Tactics available to help prove security of a new system? I believe that the accepted tactic to "prove" a system as secure is to allow the crypto-community to review it and if no vulnerabilities are found over a long period of time (5 or 6 years), then a new system can be accepted as secure until proven otherwise. Please correct me if I'm wrong.

That aside, are there any set of requirements a developer can proactively test their cipher system against before a public release?

I'm working on a large key (4096 bit), symmetric encryption system and I'm doing my best to be weary of the "anybody can make an encryption that they themselves cannot break" trick of believing that my cipher is unbreakable.

What kind of tests can I run on the process or the output? Are there good ways to measure entropy? What statistics should I expect after encrypting a section of data? I know there won't be some magic function that proves I'm secure, but I certainly don't want a public release followed immediately by being broken by a simple problem that some data analysis would have found.
Prove resistance to differential cryptanalysis.

For example, this presentation: Provable security against Impossible Differential Cryptanalysis. Or this paper:  ProvableSecurity Against a Differential Attack (1995)
Prove resistance to linear cryptanalysis.

For example: On Measuring Resistance to Linear Cryptanalysis
Run a bunch of statical tests against the cipher. 

For example: Cryptographic Randomness Testing of Block Ciphers and Hash Functions