Only the analytic side of the analytic / synthetic distinction has proof.
The synthetic side (that I call the empirical side) only has evidence.

We can know with complete logical certainty that a cat is an animal.
We cannot discern the difference between a cat and a space alien
perfectly disguised as a cat (including DNA).

My adaptation of JTB requires proof that the belief is true, with less
than proof we only have presumption and thus not knowledge.

The whole point of the Gettier argument is that unless the justification
necessitates the truth of the belief then the belief <is> insufficiently
or incorrectly justified.

My adaptation to JTB makes it a necessary truth.
Complete proof is a semantic tautology. Cats are stipulated to be animals.

That the animal in front of you seems to have all of the properties of
a cat is evidence and not proof that it is a cat.

It is certainly not impossible to know with 100% complete certainty that a dog is an animal and my adaptation to JTB specifically excludes anything that is not known on the basis of complete proof.

By addressing the problem categorically gaps in reasoning are impossible.

I have pondered this again and again for years.
"If truth is the necessary ingredient for knowledge, how do I know what I claim I know is true?"
Truth is a necessary yet insufficient condition for knowledge.

Knowledge requires:
Awareness that an expression is true on the basis of complete proof that the expression is true.

"instance where there is something outside of our ability to know"
Does not count as knowledge under my adaptation of JTB.

My adapted version of JTB does seems to perfectly divide knowledge from presumption and falsity and utterly eliminate the Gettier cases.

A space monster in disguise would not have the DNA of a cat. We could add that one of the properties of a cat is that the thing <is> a cat. Thus a perfect duplicate of a cat is not a cat.

I always view these things in terms of pure logic. If a thing in the world can be empirically validated to have all of the properties of a cat including the DNA of a cat then this thing is necessarily a cat, all opinions to the contrary are counter-factual. The belief aspect of JTB is required because unless at least one person knows X then X is not knowledge even if X is true.

JTB is partially correct in that knowledge must be a truth that is held in at least mind. If no one knows X then X is not knowledge. X must also be true. The key error is an insufficient connection between the justification and the belief. If the justification makes the belief necessarily true then the belief is impossibly false. Modal logic: □P ≡ ◇P // Necessarily(P) ≡ Not Possibly Not P

"I am skeptical towards justificationism/foundationalism.
It looks to me like human attempts at justification are always
built on intuitions which are not in themselves logically justified."

{The justification necessitates the truth of the belief} as in the
modal logic: □P ≡ ¬◇¬P // Necessarily(P) ≡ Not Possibly Not P
If Fluffy is a cat then Necessarily Fluffy is an animal.

I can't envision any element of the set of knowledge that can avoid being a necessary consequence of something else. This applies across the analytic/synthetic distinction. What specific boundary case do you have in mind?

Unless at least one mind has a belief B about subject S such that the justification of this belief necessitates its truth then B is not an element of {knowledge} because no one knows it. An expression can be true without anyone knowing it is true.

The way that I see it there are two sources of analytical truth:
(a) Some expressions are stipulated to have the semantic property of Boolean true.
We must do this when we assign meaning to otherwise totally meaningless finite strings.
Meaning is represented as relations between finite strings. The source of these relations is
the model of the world. These relations are encoded in the various language of the world.

(b) Some expressions are deduced from the first set using sound deductive inference.