Every post you make seems to suggest that you think that sandbox games are necessarily hyper-collaborative and wholly reflexive, with what you refer to as a “player-dm” fully participating in a radically non-privileged way in the game, as the players shape the world by making assertions about it that have to be authoritative or else the game is a linear one.It is better if you want a more pure sandbox game where nobody says any details.
The player-dm does not get to ask the players of characters questions: they just keep quiet and do as they are told. The sword is where the players want it to be or where the game rules or fiction or chance want it to be.
True somewhat. Though my point is very few player-dm's that run sandboxes make up very much of a world. A couple lines, maybe. But often not too much.
And their reaction is Linear or sandboxy.
Well, Quantum Sandbox might be a better term or even Qunatiumania.
Your position is not a matter of sandbox vs linear: yours is purely an assertion about playstyle. I have no doubt whatsoever that this free-form, collaborative world-building is some variation of a sandbox but your mistake is that you think that all sandboxes are like that. Your set theory is messed up, and your Venn diagrams are inverted.