Single word meaning "on its face"

I asserted that up is down. That statement is false on its face.

I feel like I've seen at least one single word that means "on its face." Words with close etymology and meaning

• Superficial. If ficial were a word that would do the job.
• Prima facie has the right meaning in actual use: "That statement is prima facie false." But that's two words. I'm also not satisfied because it literally means "on first face." What I am looking for wouldn't be so qualified: It means that the qualified thing is that way no matter how many times or how deeply we look at it.
• Close but not quite: Obviously and its synonyms – clearly, indisputably, undeniably. These all have at least some connotation that the thing so qualified could be or is in dispute, but that the dispute is rejected. I want a word that refers to the nature of the thing and whose connotation is that no dispute can even be entertained. E.g., "A false statement is false on its face."

As jsw29 said in comment, you can't have a word that is synonymous with prima facie and means ‘true no matter how deeply you investigate’, because that's not what prima facie means.

If you want the first meaning in one word, there's the legalese facially.

If you want the second meaning, see Jason Bassford's answer.

Note: This is an answer to the original version of the question that used a different example sentence. The question and example sentence has changed since I provided my answer.

I think you're looking for demonstrably:

[Merriam-Webster: demonstrable]
1 : capable of being demonstrated
2 : apparent, evident

In other words:

I asserted that the sky is green. That statement is demonstrably false. (All you have to do is look at the sky to see that it's blue.)

Another word (although it might not meet your definition of a "single word") that means almost the same thing is self-evidently:

[Merriam-Webster: self-evident]
: evident without proof or reasoning

So:

I asserted that the sky is green. That statement is self-evidently false.

As a matter of logic and reason, and in terms of a noun and something that is true not by observation but by a first-cause assertion, it would be an axiom:

1 : a statement accepted as true as the basis for argument or inference : POSTULATE sense 1
// one of the axioms of the theory of evolution
2 : an established rule or principle or a self-evident truth
// cites the axiom "no one gives what he does not have"
3 : a maxim widely accepted on its intrinsic merit
// the axioms of wisdom

In mathematics or logic, an axiom is an unprovable rule or first principle accepted as true because it is self-evident or particularly useful. “Nothing can both be and not be at the same time and in the same respect” is an example of an axiom. The term is often used interchangeably with postulate, though the latter term is sometimes reserved for mathematical applications (such as the postulates of Euclidean geometry). It should be contrasted with a theorem, which requires a rigorous proof.