In most cases, it is necessary to complete every item in the checklist. There are exceptions, however, and they're not always labeled as such. Checklist items do not need to be completed in order.
Completing a Principle advances you by one level. You can check which principles you've passed by clicking Self -> Special -> Show Details of my Level.
Citizenship, as well as the Initiation Tests into each of the seven Disciplines, are relatively simple. As such, their Principles describe the entire Test. For these Tests, passing the Test and passing its Principles are the same thing and are accomplished the same way: do every item on the checklist.
The 49 formal Tests are significantly harder. The Principles for these Tests are designed as a launching board to give you a feel for the Test and to start you on your way. The Principles here can be completed days or weeks before the Test itself is. In fact, you may never pass the Test at all.
If you manage to pass a formal Test before completing its Principle, you still receive full credit for that Principle.
Architecture
Principle of the Obelisk
Principle of the Funerary Temple
Principle of Towers
Principle of Life
Principle of the Megalopolis
Principle of Ka
Art
Principle of Khefres Children
Principle of The Raeli Mosaic
Principle of Pyrotechnics
Principle of the Prismatic Opticon
Principle of the Formal Garden
Body
Principle of the Acrobat
Principle of the Singing Cicada
Principle of Safari
Principle of the Banquet (Also known as "Feast for the Senses")
Principle of the Oyster Catcher
Principle of the Archaeologist
Principle of the Darkest Night
Harmony
Principle of Marriage
Principle of the Freeman
Principle of Souls
Principle of the Prophet
Principle of Mentorship
Principle of Reason
Principle of the Critic
Leadership
Principle of the Demi Pharaoh
Principle of Bureaucracy
Principle of True Leadership
Principle of the Plantation
Principle of the Covered Cartouche
Principle of the Coalition
Thought
Principle of the Pathmaker
Principle of the Venery
Principle of Tomb of the Immortal
Principle of the Bijou
Principle of Empty Hand
Principle of the Riddle of the Sphinx
Worship
Principle of the Vigil
Principle of Path of the Pilgrim
Principle of Beacons
Principle of Festivals
Principle of the Leavened Bread
Principle of Visions