Is there a mathematical way to discover the fastest build order required for certain unit(s)?

For a zergling rush specifically:

• Resources to consider:
  • Minerals
  • Larva
  • Supply
  • Time
• Influencing factors:
  • Queen (larva inject)
  • Drones (mineral gathering rate)
  • Overlords (add supply cap)
  • Extractor trick (temporarily 'adds' supply cap and lowers mineral rate)

For just a zergling rush, you can limit yourself to considering one of the following choices at each stage:

• Build a drone
• Build a spawning pool
• Build an overlord
• Build a queen
• Build a zergling
• Perform an extractor trick, and build a drone, queen, or zergling
• Wait until when you are able to perform an above action which you are currently unable to

Obviously you would need to check if you have the prerequisites (other buildings and/or resources) before doing any of the above.

You could certainly brute force it if you so desired. This program doesn't prove or guarantee optimal times, but if you want to see SC2 build order optimization at work, check out these:

• https://code.google.com/p/evolutionchamber/
• https://github.com/Carbon-12/SCFusion

If you are looking for a generic way of "proving" a certain build order is best, I think you will find accounting for the diminishing returns of workers, resource exhaustion, expansion, travel times, and other factors will make it a rather herculean task unless you make so many simplifications that "best" becomes largely meaningless.