Architecture Complexity: Theorem Proposal

Working on a major effort as part of my day job and I was asked about the biggest risk in a particular environment. During the course of discussing it, I tripped upon a theory of architecture - not exactly rocket science, but it occurred to me that no one ever articulates these things (or even really attempt to prove them - we simply accept them as lore), so I decided to make a point of documenting the obvious (something I believe falls squarely within the architect job description).

Any way, on to the theory:

Theorem: Given any technology environment, when the ratio of business functionality to the overall number of systems favors business functionality, there is a positive relationship to the cost and complexity of the environment.

Corollary: When the ratio favors the number of systems, there is a negative relationship with the cost and complexity of the environment.

I've purposely used relatively non-mathematical language as this approach presents a few problems:

• How do we measure "business functionality"?
• How do we normalize the business functionality measurement with the number of systems?
• How do we define the terms of the relationship between the ratio and cost/complexity?
• How do we define complexity in mathematical terms?

Regardless, it seems that this is a grand example of the "folk wisdom" that many architects believe, but never write down. All said and done, this somewhat calls out for a calculus of architecture (wasn't Djikstra working on a calculus of algorithms prior to his death?).