I recently asked a question regarding security obstacles of online voting via a web service, and one of the primary obstacles to doing this is the difficulty of proving that the program processing incoming vote data is the actual open source code shown to interested parties for verification of non malicious / unfair vote counting.
We can make the software for vote verification open source, but proving that the program actually running on the receiving end is code that should be used, running without malicious alteration seems like an interesting challenge.
So my question is: Is it fundamentally possible to provide proven (observably valid without doubt) processing information at the core of a security computing system so that expert observers / analyzers can know without doubt that specific, non-altered logic (the correct open source software) is being used to recieve, compute, and store information as promised?
More simply put, can we (fundamentally) somehow prove to people that we're performing the computational logic that we say we are? Prove to expert observers that we are receiving, processing, and storing data (for example, votes) via specificied logic, without the possibility for malicious influence?
Furthermore, if so, and this might be a difficult question, (fundamentally) can this be done in a way as to preserve the anonymity of the actual data being verified?
To be clear, such a feat might be an achievement of the utmost genius and involve writing new kernels, and and special OS - I'm not asking of the viability, I'm only asking whether it's fundamentally possible from an information security standpoint.