Remark: Family 236. The factor-of-IID threshold for free Ising states on trees

Determines when the free zero-field ferromagnetic Ising state on the infinite $d$-regular tree is a factor of independent vertex labels: exactly when $\tanh \beta \le (d-1{)}^{-1/2}$, including equality, for $d\ge 3$ and $\beta \ge 0$. The construction uses no root and is almost surely equivariant for each fixed tree automorphism, resolving the ferromagnetic case of Lyons's question.

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.