong with mutation as a source of variation of DNA, there's just no justification for the speed and pattern of variation to create all modern and extinct animals in a few centuries.
Maybe the marsupial mouse isn't really a mouse.
It's not, but it lives in a similar environment and has similar habits, but is inexplicably more like a kangaroo internally.
I don't know. Maybe there is more that these have in common than ancestry. When I hear of some certain moth whose progeny goes blind and begins to not even produce eyes as a general rule, when left in the dark for enough years as a group, then when brought back out into the light begins to develop eyes, I can't help but wonder what's going on here. Certainly that's adaptation, but the genetic disposition is already there for it to happen.
That sounds like a gene that can be switched on and off by environmental factors... like grasshoppers turning into locusts.
It's a different phenomena to a trait being gained or lost to mutation.
Some fish grow teeth, some don't. I'm not sure any of this is meaningful for or against TOE or Creation Theory.
But we're talking about birds here... birds don't have teeth.
But dinosaurs did and some had feathers too. Evolution has an explanation for this situation, and Creation does not.