If faith works how every religion says it works......

[ABierman1986]

Again, unfortunately not.

You assume that

A. A rational exploration of Christ's teachings is the only way to see if it does what is promised (ie a better life and this living water you speak of) and that this must be done with an open mind.

This makes sense, it would test whether Christ's teachings are as effective as a good self-help program, but not do a thing about divinity or the logical inconsistencies. Also the word "only" is not true 99.999% of the time.

B. Through this exploration and then using the standard of sermon on the mount you can deduce if your life or the lives of those around you have improved.

Ok sure, there may be a way to develop a comparison and statistically compare the lives of the people around you to the standards of this sermon.

C. If lives are improved, then following Christ's teachings are true and the actual way of Christianity.

Here is where your leap truly fails. The outcome of this type of exploration would only deduce if this lifestyle were a factor in quality of life, there is no reasoning to suggest this lifestyle would have been responsible for the improvement. Also, it doesn't negate alternative possibilities, including physiological changes as an explanation for why this lifestyle is effective as opposed to divinity.

You test does nothing to prove Christianity, or anything else for that matter.