BCQM_IV_b — Inertial Noise Toy Models for BCQM IV_b

This repository contains the numerical toy models and testing pipeline used in the Boundary–Condition Quantum Mechanics IV_b work on inertial noise and the coherence horizon (W_{\mathrm{coh}}).

The goal here is not to provide a final physical kernel, but to:

• validate the simulation → spectrum → amplitude → scaling pipeline,
• show that the code correctly detects a non-trivial (W_{\mathrm{coh}})-dependence when it is present, and
• provide a clean (W_{\mathrm{coh}})-blind control model (bcqm_toy_3) which yields an amplitude scaling exponent (\beta \approx 0).

For more detail, see TESTING.md and the “Numerical checks – code validation” appendix of the BCQM IV_b manuscript.

---

Repository layout

At the time of writing the repository contains:

BCQM_IV_b/
  bcqm_toy_3/              # canonical, W_coh-blind control toy
  bcqm_cluster_toy/        # independent-probe cluster COM toy
  archive/                 # archived toy models (positive/negative controls)
  outputs/
    wcoh_scan_phase1/      # example run outputs for bcqm_toy_3
  outputs_cluster/
    cluster_n_scan_v1/     # example cluster N-scan outputs
  TESTING.md               # detailed testing notes
  LICENSE                  # license information
  .gitattributes
  .DS_Store                # (macOS metadata; can be ignored)

bcqm_toy_3/ — canonical control toy

This is the current and canonical toy model used in BCQM IV_b.

• Implements a simple, stationary acceleration kernel on a single thread:
  • an Ornstein–Uhlenbeck–type update (a_{n+1} = (1-\gamma)a_n + \sigma s \xi_n), with fixed (\gamma), (\sigma), and sign mode (s), and Gaussian kicks (\xi_n).
• Crucially, the kernel is independent of (W_{\mathrm{coh}}):
  • (W_{\mathrm{coh}}) appears only in the scan, as a label for different ensembles.
• The code:
  • generates ensembles of acceleration trajectories for each (W_{\mathrm{coh}}) in a scan list,
  • computes one-sided power spectral densities (S_a(\omega)),
  • extracts a total amplitude (A(W_{\mathrm{coh}}) = (\int S_a(\omega),d\omega)^{1/2}) and a characteristic frequency (\omega_c),
  • saves the results to outputs/,
  • and performs a log–log fit (A(W_{\mathrm{coh}}) \propto W_{\mathrm{coh}}^{-\beta}).

In a representative configuration (as described in TESTING.md), the measured amplitudes are essentially constant and the fit returns (\beta) consistent with zero, as expected for a (W_{\mathrm{coh}})-blind kernel.

bcqm_cluster_toy/ — cluster COM toy (independent baseline)

This sibling package implements a simple independent-probe cluster toy used in BCQM IV_b Section6 and AppendixA to test how the centre-of-mass (COM) inertial-noise amplitude scales with the number of probes.

• Each probe thread follows the same $W_{\mathrm{coh}}$-blind Ornstein–Uhlenbeck–type kernel as in bcqm_toy_3.
• A cluster of size $N$ is modelled as $N$ such probes evolved in parallel with independent Gaussian kicks.
• The only collective observable is the COM acceleration, $$ a_{\mathrm{COM},n} = \frac{1}{N} \sum_{i=1}^{N} a_{i,n}, $$ so this branch provides a clean independent-probe baseline: any suppression of COM noise arises purely from averaging.

The cluster_simulate.py driver:

• takes a list of cluster sizes $N \in {2,4,8,16,32,64,128}$,
• generates COM trajectories using the same $(\Delta t, N_\text{steps}, n_\text{ensembles})$ as the single-probe scan,
• computes ensemble-averaged COM spectra and extracts a COM amplitude $A_{\mathrm{COM}}(N)$ and centroid $\omega_{c,\mathrm{COM}}$,

and writes:

• cluster_N{N}.npz files with $(\omega, S_{a,\mathrm{COM}})$, and
• an amplitude_scaling_COM.csv summarising $(N, A_{\mathrm{COM}}, \omega_{c,\mathrm{COM}})$.

In a representative configuration (matching the IV_b figures), a log--log fit of $A_{\mathrm{COM}}(N)$ vs.~$N$ gives a slope $\alpha \simeq -0.495$ with $R^2 \simeq 0.99997$, and the product $A_{\mathrm{COM}}(N) \sqrt{N}$ is constant at the $\sim 1,%$ level across $N=2$--$128$, confirming the expected $N^{-1/2}$ COM suppression for independent probes.

archive/ — historical toy models (optional controls)

The archive/ directory is reserved for historical toy models that were used to validate the pipeline, but are not part of the canonical IV_b narrative. Typical contents include:

• a toy dynamics branch in which (W_{\mathrm{coh}}) is built directly into the step statistics, producing a non-trivial power-law (A(W_{\mathrm{coh}})) scaling (positive control);
• a rudder toy that implements a minimal BCQM-style memory rule with interruptions, showing that not every appearance of (W_{\mathrm{coh}}) produces a clean amplitude power law (negative control).

These models are documented in TESTING.md and the IV_b appendix, but new users only need bcqm_toy_3 to reproduce the main control result.

outputs/wcoh_scan_phase1/

This directory contains example outputs from a standard scan run of the canonical toy:

• individual .npz files for each (W_{\mathrm{coh}}) value (averaged spectra and summary statistics),

• an amplitude_scaling.csv file with columns

  Wcoh, A, omega_c
  

You can delete or regenerate these files as needed.

---

Installation

You need Python 3 with a minimal set of scientific packages. On macOS or Linux, a typical setup might be:

python3 -m venv venv
source venv/bin/activate
pip install numpy pyyaml matplotlib

(Additional packages such as scipy can be installed if you wish to extend the analysis, but they are not strictly required for the basic pipeline.)

---

Running the canonical control scan

From the repository root (BCQM_IV_B/), a typical run looks like:

python3 -m bcqm_toy_3.cli run bcqm_toy_3/configs/wcoh_scan_phase1.yml

This will:

1. Load the specified YAML configuration.
2. For each (W_{\mathrm{coh}}) in the wcoh_values list:
   • generate an ensemble of trajectories,
   • compute and average the acceleration power spectra,
   • extract (A) and (\omega_c),
   • save the results under outputs/wcoh_scan_phase1/.
3. Collect the amplitudes into outputs/wcoh_scan_phase1/amplitude_scaling.csv.
4. Perform a log–log fit (A(W_{\mathrm{coh}}) \propto W_{\mathrm{coh}}^{-\beta}) and print the fitted (\beta) and an uncertainty estimate.

Example output (numbers indicative only):

Running ensemble for W_coh = 5.0 ...
  A = 2.30, omega_c = 0.26
...
Fitted beta: 3.8e-04
Estimated error: 7.2e-04

Running the cluster N-scan

To reproduce the simple cluster results used in BCQM IV_b Section~6, run the cluster driver from the repository root:

python3 -m bcqm_cluster_toy.cli run bcqm_cluster_toy/configs/cluster_n_scan.yml

This will:

1. Load the specified YAML configuration for the cluster scan.
2. For each cluster size $N$ in the N_values list:
   • generate ensembles of COM acceleration trajectories,
   • compute and average the COM power spectra,
   • extract $A_{\mathrm{COM}}(N)$ and $\omega_{c,\mathrm{COM}}$,
   • save the spectra to outputs_cluster/cluster_n_scan_v1/cluster_N{N}.npz.
3. Collect the COM amplitudes into outputs_cluster/cluster_n_scan_v1/amplitude_scaling_COM.csv.

In the configuration used for the IV_b figures, a log--log fit of $A_{\mathrm{COM}}(N)$ against $N$ yields a slope $\alpha \approx -0.495$ with $R^2 \approx 0.99997$, and the product $A_{\mathrm{COM}}(N)\sqrt{N}$ is constant at the $\sim 1,%$ level across $N=2$--$128$, as expected for an independent-probe COM baseline.

---

Relation to BCQM IV_b

This repository is intended as a companion code base for the BCQM IV_b manuscript on inertial noise and the coherence horizon:

• bcqm_toy_3 corresponds to the canonical control toy described in the main text and in the appendix “Numerical checks – code validation”.
• The archived toys in archive/ correspond to the positive and negative controls also described in that appendix.
• TESTING.md mirrors and expands on the appendix, providing a prose description of:
  • the purpose of each toy,
  • the pipeline structure,
  • and the expected qualitative behaviour of (A(W_{\mathrm{coh}})).

Once BCQM IV_b is publicly available, you may wish to add its DOI and citation here.

---

License

See LICENSE for full details.

---

Contact

For questions, comments, or bug reports related to this code or the BCQM IV_b work, please contact the author of the BCQM series.