Remark: Family 229. Sharp three- and four-state reconstruction thresholds

Proves the exact reconstruction threshold $d{\lambda }^2>1$, with nonreconstruction at equality, for three-state symmetric and four-state ferromagnetic broadcasting on regular trees ($d\ge 2$) and observed Poisson trees (mean $d>1$ and $d>0$, respectively), with Poisson advantage averaged without conditioning on survival. The three-state theorem allows both signs of $\lambda$ and gives the exact weak-recovery threshold for the symmetric three-community stochastic block model.

The authors are OpenAI. This companion does not verify the proof. The claim is not a theorem of the Enter series and not a premise of any registered sheet. It does not close O.1. O.1 stays NOT_CLOSED. It is not a de Sitter state and not the QdS external-leg scheme. No quantum coefficient is admitted. No compilation is asserted.