Boundary-Condition Quantum Mechanics III: A Stochastic Growth Model for Causal Event Chains and the Emergence of Inertia This paper develops the third stage of the Boundary-Condition Quantum Mechanics (BCQM) programme, in which spacetime is modelled as an emergent causal graph of irreversible quantum events. The central object is the q-wave: an informational propensity field that guides the stochastic growth of a particle’s event chain. Building on BCQM II (event ontology and emergent spacetime) and the separate Analytical Proofs note, this work: defines a concrete, Lorentz-respecting mathematical form for the retarded q-wave \psi^+ on a future boundary; specifies a hop-bounded, retarded graph-growth rule (Algorithm 1) that realises a particle’s worldline as a sequence of stochastic “ticks”; ties the lattice regulator and environment window to a finite coherence horizon W_{\mathrm{coh}}, and shows how to take a clean continuum limit; demonstrates numerically that the classical principle of inertia is an emergent statistical consequence of phase coherence: the coarse-grained trajectory follows the path of stationary action, with jitter set by W_{\mathrm{coh}}; shows that the effective inertial parameter scales as m_{\mathrm{eff}} \propto W_{\mathrm{coh}}^{-2}, supported by simulation and an operator-theoretic sketch. Particular care is taken to clarify the role of the “advanced” contribution: in this paper, “advanced” refers only to the advanced Green’s-function branch used to maintain time-symmetric amplitude bookkeeping. The advanced factor enters as the conjugate co-contribution in the t^+/t^- pairing at an event and is absorbed into the normalisation at the probability step. It does not represent backwards-in-time dynamics and cannot be used for signalling; all realised events are ordered along the usual chronological time. The record includes simulation details and parameter tables (Appendix A–C), normalisation and advanced-branch conventions (Appendix D–E), and a brief complexity estimate. A reference implementation of the stochastic event-chain simulations used for Figs. 1–4 is available in the public GitHub repository: Code: https://github.com/PMF57/BCQM_III Archived code and figure-generation scripts: 10.5281/zenodo.17632820This paper is part of an ongoing series on BCQM: BCQM I – Foundations and collapse horizon: 10.5281/zenodo.17191306 Analytical Proofs for BCQM: 10.5281/zenodo.17242311 BCQM II – From quantum events to spacetime: 10.5281/zenodo.17398294 BCQM Primitives – hop-bounded selection rule: 10.5281/zenodo.17495038 BCQM III provides the “engine” linking the event-based ontology to classical inertial motion. The follow-on work (BCQM IV) uses the same framework to analyse the inertial noise spectrum and its prospective experimental signatures.