No, an anarchist system as described isn't functional
Because there's nothing to function. It would be like a permanent Purge. Keeping records requires organization and making it worth anybody's time requires some form of enforcement/punishment.
The truth is, no large population can exist without government. It's nice to believe that's not true, but you need to coordinate transportation of goods (like food and sewage) and the provision of services (like financial and legal) and that coordination is only possible with a government. It doesn't matter how you try to invent it, what you base it on, or the specifics of its operation, it's a government. And the moment you have one, people want to use it to complain about how loud their next door neighbor's daughter is playing her music.
You obviously had a government because it would take one whomping well organized society just to build the dyson sphere. That government would have set up procedures for mantaining the sphere (maintenance via Clarkean magic is possible, but not believable). Those maintenance services would be a huge catalyst to maintaining a continuing government.
And just as our own planet doesn't simply fall apart when one nation becomes chaotic, your entire dyson sphere population wouldn't become chaotic, either. Justifying a non-government condition with that many people is well beyond my own ability to suspend my disbelief.
Let's take a tangent...
So, just for fun, let's organize the government so we know what could be forced to anarchy.
Because I live here, let's look at the scalability of the United States. City governments work with county governments, which work with state governments, which work with the federal government, which works with the United Nations (the closest thing we have to a planetary government).
It's not perfect, but it works fine. Why wouldn't this work for a gigantic population?
- Note that a Dyson Sphere can host a breathtaking population. It has 5.5x108 times the surface area of Earth.1 Scaled from today's estimated population, that's 4.2x1018 people. It's a bit crowded for my tastes.
We have national governments on Earth today that manage to deal with 1-billion+ populations (Yo! India & China! WOOO!). They have subdivided governing regions just like the U.S. But, let's assume 1-billion is the practical max for a "national" governing unit. How many levels of government would be required for our megapopulation? Well, I'm back to using the familiar U.S. structure. So...
- 50 "states" per "nation" leading us to
- 50 "nations" per "planetary government" (5x1010 people)
- 50 "system governments" (where are the Goa'uld when you need them? Anyway, 2.5x1012)
- 50 "cluster governments" (1.25x1014)
- 50 "sector governments" (6.25x1015)
- 50 "quadrant governments" (3.1x1017)
- 13.5 "spiral arm governments" (4.2x1018)
- 1 "overlord government" aka, "big brother"
It's unlikely that's a galactic population living on the surface of one sphere, but it feels like it. Nevertheless, from city to spiral arm we're talking 11 levels of government.
That's plenty of room for chaos despite enormous organization.
1 The surface area of a 1AU sphere divided by the surface area of Earth. Hopefully I got it right. Dang Windows 10....