Boundary-Condition Quantum Mechanics V: Emergent Spacetime from Lockstep Bundles

This paper is the fifth part of an ongoing series developing Boundary-Condition Quantum Mechanics (BCQM) as a pre-spacetime, event-graph framework for quantum theory and emergent gravity. BCQM I–IV introduced the core primitives (events, directed edges, propensity kernels) and a finite coherence horizon W_{\mathrm{coh}}, and showed how inertial noise and effective mass can emerge from stochastic event chains without postulating a background manifold. BCQM V takes the first explicit step towards emergent spacetime and gravity by introducing the lockstep hierarchy: single primitive threads with diffusive inertial noise, local bundles of threads with partial lockstep, and a dominant/global lockstep band that can act as an effective “background” clock. Using one-dimensional bundle simulations (built on the BCQM IV_d “soft-rudder” model and an extended glue-axescode), the paper studies how four phenomenological glue axes—shared bias, phase locking, domains, and cadence disorder—control the formation and stability of lockstep bundles. It defines operational diagnostics for: clock quality Q_{\text{clock}} from tick statistics, lockstep length \ell_{\text{lock}}, and an effective mass m_{\text{eff}} \propto 1/A_a(W_{\mathrm{coh}}) from the centre-of-mass inertial noise spectrum. The numerical results map out regimes where bundles behave in a particle-like way (stiff, low COM noise), clock-likeway (high Q_{\text{clock}}), and background-like way (large-scale, long-lived lockstep with relatively low residual noise). This supports the central BCQM V picture: spacetime and matter are two phases of the same lockstep mechanism, with massive particles interpreted as soft defects in a more rigid lockstep background. Finally, the paper sketches how a dominant lockstep band could be projected to an emergent time parameter \tau and a coarse-grained spacetime geometry (for example via diffusion-metric or spectral-embedding constructions), and outlines how bundle-induced distortions of the lockstep pattern provide the bookkeeping for later, fully dynamical gravity models. BCQM V itself remains a Stage-1, pre-spacetime, phenomenology and phase-diagram paper: it does not introduce field equations or a full projector, but instead documents the numerical evidence and conceptual structure that BCQM VI and later work will build on. The simulations underlying this paper are fully reproducible. All configuration files, outputs, and run reports for the bundle and glue-axes models are available in the accompanying BCQM code repositories (linked in the paper and via their own Zenodo records).