Action - Practice Siege Warfare
Using the siege weapons and watching them used increases the inherent understanding of your Regiment/Brigade. Players can visit the Siege Works (if their Sovereign has built one) and show off how to use these impressive machines. This action increases the Arithmetic by a percent for the Regiment/Brigade.
Main - Siege HQ
The Siege HQ handles everything that it should. It makes Boards, it hires Engineers, and it constructs/trains Siege units. On top of all this great stuff it also increases local Education, so any housing buildings constructed in the Siege Works will be particularly well educated, and really good at throwing rocks at each other. Constructing a Siege works and keeping it running requires Beer and Magnets. The Engineers will accept no alternatives.
Exclusive - Siege Workshop
The Siege Workshop is where they practice building and firing your Siege weapons. As many can be built in the Siege Works district as you can cram in but they're not really useful anywhere else. Every level of Siege Workshop decreases local training costs and always has a catapult lying around that will join the settlement Garrison forces for combat. They also increase Education across the entire settlement.
Linked - Sawmill
Sawmills are vital to a growing settlement, as they produce Boards in every district. While they are most efficient in the Siege Works, it's likely that many (peaceful) settlements may choose to forgo the Siege Works entirely and concentrate a few Sawmills in another district when they need to construct one of the many buildings that requires Boards.
Minister - Chief Wrecker
The Chief Wrecker is possibly the coolest seat in the Education department. Sure, they have to do a bunch of math but the fun part is doing shit-tons of damage. They are backed by the Ministry of Demolition, a similarly smug set of badasses among their administerial counterparts in other Educational Ministries.