Yeah, that was my fear.

Is there any way of bounding it though? I mean, you're right that you won't have 180 degree wrap, but you also don't need the chain to be exactly the same length for the two gears to work, especially because this is a singlespeed, not a fg.

Also, I'd be interested in gaining some insight into how you determine the extent of wrap given a gear ratio and distance.

If an analytic solution proves messy, is there any way of just rephrasing this as a constrained minimization problem and solving numerically? Ultimately, you're just looking for the smallest amount of chain subject to the constraints that the chain is always a certain distance away from two poles, those distances determined by the radii of the chainring and cog and the position of the poles represented by the variable d above.

Unfortunately, I dropped physics in college and am therefore no good with this sort of thing.