Supplementary Material: Full Proofs and Auxiliary Lemmas for Safety and Liveness Failure Under Budget-Constrained Mining: Composing Backbone Proofs with Endogenous Honest Majority

This document is the supplementary appendix to the SmartNets 2026 conference paper “Safety andLiveness Failure Under Budget-Constrained Mining: Composing Backbone Proofs with EndogenousHonest Majority.” The conference page limit forces the main paper to present results in compressed form.This supplement contains complete proofs of all original results, restated for self-contained reading;auxiliary lemmas used inside those proofs; technical discussion of edge cases (single-honest-miner zerodriftrecurrence, Δ = 0 instantaneous-propagation degeneracy, unbounded-budget tails, ties at threshold);and detailed verification of the constants invoked. Imported results from Garay–Kiayias–Leonardos [3]are quoted with full hypotheses but not re-proven; their proofs are in [2, 3]. The conventions and notationmatch those of the main paper.