I just explained three different models to represent side-effects without leaving the purity. I'm not sure if you are trolling, throwing random big words or simply have a really misguided impression about how computer science connects with math.

Your monad(WorldState, Value) example assumes shared global state. So not a networked system then?

That's not a train-smash for your argument if you can show me a Mathematical model for a global mutex.

When the discussion turns to control-flow you are inevitably in the land of well-ordered imperatives, not Mathematical, lazy-evaluated declaratives.