This document is the standalone supplementary appendix to the conference paper “The DecentralisationThreshold: When More Validators Reduce Net Security” (SmartNets 2026). Itcontains complete proofs of every theorem and proposition stated in the paper, full derivationsof the closed-form approximation including the Lambert W algebra, a self-contained proof ofthe binomial anti-concentration lemma at the BFT threshold, the Jensen / AM-GM derivationof the heterogeneous Chernoff bound, comparative-statics derivatives, and reproducibility codelistings for every figure, every numerical table, and the heterogeneous Poisson-binomial example.The appendix is self-contained: it restates the model and all definitions in full, so it can be readindependently of the conference paper. It is archived at Zenodo, doi:10.5281/zenodo.20078799.