Methods & Reproducibility

One-command run

From the repository root:

python3 numerics/scripts/run_all.py
quarto render numerics/docs

Run the T15a Bianchi I geometry-labelled benchmark independently:

python3 numerics/scripts/run_t15a_bianchi_i.py

T15a and T15b standalone runs are preserved in numerics/results/ and are merged into the dashboard aggregate when the main suite is regenerated. T15b has a seeded Euclidean persistent-walk run and two Masoliver–Lindenberg comparison protocols; v2 is the longer-horizon follow-up to v1.

Output contract

The runner writes:

  • numerics/results/all-results.json — complete arrays and diagnostics;
  • numerics/results/validation-summary.csv — flat refinement summary;
  • numerics/results/registry.json — run IDs, status, provenance, and claim gates;
  • numerics/docs/data/*.json — dashboard-local copies generated from the same run.

The dashboard aggregate includes the complete T15a case/replicate diagnostics and the full T15b moment, radial-snapshot, transform, Fourier-mode, check, and provenance records. Individual standalone JSON files are also copied into numerics/docs/data/ for direct access from the rendered site.

Every run records the seed, Git revision, Python and NumPy versions, platform, timestamp, and wall time.

NUM-7 freezes its profile definitions, ordering cases, domain, reference grid, output times, population levels, independent final seeds, and density-error gate in numerics/docs/tasks/t15a-bianchi-i-benchmark-v1.md. It evolves the coupled Fourier equations for both density and current, so the nonzero-current initial case does not reuse the zero-current-only NUM-1 helper. Its event-driven sampler draws each initial direction from the prescribed local directional population and flips at the selected Poisson rate.

The T15b Euclidean run samples exact Poisson event counts and conditional Dirichlet segment fractions for an unbounded constant-speed planar flight with uniform heading resets. It checks the exact mean-square displacement and causal-radius bound. Separate v1/v2 runs compare the exact Fourier–Laplace transform and seeded Fourier modes with the Masoliver–Lindenberg fluid-limit telegraph approximation; v2 extends the observation horizon through \(\lambda t=128\).

Numerical methods

Scalar telegraph PDE

The periodic-box PDE is solved mode-by-mode in Fourier space. For each wave number \(k\),

\[ \widehat u_{tt}+2a\widehat u_t+v^2|k|^2\widehat u=0, \]

with \(\partial_tu(0)=0\). This avoids time-step error in NUM-1 through NUM-3.

Velocity-jump ensembles

The number of Poisson events is sampled exactly. Conditional on the event count, event spacings are Dirichlet distributed. The one-dimensional process alternates velocity sign; the two-dimensional process samples a fresh isotropic direction for every segment.

Bianchi IX amplitude

NUM-4 uses a second-order finite-difference telegraph update with an absorbing computational boundary and a CFL-limited \(\alpha\) step. Grid and step size are refined together in the central well. A second full-box calculation rewrites the PDE as a first-order field system and uses adaptive DOP853 integration for the exponentially stiff outer walls. Signed minima, source extrema, negative-cell fraction, and signed mass are reported. The outer-wall study now uses the fixed-box sequence \(48,56,64,72\). Its largest adjacent signed-mass change is below 5%, while the common-interior signed-field \(L^1\) difference is also reported and decreases under refinement. Since the finest field difference remains appreciable, the solver remains provisional rather than a converged microscopic Bianchi IX realization.

Signed-particle Bianchi IX prototype

NUM-6 rewrites the central-well second-order update as a sparse recurrence on the state pair \((\Psi_n,\Psi_{n-1})\). Each walker samples one signed matrix contribution and carries a real signed weight. After each \(\alpha\) step, walkers in the same state are cancelled and resampled. The deterministic finite-difference update at the same grid and step size is the reference. This construction tests an amplitude-level estimator; it does not turn the sign-changing field into a positive probability distribution.

Intertwiner diffusion

The state graph contains the intermediate spins admitted by both pair couplings of a four-valent SU(2) intertwiner. The reference solution is its exact spectral heat kernel. The Monte Carlo process uses uniformization with an exact Poisson event count, including the required endpoint self-loops.

Acceptance rules

  • A causal front is reported separately from density convergence.
  • A passing numerical implementation does not validate a microscopic model.
  • Failed target claims remain visible in the registry.
  • The unit-round-\(S^3\) normalization is fixed in the manuscript, but Bianchi IX remains provisional until the outer-wall field diagnostic converges and a signed microscopic estimator meets its fixed-grid \(L^1<0.10\) gate.
  • The single-node intertwiner model cannot be promoted as a many-node spin-network continuum result.