BCQM IV_c — Soft-Rudder Single-Thread Inertial Noise

This repository contains the reference implementation for

Boundary-Condition Quantum Mechanics IV_c:
Diffusive inertia and the limits of binary hop kernels

It implements the single-thread, binary soft-rudder model used in BCQM IV_c to show that, for natural slip laws, the inertial-noise amplitude scales with the coherence horizon as [ A(W_{\mathrm{coh}}) \sim W_{\mathrm{coh}}^{-\beta}, \qquad \beta \approx \tfrac{1}{2}, ] i.e. single primitive threads live in a diffusive universality class.

The code reproduces the numerical results and plots in the paper:

• the amplitude vs. (W_{\mathrm{coh}}) scaling,
• the fitted exponent (\beta \approx 0.5) for the soft-rudder kernel.

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Repository layout

bcqm_soft_rudder/
    __init__.py
    model.py          # soft-rudder kernel and trajectory generator
    spectra.py        # PSD estimation and amplitude extraction
    cli.py            # command-line interface for scans
    fit_beta.py       # standalone beta-fit helper

configs/
    wcoh_scan_soft_rudder.yml  # reference W_coh scan configuration

outputs_soft_rudder/
    (created on first run)
    amplitude_scaling_soft_rudder.csv
    Wcoh_5.npz
    Wcoh_10.npz
    ...
    Wcoh_160.npz

README.md
TESTING.md                # numerical checks and validation notes
pyproject.toml / setup.cfg (if present)

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Dependencies

Tested with:

• Python 3.10+
• numpy
• scipy
• pyyaml
• matplotlib (only needed if you want to make plots yourself)

You can install everything into a local virtual environment, for example:

python3 -m venv .venv
source .venv/bin/activate  # on macOS / Linux
# .venv\Scripts\activate   # on Windows (PowerShell / cmd)

pip install -r requirements.txt
# or, if using pyproject.toml:
# pip install .

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Quick start: reproduce the reference W_coh scan

From the repository root:

source .venv/bin/activate    # if using a virtual environment

python3 -m bcqm_soft_rudder.cli run configs/wcoh_scan_soft_rudder.yml

You should see output similar to:

Scanning W_coh values: [  5.  10.  20.  40.  80. 160.]
W_coh = 5:   A = 0.02478, omega_c = 2.08
W_coh = 10:  A = 0.01754, omega_c = 1.84
W_coh = 20:  A = 0.01235, omega_c = 1.71
W_coh = 40:  A = 0.00873, omega_c = 1.64
W_coh = 80:  A = 0.00620, omega_c = 1.61
W_coh = 160: A = 0.00440, omega_c = 1.59

This run will create:

• outputs_soft_rudder/amplitude_scaling_soft_rudder.csv
• outputs_soft_rudder/Wcoh_*.npz (ensemble PSD data for each (W_{\mathrm{coh}}))

To fit the exponent (\beta) from the CSV file:

python3 -m bcqm_soft_rudder.fit_beta outputs_soft_rudder/amplitude_scaling_soft_rudder.csv

Expected output (up to statistical fluctuations):

Fitted beta (A ~ W^(-beta)): beta = 0.499

which is the diffusive scaling reported in BCQM IV_c.

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Configuration file

The scan is controlled by a simple YAML file, e.g.:

simulation:
  dt: 1.0
  n_steps: 16384
  n_ensembles: 64
  seed: 12345

scan:
  wcoh_values: [5.0, 10.0, 20.0, 40.0, 80.0, 160.0]
  label: soft_rudder_wcoh_scan

• dt — time step (fixed to 1.0 in the IV_c paper; kept configurable for convenience).
• n_steps — number of time steps per trajectory; higher values improve spectral resolution at the cost of CPU time.
• n_ensembles — number of independent trajectories in the ensemble; higher values reduce statistical noise.
• seed — random seed for reproducibility.
• wcoh_values — list of (W_{\mathrm{coh}}) values to scan.
• label — used to tag output files in outputs_soft_rudder/.

You can create additional config files (e.g. to test different (W_{\mathrm{coh}}) ranges or ensemble sizes) and pass them to the CLI in the same way.

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Model details (soft-rudder kernel)

The core of the model (in bcqm_soft_rudder/model.py) is:

• A single binary velocity thread (v_n \in {+1, -1}).
• A fixed-step position update (x_{n+1} = x_n + v_n).
• A slip law: [ q(W_{\mathrm{coh}}) = 1 - p_{\mathrm{stay}}(W_{\mathrm{coh}}) = \frac{k}{W_{\mathrm{coh}}}, ] with (k = 2.0), so that the direction flips with probability (q(W_{\mathrm{coh}})) on each tick.

Acceleration is defined as the second difference, [ a_n = x_{n+1} - 2 x_n + x_{n-1}, ] and the acceleration power spectral density is estimated via Welch’s method (see spectra.py). The amplitude [ A(W_{\mathrm{coh}}) = \sqrt{\int_0^\infty S_a(\omega), d\omega} ] is obtained by integrating the ensemble-averaged PSD.

The pipeline is identical to that used in the BCQM IV_b control model, so the two codes can be compared directly.

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Output files

A typical run generates:

• outputs_soft_rudder/amplitude_scaling_soft_rudder.csv

  CSV with columns:

  Wcoh, A, omega_c
  5.0, 0.02478188, 2.08026
  10.0, 0.01753947, 1.83590
  ...
  

• outputs_soft_rudder/Wcoh_XX.npz

  For each (W_{\mathrm{coh}}), an .npz file containing:

  • the ensemble of acceleration time series, and/or
  • the ensemble-averaged PSD used to compute (A) and (\omega_c).

The precise contents are documented in TESTING.md.

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Numerical checks

A separate TESTING.md file documents:

• Convergence checks in n_steps and n_ensembles.
• Sensitivity of (\beta) to the slip prefactor (k).
• Comparisons with known telegraph-noise scaling predictions ((\beta \approx \alpha/2) for (q(W_{\mathrm{coh}}) \propto W_{\mathrm{coh}}^{-\alpha})).

If you modify the kernel or the YAML configuration, you can re-run the tests and add short notes to TESTING.md describing any changes.

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Relation to the BCQM IV_c paper

This code is intended as the companion implementation to the BCQM IV_c paper:

Boundary-Condition Quantum Mechanics IV_c: Diffusive inertia and the limits of binary hop kernels,
Peter M. Ferguson, (Year, DOI to be added).

Every figure and numerical claim involving the single-thread soft-rudder model in IV_c should be reproducible by:

1. editing or copying the reference YAML configuration(s),
2. running the CLI as above, and
3. plotting the resulting CSV / .npz outputs.

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Citation

If you use this code in academic work, please cite the BCQM IV_c paper and, if applicable, the Zenodo record for this repository (once minted). A template BibTeX entry might look like:

@misc{Ferguson_BCQM_IVc_code,
  author       = {Peter M. Ferguson},
  title        = {{BCQM IV\_c soft-rudder single-thread inertial-noise code}},
  year         = {2025},
  howpublished = {\url{<Zenodo-URL-or-github-URL>}},
  note         = {Companion code to "Boundary-Condition Quantum Mechanics IV\_c: Diffusive inertia and the limits of binary hop kernels"}
}

Update the year, URL and (when available) DOI as appropriate.

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License

This code is released under the MIT License. See LICENSE for details.