Rank 5 settlements are really getting somewhere. Sovereigns who have reached Rank 5 themselves have got their world locked down. There is a steady flow of Importance from their pledges and bonds and they put a lot of personal effort continuing the improvement of themselves and their settlement. In order to move your settlement to Rank 5, you must build the Road Infrastructure of the Internal Roads. This requires the Masonry science which is behind Mathematics so starting this research chain is advisable.
Building your Internal Roads requires not only Masonry (and Internal Roads) research but a supply of Bricks and Pack Animals. Fortunately these items are in common enough demand you should already have infrastructure to produce them. The Road Infrastrcture increases building storage and decreases Brick upkeep for every building in the settlement. It also crafts new Carts. The exclusive Rest Stop requires Furs and Groceries to build but increases Toughness for every settlement citizen. The Post Office is most efficient in the Internal Roads but serves it's best purposes increasing Prosperity in suffering districts.
This district also requires Masonry in order to build the Keep, it's main building. Each level of Keep inspires free Guards to join, provides a couple Prison cells and increases settlement security. But it will require more Mathematics, Masonry and then Keep research. The Moat and Guardhouse on the other hand don't require any research but their own to build and upgrade. Exclusive Moats will increase settlement Security and decrease attacking Peer Celerity and Might. Guardhouses are powerful local Security providers that are most efficient in the Inner Walls but probably more useful scattered throughout your district.
Constructing the Grand Pavilion, main building of this district, you will need Slabs, Gold Florins and to research Aesthetics. This fortunately shallow science allows you to build pretty things. This is a district of wealth and play. The Grand Pavilion offers a bonus to Commerce districts, processes Grapes into Wine and sells Canvas as if it were an Art Gallery. The Exclusive Trading Tent produces silver Thalers for every Trade conducted in the settlement and can store carts. Vineyards can be built anywhere but are most efficient in the Champaign Faire.
Constructing and improving your Minstrel Stage will require knowledge of Carpentry, which requires the science of Mathematics. To upgrade the main building/ district, you'll have to research Mathematics, Carpentry and the Minstrel Stage itself. The Minstrel Stage itself increases the Encounter success chance and increases the Education of the entire settlement. It also sells various items as an Income building. The Exclusive Stocks & Pillory provide prison slots and a bonus to Settlement Security. The Linked Stables process Livestock into Horses.