This supplement collects the long technical derivations referenced by the main paper. It is organised into five appendices that mirror the conceptual ladder of the main text. Appendix A develops the finite-state CTMC and the linear birth–death–immigration process, derives the closed-form bridge-expectation and Fisher-score sufficient statistics for TKF91 and TKF92, resolves the L’Hôpital singularity at \(\insrate {}={}\delrate \), and discusses the relationship between TKF92 and the latent-boundary-free General Geometric Indel (GGI) model. Appendix B collects the closed-form substitution M-steps for TKF91 / TKF92 / MixDom in their many GTR specialisations, the stochastic-variational Baum–Welch loop with its convergence theorem and linearised analysis at stationarity, Maraschino (the TKF92 cherry-distilled generalisation of CherryML) and its tree-level inference algorithms (FSA, BeamASR, VarAnc, svi-VarAnc), the mixture-of-trees variational ancestral presence/absence inference, and the structural bias of the BP cumulant under a column-factorised variational field. Appendix C develops the recursive TKF family: MixDom, the hierarchical-mixture-of-domains generalisation of TKF92, with its exact closed-form M-step via six-step chain restoration through a fully exploded null-state model; the order-1 Maraschino adjacency distillation; algebraic distillation of MixDom; the MixDom-specific SVI-BW convergence considerations; the tree-level VEM and ancestral-reconstruction algorithms; the generalised phylo-HMM; the labeled-MixDom Singlet and WFST; and the recursive-grammar-elaboration rules together with worked recursive examples (L-TKF, TKFST, TKFStack, TKF-Genome). Appendix D develops the TKF-DP generative model, the class-level path-measure variational likelihood with pairwise bridge expectations, the time-indexed gravestone-augmented pair SCFG, the SVI inference loop, and the pairwise alignment postprocessing landscape. Appendix E develops the infinite Pair HMM as the principled fixed point and the Gibbs+MH+replica-exchange MCMC sampler that draws from it.