Boundary-Condition Quantum Mechanics III: A Stochastic Growth Model for Causal Event Chains and the Emergence of Inertia

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.

v2 -  note (updated BCQM III) 2025-11-18: This version clarifies the scope and assumptions of the paper. In particular, it makes explicit that the phase structure \Delta S \approx p_\mu \Delta x^\mu is used at the emergent spacetime level as an effective description, and that deriving this structure (and the underlying Lorentzian geometry) from the primitive BCQM event-graph rules is deferred to future work (BCQM IV–V). No results, simulations, or numerical conclusions have been changed; the update is purely conceptual and expository.