Does logos should be always reproducible via Compass and straightedge construction?
CSC is an ancient technique utilized by mathematicians to define geometric shapes. Basically you can reproduce a geometric shape just by using a straightedge (without measurement marks) and a compass.
In graphic design, grid is a common construct used to organize information. CSC is just a generalized version of the grid. You can have diagonal lines, curves and other features more essential for logo designs.
One of the best example I’ve seen is the design of Radon identity by Sebastian Gram.
CSC is especially good for geometric shaped designs, which is common place for logo designs less so in graphical designs.
Here you can see a video example regarding Twitter logo:
In conclusion, probably integrating CSC into your design process is a great way to better your logo’s specification.