I feel the need to declare this, so I'll make a post.
I know my church is true. I follow a living prophet, read His Word from both the Bible and Book of Mormon, and he answers his prayers. The proof can be given, provided a few logical assumptions and methods.
The scientific method provides a path to verify a hypothesis.
-make an observation, do some thinking, or do research. This leads to an idea of how it works.
-state the idea as a hypothesis, providing an expectation.
-form an experiment, or observational guidelines(some things can't be lab controlled), which can be reproduced with certainty.
-Confirm or revise the hypothesis, depending.
I'm sure I missed something, IIRC the method has 8 steps, but I'll not worry about that. At the end of the BoM, in the book of Moroni, 10:4, Moroni states the start of a sound hypothesis by stating that if the things in the book are true, when you pray to God, he will answer you and tell you if it's true. This is a bit more to-the-point than James 1:5, which says that God will give wisdom to all men, who ask in faith.
We can think a bit, pondering the complexity of our world, and we can read the scriptures, and other books, etc. This will lead to questions about the underlying nature, and in this case we get the theory that God created the world, etc.
Thus we get a hypothesis--"If God exists, and he's the same God as in the Bible and BoM, if you ask in faith, he'll answer in a way you'll know, but is subtle." Subtle, because it's still somewhat of a trial of faith--looking at examples like the Pharisees, and Nephi's older brothers, miracles and visions can't convert people unless faith already exists.
Thus we get our experiment. If God exists, he'll answer. If not, you'll get nothing. Either way, you have experimental results that conclusions can be drawn from. This is also where things break down a bit, and I get yelled at by the scientist type crowd.
Spiritual matters are quite solidly an individual issue. I can repeat the experiment for myself, as many times as I like. Spiritual manifestations, especially subtle ones, are different in varying degrees for different people. Allowances must be taken for this, and the fact that you therefore can't peer review the experiment. This experiment says however, that this process is possible, and anyone who says otherwise is a fool. Because of the personal nature, you can't take a man's word for it--the experiment must be done personally. Only then will you have the knowledge to declare it true or false. This also means that I cannot condemn you if you fail to get the message I believe you should have gotten. This gives me trouble in proselytizing, since I am not a teacher, and especially not on this topic. The research and verification I do is for me personally, and so frequently, I do not have references, quotes, and other such stuff needed to talk to other people.
I say that God exists; I have felt his Spirit, had him answer my prayers, felt his comfort, and felt his condemnation for my sins. I have confirmed that the Church of Jesus Christ of Latter-Day Saints is true, and that I am on the right path, at least when I'm not laying in the mud. I've been getting betting about staying on my feet, and that has only strengthened the confirmation that the path is right.