Boundary-Condition Quantum Mechanics (BCQM): Primitives and Emergence This note proposes a minimal, amplitude-first foundation for BCQM built from three primitives—events, directed edges, and complex edge amplitudes—and a single hop-bounded, locally windowed selection rule that realizes the next event with probability proportional to |K_r|^2. Two operational scales anchor the construction: a short-window envelope (w_env) that controls locality in path sums, and a coherence horizon (W_coh) that reflects an empirical floor on recoverable interference. We give operator-level guarantees (Neumann-series control for the truncated kernel; LLN/CLT for hop counts under a local-boundedness principle), show that no-signalling is preserved, and motivate the envelope from GKLS-type local dephasing. To encode persistent records, inertia-like drift, and curvature from inhomogeneities, we introduce a strictly local back-reaction (Axiom 6) that remodels edge amplitudes near each realized event; we provide concrete, equivariant examples and a small menu of remodeling maps. An appendix outlines simulations with falsifiable hooks: S1: straight-line drift from phase gradients (inertia proxy) S2: PSD knee ≈ 1/W_coh in tick/jitter spectra S3: Zeno-like stall as w_env → 0 S4: stochastic anisotropy with 1/√M self-averaging S5: efficient truncated-kernel computation for hop balls This is a concise, testable proposal; continuum limits and full gravitational dynamics are deferred to follow-ups. v2 note 2025-11-18: This version adds an explicit “Outlook: entangled clusters” paragraph in the Primitives section. It clarifies that the current primitives are intentionally single-particle and that a configuration-space primitive for entangled clusters—consistent with BCQM II’s joint q-wave structure—is deferred to BCQM IV–V. No core definitions or results are changed; the update is conceptual and expository. Build notes: Compiles with biblatex/biber (Overleaf: set the main .tex and recompile twice).
Boundary-Condition Quantum Mechanics (BCQM): Primitives and Emergence This note proposes a minimal, amplitude-first foundation for BCQM built from three primitives—events, directed edges, and complex edge amplitudes—and a single hop-bounded, locally windowed selection rule that realizes the next event with probability proportional to |K_r|^2. Two operational scales anchor the construction: a short-window envelope (w_env) that controls locality in path sums, and a coherence horizon (W_coh) that reflects an empirical floor on recoverable interference. We give operator-level guarantees (Neumann-series control for the truncated kernel; LLN/CLT for hop counts under a local-boundedness principle), show that no-signalling is preserved, and motivate the envelope from GKLS-type local dephasing. To encode persistent records, inertia-like drift, and curvature from inhomogeneities, we introduce a strictly local back-reaction (Axiom 6) that remodels edge amplitudes near each realized event; we provide concrete, equivariant examples and a small menu of remodeling maps. An appendix outlines simulations with falsifiable hooks: S1: straight-line drift from phase gradients (inertia proxy) S2: PSD knee ≈ 1/W_coh in tick/jitter spectra S3: Zeno-like stall as w_env → 0 S4: stochastic anisotropy with 1/√M self-averaging S5: efficient truncated-kernel computation for hop balls This is a concise, testable proposal; continuum limits and full gravitational dynamics are deferred to follow-ups. v2 note 2025-11-18: This version adds an explicit “Outlook: entangled clusters” paragraph in the Primitives section. It clarifies that the current primitives are intentionally single-particle and that a configuration-space primitive for entangled clusters—consistent with BCQM II’s joint q-wave structure—is deferred to BCQM IV–V. No core definitions or results are changed; the update is conceptual and expository. Build notes: Compiles with biblatex/biber (Overleaf: set the main .tex and recompile twice).
This BCQM IV manuscript continues the Boundary-Condition Quantum Mechanics (BCQM) research programme initiated in BCQM I–III and the “Primitives and Emergent Spacetime” note. BCQM treats the \qwave\ as an informational (propensity) field that constrains a single realised particle thread along the chronological axis t^{+}, while a finite coherence horizon \(\Wcoh\) encodes the recoverability of interference along a coherence axis t^{-}. BCQM III introduces a hop-bounded event-growth engine and shows how an effective inertial response \(\meff(\Wcoh)\) emerges in the coarse-grained spacetime regime. BCQM IV asks what irreducible fluctuations remain around this emergent inertial motion once ordinary environmental noise is idealised away. It defines inertial noise as the residual stochastic departure of realised trajectories from the BCQM inertial drift and characterises it via the power spectral density of velocity or acceleration. Within a broad class of local, hop-bounded graph-phase models with finite coherence horizon, the manuscript argues that the inertial noise spectrum has a universal scaling structure, fixed by the same effective relativistic action that governs the deterministic drift. A second objective is to extend the BCQM primitives to entangled clusters, introducing a configuration-space event graph that reproduces the BCQM II joint \qwave\ law while preserving no-signalling. Within this framework the work studies how centre-of-mass inertial noise of a cluster arises from the same graph-based mechanism as for single threads, and outlines numerical experiments and possible observational signatures. This is a technical research document aimed at physicists and mathematically inclined readers familiar with quantum foundations, open quantum systems, and stochastic dynamics. It is part of an evolving series and should be read alongside BCQM I–III and the associated technical note on emergent inertia and invariant action.
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.
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).
This record hosts the preprint “Boundary-Condition Quantum Mechanics IV_b” (BCQM IV_b), a companion paper to BCQM IV in the Boundary-Condition Quantum Mechanics series. The BCQM programme develops a discrete, event-based extension of quantum mechanics with a finite coherence horizon W_coh, from which spacetime structure, inertial response, and an intrinsic inertial-noise floor are intended to emerge. BCQM IV_b focuses on the numerical side of this programme. It develops and tests a minimal inertial-noise pipeline built from a simple, W_coh-blind control kernel (“bcqm_toy_3”) defined at the level of a single event thread. The paper describes how acceleration time series are generated, how one-sided power spectra are estimated, and how a noise amplitude A and characteristic frequency omega_c are extracted and fit to log–log scaling laws. A first set of results demonstrates that, for a deliberately W_coh-independent kernel, the fitted exponent β in the scaling law A(W_coh) ∝ W_coh^{−β} is statistically compatible with zero, as required for a clean control model. The paper then extends the analysis to simple configuration-space cluster toys. An independent-probe cluster model (“bcqm_cluster_toy”) is used to study how the centre-of-mass (COM) inertial-noise amplitude scales with the number of probes N. The measured behaviour A_COM(N) ∝ N^{−1/2} and an almost N-independent spectral centroid confirm the expected COM suppression for independent probes and provide a baseline against which more realistic, correlated cluster kernels can be compared in future work. The dimensionless results are finally mapped into SI units for representative mechanical and atomic benchmark platforms, yielding order-of-magnitude estimates for the BCQM inertial-noise floor and associated thermal crossover temperatures. Taken together, these results provide a numerically validated control model, a documented pipeline from event-level dynamics to inertial-noise spectra, and simple few-body benchmarks that will be used in the gravitational and emergent-curvature analysis developed in BCQM V. The companion simulation and analysis code for this paper is archived separately on Zenodo under DOI 10.5281/zenodo.17815304 and is referenced in the text.
This record contains BCQM VII (v1.0) — Boundary-Condition Quantum Mechanics VII: Stage–2 “cloth” results. The paper advances the BCQM programme by moving beyond the short, fast-mixing active slice (Stage–1) and defining a persistent coarse spatial cloth that can be analysed with stable, reproducible diagnostics without assuming an underlying manifold. Time and ordering (the clock) BCQM retains a mechanistic separation between (i) an operational time/ordering signal and (ii) emergent spatial adjacency. The v_glue module supplies an operational clock-quality observable Q_{\mathrm{clock}} that tracks ordered time/temporal coherence. In Stage–2 we use Q_{\mathrm{clock}} as a stability and sanity check: the clock signal remains robust while the spatial cloth develops a structured core + halo organisation at larger N (notably N=32). What the paper does Builds a Stage–2 cloth object using a Path A cross-link layer on top of the v_glue clock engine (primitive ontology remains pre-spacetime: events + directed connections). Shows why strict edge-level persistence tends to isolate motifs rather than a spanning backbone, motivating a community coarse-graining and a community super-graph representation. Establishes stability of the cloth and super-graph across seeds and through scale-ups to N=32 in the high-coherence regime (n=0.8), including stable geometry diagnostics alongside a growing halo. Demonstrates that thread trajectories are local on the emergent coarse geometry (small hop distances on the community super-graph), supporting the interpretation that particle-like threads “respect” the cloth. What this is not yet This paper establishes a spatial cloth (persistent adjacency and robust diagnostics) with an independent operational ordering/clock signal. It does not yet extract a unified spacetime metric, causal structure, or a continuum limit; those are deferred to later stages. Reproducibility Repository artefacts (configs, CSV tables, and figure-generation scripts) are archived separately as a code release: Code DOI (v1.0): 10.5281/zenodo.18505934 Concept DOI (all versions): 10.5281/zenodo.18497216
“Boundary-Condition Quantum Mechanics IV_c: Diffusive inertia and the limits of binary hop kernels” by Peter M. Ferguson. BCQM IV_c continues the inertial-noise line opened in BCQM IV and IV_b, but now at the level of a single primitive event thread. The paper introduces a “soft-rudder” binary hop kernel, in which a one-dimensional thread moves with velocity ±1 and occasionally flips direction with a coherence-horizon–dependent slip probability. Using the same acceleration power-spectral-density pipeline developed in BCQM IV_b, the work shows that natural single-thread models robustly produce an inertial-noise amplitude scaling A(W_coh) ∝ W_coh^(−β) with β ≈ 1/2, i.e. a fundamentally diffusive universality class. This establishes an important negative result: if BCQM is to reproduce Newtonian inertia and the β ≈ 2 scaling suggested by the broader programme, it cannot do so with isolated primitive threads alone. Instead, long-lived, low-noise “bundles” of threads (shared-event clusters) are required, pushing the next papers (BCQM IV_d and V) towards modelling thread bundles, event-graph defects, and emergent inertial rigidity. BCQM IV_c therefore serves as both a technical baseline and a conceptual waypoint: it closes off overly simple single-thread routes, clarifies where the diffusive floor really sits, and sharpens the motivation for bundle-based models and emergent spacetime structure in the later BCQM stages. A companion open-source Python implementation of the soft-rudder model, including configuration files and numerical checks, is released in a separate Zenodo record and is linked here under “Related/Associated” resources.