AccretedMinutiae,

Your passionate treatise is a statement of your beliefs. Nothing wrong with that. I'm trying to think of how to improve on it, to make it more articulate and believable. You seem to be objecting to the idea that observation AND reason are two components of the foundation of science. You wish to show that ONLY observation is the basis of knowledge and science.

I'm trying to think of a way to prove your thesis. To prove it we would have to have immutable observables as our foundation, but not reasons as part of our foundation. That would seem to eliminate thought experiments such as a brain in a vat, since neither you nor I can observe a brain in a vat. the brain in a vat has to go, be cause we can't observe a brain in a vat. The Thought experiment part has to go too, because we want to prove that ONLY immutable observation is the foundation of science and knowledge. You see, if you include "thought experiments" in your proof, you are admitting to reason as part of the foundation of knowledge and science. We don't want to admit that reason is part of the foundation, otherwise, bad things will follow. For example, a rationalist might give reasons for God. There are many rational proofs for God, after all. And if we accepted reason as part of the foundation of knowledge, we could be in trouble. It is better to stick to pure observation as the foundation of knowledge. That way we can always say we can't see God. The rationalist can't get us that way.

Given that basis, we must, I think stick to pure observation to explain and produce knowledge. Our task then will be to catalog immutable observables and go from there. I hope the job is not too tedious.

Right now that's the best I can do to help you. You seem like a thoughtful individual and I encourage you to keep thinking about these issues. In saying so, I just noticed I contradicted the program of collecting observables - I encouraged you to think about it, when really we should be getting our foundational observables in order first.

Grinseed,

1.
Yes. A coherent system of beliefs, isn't proof, as the opposite is still coherent. A coherent system and its opposite can't both be right even though both are coherent. But people incorporate the empirical into their coherent system. Many of their beliefs are tied to empirical observations. For example, I believe that the "Little Foot" creature exhibited in a fossil collection is the ancestor of apes and humans. I used to believe in the fact that humans came from the apes, but that fact is now non-operative. In the coherence theory, a candidate belief, to be incorporated into my belief system must imply a belief I already hold, and the belief I already hold must imply the new belief (about Little Foot, for example). It is mutual implication. Sometimes new beliefs contradict old ones, so the old one has to be dropped. Not all my beliefs need to be empirical, but the non-empirical ones need to be coherent with the empirical ones. In that manner, both reason and the empirical form my knowledge. Its being empirical as well as rational but it isn't positivist/empiricist.
2.
Core beliefs: An atheist who is aware they want coherent knowledge, would want a mutually implying system that coheres with their disbelief in God. (Mutually implying is what you describe as circular.) Nothing wrong with that. Everyone has a core position. A theist would want a coherent system that is coherent with belief in God. In principle, both theist and atheist can each have a coherent system of beliefs. In part, that's because there is no autonomous knower who is in possession of absolute truth.
3.
Your example of physical experience is quite simple and understandable: a belief in a mountainous planet being abandoned after photos show it to be smooth. Noting wrong with that. I change my beliefs too. But not all seeing is that simple, so it isn't representative of other types of seeing. For example, in the middle ages, people (the intellectuals anyway, if not people in general) saw the physical world in Aristotelian terms. Post Newton, seeing changed. People now saw the world through a Newtonian lens, or paradigm. So seeing is always so simple as the terrain of a planet. Believing in a Newtonian framework contributes to seeing and knowing what he believed. But if someone of that era disbelieved Newtons views, that person could not see or know the universe the way Newton did. They would still be stuck in the Aristotelian perspective. These days Einstein's views changed the way we see things once again. Seeing is seeing from within a model of the universe. It isn't as simple as seeing if a planet is smooth or not. Too, believing is seeing. While many people believe Einsteins' theory, few people actually know his theory. E=Mc2 looks simple on the surface but the actual justification for it is beyond what most people are capable of. So many people believe without knowing. Some people who believe it as children, for example, then study mathematics to a level in which they can know it as well as believe it. But someone who disbelieved E=mc2 might not bother to learn the required math to know it. Disbelief can contribute to lack of knowledge. Believing can contribute to knowledge.
4.
Compartmentalization: In principle, its possible that in Newton's time some people held to both Aristotelian perception, and Newtonian perception, but not at the same time. They might switch between the two from moment to moment or day to day. Perhaps a clearer example of compartmentalization is a fundamentalist christian who on Sunday sees the universe in an ancient way where heaven is just above the clouds and mountain tops, and that's where God, Jesus, and the angels reside, while Monday to Saturday they switch to a more modern view and function as a modern person. On Sunday they might believe in healing by faith, and prayer, but on Monday visit the doctor as faith in the doctor is sometimes more reliable. This relates to the coherent theory of truth: due to compartmentalization, they don't have a coherent worldview they have two mutually exclusive worldviews that they switch between. Although I find that strange, the switching between ways of seeing might contribute to understanding that believing is seeing, and switching beliefs changes what one sees. Positivists also compartmentalize. One compartment for them is things like neutrons, protons, electrons. they don't actually see them as one might directly see the terrain of a planet. Electrons and so on are abstract theoretical models of some phenomenon. Strangely, positivists don't use abstract models as examples of seeing. Instead they use very simple examples such as is a planet smooth or not. So we have one compartment of everyday seeing the terrain, and another compartment in which seeing is seeing thorough the lens of a theoretical model. Positivism is in some sense in denial of theory and the assumptions that the theory is tied to. In that way they misrepresent believing, seeing and knowing in science. The incoherence between the positivist's simple examples of seeing, and the reality of theoretical science was a serious problem that could not be resolved. That's why Scientists such as Polanyi, Kuhn, Popper and others went after those erroneous ideas of knowledge until the positivist/empiricists gave up.
5.
What about the propositions being independent true first? I'm not convinced there are any propositions being independently true. Independence of what? Its conceivable that everything everyone and anyone believes is false.
6.
Cure for cancer: One of my beliefs is science will cure cancer. But it isn't as simple as the planet is smooth, the grass is green, I see a tree...If you want to confine yourself to such observable s, nothing will get cured.
7.
I think you misunderstand the word "belief" as I use it. It can be, and often is about empirical observations. But the word "belief' acknowledges in part, that some facts change. It used to be a fact that humans came from the apes. It is currently a fact that humans and the apes came from whatever Little Foot is/was. "Beliefs" is a fairly general term. Some of my beliefs are facts, while others beliefs are not. The essential thing is that my beliefs that are not facts are coherent with my beliefs that are. I believe in atomic theory. But do I know it as a fact? No. In order to know it in that way, I would have to be deeply educated in atomic physics. Only such educated people really believe it and know it. most people just believe it without knowing it.
8.
I have long suspected that knowledge is beliefs that are true, and that believing contributes to knowing. When I was young I disbelieved in evolution, and declined, therefore, to actually read any of it. Consequentially, disbelief contributed to hampering education and knowledge. In my disbelief, I was being a a positivist/empiricist. No one ever saw a human come from an ape, so why should it be taken as fact? Once I became open to it, believed it, I read about it in a different way. I had to remove the shackles of positivism/empiricism in order to accept that humans came from the apes. Despite the fact that I and no one could see the first human emerge from an ancestor, I believe it. If you confine yourself to observables such as the planet is smooth and disbelieve evolution because no one can see the first human emerge, you get nowhere. Believing contributed to knowing.
9.
Recently there is an atheist cosmologist who has a perspective on the origin of the universe coming from nothing. He admits he can't prove it, and obviously there are no observable employed. Just as positivism/empiricism would throw the Atheist Hawking under the bus, they would tie this other guy to the railroad tracks. Positivism/empiricism can't help them. He might end up employing the coherence theory of knowledge.