Benchmarks

MHX benchmark workflows are intentionally command-line reproducible. The active catalog now includes a FAST reduced-MHD smoke run, exact linear physics gates, Harris tearing eigenvalue checks, nonlinear energy-budget and duration gates, Orszag–Tang and turbulence media, seed-QI, neural-ODE reproducibility, and restartable Rutherford-executor artifacts.

mhx benchmark run \
  --config examples/linear_tearing.toml \
  --outdir outputs/benchmarks/linear_tearing_fast \
  --gif
mhx benchmark validate outputs/benchmarks/linear_tearing_fast

Expected files include:

  • manifest.json

  • diagnostics.json

  • trajectory.npz

  • figures/energy_history.png

  • figures/flux_final.png

  • figures/mode_amplitude.png

  • figures/flux_movie.gif

  • report.json

  • report.md

  • validation.json

Detailed command index

Use this section as the detailed command catalog that the README intentionally does not carry. Most benchmark commands write claim_level = "validation"; mhx benchmark run writes smoke, and campaign planning commands can write production_template.

mhx benchmark run --config examples/linear_tearing.toml --outdir outputs/benchmarks/linear_tearing_fast --gif
mhx benchmark validate outputs/benchmarks/linear_tearing_fast
mhx benchmark decay --outdir outputs/benchmarks/resistive_decay
mhx benchmark scaling --outdir outputs/benchmarks/reconnection_scaling
mhx benchmark fkr-window --outdir outputs/benchmarks/fkr_window
mhx benchmark fkr-growth --outdir outputs/benchmarks/fkr_growth_rate
mhx benchmark harris-delta-prime --outdir outputs/benchmarks/harris_delta_prime
mhx benchmark linear-tearing-eigenvalue --outdir outputs/benchmarks/linear_tearing_eigenvalue
mhx benchmark linear-tearing-dispersion --outdir outputs/benchmarks/linear_tearing_dispersion
mhx benchmark linear-tearing-layer --outdir outputs/benchmarks/linear_tearing_layer
mhx benchmark linear-tearing-timedomain --outdir outputs/benchmarks/linear_tearing_timedomain
mhx benchmark linearized-rhs --outdir outputs/benchmarks/linearized_rhs
mhx benchmark reduced-mhd-eigenmode --outdir outputs/benchmarks/reduced_mhd_eigenmode
mhx benchmark cosine-equilibrium-linearization --outdir outputs/benchmarks/cosine_equilibrium_linearization
mhx benchmark current-sheet-eigenvalue --outdir outputs/benchmarks/periodic_current_sheet_eigenvalue
mhx benchmark current-sheet-timedomain --outdir outputs/benchmarks/periodic_current_sheet_timedomain
mhx benchmark current-sheet-nonlinear-bridge --outdir outputs/benchmarks/periodic_current_sheet_nonlinear_bridge
mhx benchmark double-harris-growth --outdir outputs/benchmarks/periodic_double_harris_nonlinear_growth
mhx benchmark double-harris-long-run --outdir outputs/benchmarks/periodic_double_harris_seeded_long_run --movies
mhx benchmark double-harris-convergence --outdir outputs/benchmarks/periodic_double_harris_convergence
mhx benchmark nonlinear-energy-budget --outdir outputs/benchmarks/nonlinear_energy_budget
mhx benchmark orszag-tang --outdir outputs/benchmarks/orszag_tang_vortex --movies
mhx benchmark decaying-turbulence --outdir outputs/benchmarks/decaying_mhd_turbulence --movies
mhx benchmark forced-turbulent-reconnection --outdir outputs/benchmarks/forced_turbulent_reconnection --movies
mhx benchmark forced-turbulent-reconnection-readiness-check outputs/benchmarks/forced_turbulent_reconnection
mhx benchmark nonlinear-duration-audit --outdir outputs/benchmarks/nonlinear_duration_audit
mhx benchmark duration-policy --outdir outputs/benchmarks/duration_policy
mhx benchmark diffusion-eigenvalue --outdir outputs/benchmarks/diffusion_eigenvalue
mhx benchmark power-iteration --outdir outputs/benchmarks/power_iteration
mhx benchmark arnoldi --outdir outputs/benchmarks/arnoldi
mhx benchmark timing --outdir outputs/benchmarks/timing --repeats 3 --warmups 1
mhx benchmark catalog --outdir outputs/benchmarks/catalog
mhx benchmark seed-robust-qi --outdir outputs/benchmarks/seed_robust_qi
mhx benchmark seed-robust-qi-sweep --outdir outputs/benchmarks/seed_robust_qi_sweep
mhx neural-ode dataset --outdir outputs/neural_ode/seed_qi_fast
mhx neural-ode train --outdir outputs/neural_ode/latent_ode_fast
mhx campaign rutherford-template --outdir outputs/campaigns/rutherford_template
mhx campaign rutherford-run-fast --outdir outputs/campaigns/rutherford_fast
mhx campaign rutherford-plan-production --outdir outputs/campaigns/rutherford_production_plan
mhx campaign rutherford-resume-plan outputs/campaigns/rutherford_production_plan
mhx campaign rutherford-execute outputs/campaigns/rutherford_production_plan --max-steps 128
python examples/make_rutherford_production_plan.py --outdir outputs/examples/rutherford_production_plan
python examples/run_rutherford_production_chunk.py --outdir outputs/examples/rutherford_chunk --movies
python examples/run_orszag_tang.py --outdir outputs/examples/orszag_tang --nx 64 --ny 64 --t-end 6
python examples/train_latent_ode_fast.py --outdir outputs/examples/latent_ode_fast
mhx validate all --outdir outputs/validation_suite
mhx validate readiness --suite outputs/validation_suite --outdir outputs/validation_readiness
mhx validate paper-pipeline --outdir outputs/paper_pipeline

mhx validate all includes the forced-turbulent-reconnection readiness check as a separate required public-release gate after generating a FAST forced run. This accounts for the validation-only claim boundary in release readiness while still leaving nonlinear publication claims blocked pending production evidence. mhx validate paper-pipeline wraps that suite, the readiness report, and a top-level recursive artifact manifest into the bundle documented in paper_pipeline.md.

The equation-heavy physics gates and still figures are on validation.md. Campaign scaffolds and restartable execution are documented in campaigns.md and campaign_runner.md. Neural-ODE baseline comparison and fitted latent-ODE outputs are documented in neural_ode_reproducibility.md.

Orszag–Tang nonlinear vortex example

mhx benchmark orszag-tang \
  --outdir outputs/benchmarks/orszag_tang_vortex \
  --nx 64 --ny 64 --t-end 6 --save-every 40 --movies

Expected files include:

  • manifest.json

  • diagnostics.json

  • validation.json

  • orszag_tang_vortex.npz

  • figures/orszag_tang_summary.png

  • figures/orszag_tang_flux.gif

  • figures/orszag_tang_current.gif

  • figures/orszag_tang_vorticity.gif

This is an incompressible reduced-MHD adaptation of the Orszag–Tang 1979 vortex. It is intended as a nonlinear morphology, energy-decay, divergence, and high-\(k\) cascade gate, not as a compressible full-MHD shock benchmark.

Decaying turbulence and forced turbulent reconnection

mhx benchmark decaying-turbulence \
  --outdir outputs/benchmarks/decaying_mhd_turbulence \
  --nx 64 --ny 64 --t-end 8 --save-every 20 --movies

mhx benchmark forced-turbulent-reconnection \
  --outdir outputs/benchmarks/forced_turbulent_reconnection \
  --nx 64 --ny 64 --t-end 80 --save-every 100 --movies

The decaying case starts from a deterministic band-limited reduced-MHD state and gates finite fields, dissipative total energy, current-density growth, and high-wavenumber transfer. The forced case adds a periodic current sheet, broadband perturbations, weak large-scale vorticity forcing, an X/O-aware reconnection proxy when critical points are found, and a conservative fallback proxy when they are not.

Expected files include:

  • diagnostics.json

  • validation.json

  • decaying_mhd_turbulence.npz or forced_turbulent_reconnection.npz

  • figures/decaying_mhd_turbulence_summary.png or figures/forced_turbulent_reconnection_summary.png

  • optional figures/*_flux.gif and figures/*_current.gif with --movies

These examples are anchored to 2-D MHD turbulence/current-sheet studies and turbulent reconnection ideas, but they remain validation media. They do not claim 3-D Lazarian–Vishniac fast reconnection or converged turbulence statistics.

Exact-decay validation benchmark

The first physics-gated numerical validation is a single Fourier mode in the resistive induction limit:

mhx benchmark decay --outdir outputs/benchmarks/resistive_decay

This checks the literature-standard diffusion law \(\psi_k(t)=\psi_k(0)\exp(-\eta |k|^2t)\) and \(E_B(t)=E_B(0)\exp(-2\eta |k|^2t)\). It writes:

  • diagnostics.json

  • validation.json

  • decay_history.npz

  • figures/decay_amplitude.png

  • figures/decay_energy.png

  • figures/decay_relative_error.png

The same benchmark is documented with figures on the validation page.

Reconnection and linear-operator gates

The benchmark roadmap includes analytic power-law gates for expected FKR, Sweet-Parker plasmoid, and ideal-tearing exponents plus linear operator and direct eigenvalue gates:

mhx benchmark scaling --outdir outputs/benchmarks/reconnection_scaling
mhx benchmark fkr-window --outdir outputs/benchmarks/fkr_window
mhx benchmark fkr-growth --outdir outputs/benchmarks/fkr_growth_rate
mhx benchmark harris-delta-prime --outdir outputs/benchmarks/harris_delta_prime
mhx benchmark linear-tearing-eigenvalue --outdir outputs/benchmarks/linear_tearing_eigenvalue
mhx benchmark linear-tearing-dispersion --outdir outputs/benchmarks/linear_tearing_dispersion
mhx benchmark linear-tearing-layer --outdir outputs/benchmarks/linear_tearing_layer
mhx benchmark linear-tearing-timedomain --outdir outputs/benchmarks/linear_tearing_timedomain
mhx benchmark linearized-rhs --outdir outputs/benchmarks/linearized_rhs
mhx benchmark reduced-mhd-eigenmode --outdir outputs/benchmarks/reduced_mhd_eigenmode
mhx benchmark cosine-equilibrium-linearization --outdir outputs/benchmarks/cosine_equilibrium_linearization
mhx benchmark current-sheet-eigenvalue --outdir outputs/benchmarks/periodic_current_sheet_eigenvalue
mhx benchmark current-sheet-timedomain --outdir outputs/benchmarks/periodic_current_sheet_timedomain
mhx benchmark current-sheet-nonlinear-bridge --outdir outputs/benchmarks/periodic_current_sheet_nonlinear_bridge
mhx benchmark double-harris-growth --outdir outputs/benchmarks/periodic_double_harris_nonlinear_growth
mhx benchmark double-harris-long-run --outdir outputs/benchmarks/periodic_double_harris_seeded_long_run
mhx benchmark double-harris-convergence --outdir outputs/benchmarks/periodic_double_harris_convergence
mhx benchmark nonlinear-energy-budget --outdir outputs/benchmarks/nonlinear_energy_budget
mhx benchmark nonlinear-duration-audit --outdir outputs/benchmarks/nonlinear_duration_audit
mhx benchmark duration-policy --outdir outputs/benchmarks/duration_policy
mhx benchmark diffusion-eigenvalue --outdir outputs/benchmarks/diffusion_eigenvalue
mhx benchmark power-iteration --outdir outputs/benchmarks/power_iteration
mhx benchmark arnoldi --outdir outputs/benchmarks/arnoldi
mhx benchmark catalog --outdir outputs/benchmarks/catalog

This writes:

  • diagnostics.json

  • validation.json

  • scaling_history.npz

  • figures/fkr_scaling.png

  • figures/plasmoid_scaling.png

  • figures/ideal_tearing_scaling.png

The FKR-window command writes:

  • diagnostics.json

  • validation.json

  • fkr_window.npz

  • figures/fkr_constant_psi_window.png

The FKR growth-rate command writes:

  • diagnostics.json

  • validation.json

  • fkr_growth_rate.npz

  • figures/fkr_growth_rate.png

The Harris Delta-prime command writes:

  • diagnostics.json

  • validation.json

  • harris_delta_prime.npz

  • figures/harris_delta_prime.png

The direct Harris-sheet tearing eigenvalue command writes:

  • diagnostics.json

  • validation.json

  • linear_tearing_eigenvalue.npz

  • figures/linear_tearing_eigenvalue.png

The finite-domain tearing dispersion command writes:

  • diagnostics.json

  • validation.json

  • linear_tearing_dispersion.npz

  • figures/linear_tearing_dispersion.png

The Harris eigenfunction-layer command writes:

  • diagnostics.json

  • validation.json

  • linear_tearing_layer.npz

  • figures/linear_tearing_layer.png

The time-domain Harris eigenmode replay command writes:

  • diagnostics.json

  • validation.json

  • linear_tearing_timedomain.npz

  • figures/linear_tearing_timedomain.png

The linearized-RHS command writes:

  • diagnostics.json

  • validation.json

  • linearized_rhs.npz

  • figures/linearized_rhs_errors.png

The reduced-MHD eigenmode command writes:

  • diagnostics.json

  • validation.json

  • reduced_mhd_linear_eigenmode.npz

  • figures/reduced_mhd_linear_eigenmode_errors.png

The cosine-equilibrium linearization command writes:

  • diagnostics.json

  • validation.json

  • cosine_equilibrium_linearization.npz

  • figures/cosine_equilibrium_linearization_errors.png

The periodic current-sheet eigenvalue command writes:

  • diagnostics.json

  • validation.json

  • periodic_current_sheet_eigenvalue.npz

  • figures/periodic_current_sheet_spectrum.png

The periodic current-sheet time-domain replay command writes:

  • diagnostics.json

  • validation.json

  • periodic_current_sheet_timedomain.npz

  • figures/periodic_current_sheet_timedomain.png

The nonlinear current-sheet differentiability bridge command writes:

  • diagnostics.json

  • validation.json

  • periodic_current_sheet_nonlinear_bridge.npz

  • figures/periodic_current_sheet_nonlinear_bridge.png

The periodic double-Harris nonlinear-growth command writes:

  • diagnostics.json

  • validation.json

  • periodic_double_harris_nonlinear_growth.npz

  • figures/periodic_double_harris_nonlinear_growth.png

The periodic double-Harris seeded long-run command writes:

  • diagnostics.json

  • validation.json

  • periodic_double_harris_seeded_long_run.npz, including reconnected_flux, seed_mode_reconnected_flux, rutherford_island_width, dominant low-mode indices, X/O counts, energies, and saved base/seeded fields

  • figures/periodic_double_harris_seeded_long_run.png

  • optional figures/periodic_double_harris_flux.gif and figures/periodic_double_harris_current.gif with --movies

The periodic double-Harris convergence command writes:

  • diagnostics.json

  • validation.json

  • periodic_double_harris_convergence.npz

  • figures/periodic_double_harris_convergence.png

The periodic double-Harris parameter-sweep command writes:

  • diagnostics.json

  • validation.json

  • periodic_double_harris_parameter_sweep.npz

  • figures/periodic_double_harris_parameter_sweep.png

The periodic double-Harris promotion checker writes:

  • promotion_readiness.json

  • validation.json

  • figures/promotion_matrix.png

  • manifest.json and artifact_manifest.json

The nonlinear energy-budget command writes:

  • diagnostics.json

  • validation.json

  • nonlinear_energy_budget.npz

  • figures/nonlinear_energy_budget.png

The nonlinear duration-audit command writes:

  • diagnostics.json

  • validation.json

  • nonlinear_duration_audit.npz

  • figures/nonlinear_duration_audit.png

The duration-policy command writes:

  • duration_policy.json

  • duration_policy.md

  • validation.json

  • manifest.json

The diffusion-eigenvalue command writes:

  • diagnostics.json

  • validation.json

  • diffusion_eigenvalue.npz

  • figures/diffusion_eigenvalue_errors.png

The power-iteration command writes:

  • diagnostics.json

  • validation.json

  • power_iteration_history.npz

  • figures/power_iteration_history.png

The Arnoldi command writes:

  • diagnostics.json

  • validation.json

  • arnoldi_spectrum.npz

  • figures/arnoldi_ritz_values.png

The catalog command writes:

  • benchmark_catalog.json

  • benchmark_catalog.md

  • manifest.json

CI artifacts

Every push runs a benchmark-artifacts CI job. It executes deterministic FAST pipelines:

mhx benchmark run --config examples/linear_tearing.toml --outdir outputs/ci/linear_tearing_fast --gif
mhx benchmark validate outputs/ci/linear_tearing_fast
mhx benchmark decay --outdir outputs/ci/resistive_decay
mhx benchmark scaling --outdir outputs/ci/reconnection_scaling
mhx benchmark fkr-window --outdir outputs/ci/fkr_window
mhx benchmark fkr-growth --outdir outputs/ci/fkr_growth_rate
mhx benchmark harris-delta-prime --outdir outputs/ci/harris_delta_prime
mhx benchmark linear-tearing-eigenvalue --outdir outputs/ci/linear_tearing_eigenvalue
mhx benchmark linear-tearing-dispersion --outdir outputs/ci/linear_tearing_dispersion
mhx benchmark linear-tearing-layer --outdir outputs/ci/linear_tearing_layer
mhx benchmark linear-tearing-timedomain --outdir outputs/ci/linear_tearing_timedomain
mhx benchmark linearized-rhs --outdir outputs/ci/linearized_rhs
mhx benchmark reduced-mhd-eigenmode --outdir outputs/ci/reduced_mhd_eigenmode
mhx benchmark cosine-equilibrium-linearization --outdir outputs/ci/cosine_equilibrium_linearization
mhx benchmark current-sheet-eigenvalue --outdir outputs/ci/periodic_current_sheet_eigenvalue
mhx benchmark current-sheet-timedomain --outdir outputs/ci/periodic_current_sheet_timedomain
mhx benchmark current-sheet-nonlinear-bridge --outdir outputs/ci/periodic_current_sheet_nonlinear_bridge
mhx benchmark double-harris-growth --outdir outputs/ci/periodic_double_harris_nonlinear_growth
mhx benchmark double-harris-convergence --outdir outputs/ci/periodic_double_harris_convergence
mhx benchmark nonlinear-energy-budget --outdir outputs/ci/nonlinear_energy_budget
mhx benchmark orszag-tang --outdir outputs/ci/orszag_tang_vortex
mhx benchmark decaying-turbulence --outdir outputs/ci/decaying_mhd_turbulence
mhx benchmark forced-turbulent-reconnection --outdir outputs/ci/forced_turbulent_reconnection
mhx benchmark forced-turbulent-reconnection-readiness-check outputs/ci/forced_turbulent_reconnection
mhx benchmark nonlinear-duration-audit --outdir outputs/ci/nonlinear_duration_audit
mhx benchmark duration-policy --outdir outputs/ci/duration_policy
mhx benchmark diffusion-eigenvalue --outdir outputs/ci/diffusion_eigenvalue
mhx benchmark power-iteration --outdir outputs/ci/power_iteration
mhx benchmark arnoldi --outdir outputs/ci/arnoldi
mhx benchmark timing --outdir outputs/ci/timing --repeats 1 --warmups 0
mhx benchmark catalog --outdir outputs/ci/catalog
mhx benchmark seed-robust-qi --outdir outputs/ci/seed_robust_qi
mhx benchmark seed-robust-qi-sweep --outdir outputs/ci/seed_robust_qi_sweep
mhx neural-ode dataset --outdir outputs/ci/neural_ode_dataset
mhx neural-ode train --outdir outputs/ci/neural_ode_latent_fit
mhx campaign rutherford-plan-production --outdir outputs/ci/rutherford_plan
mhx campaign rutherford-execute outputs/ci/rutherford_plan --max-steps 4
mhx validate all --outdir outputs/ci/validation_suite
mhx validate readiness --suite outputs/ci/validation_suite --outdir outputs/ci/readiness
python examples/make_readme_media.py
mhx run examples/linear_tearing_twofluid_toy.toml --outdir outputs/ci/twofluid_toy
mhx figures outputs/ci/twofluid_toy --gif
mhx report outputs/ci/twofluid_toy
mhx run examples/linear_tearing_plugin_demo.toml --outdir outputs/ci/plugin_demo
mhx figures outputs/ci/plugin_demo --gif
mhx report outputs/ci/plugin_demo
mhx artifact-manifest outputs/ci

The job uploads outputs/ci as the mhx-fast-artifacts GitHub Actions artifact. Readers can download it to inspect manifests, reports, PNG figures, GIF movies, and artifact_manifest.json checksums generated from the exact commit under test.

The catalog file outputs/ci/catalog/benchmark_catalog.md gives readers a single table of active FAST gates, schemas, commands, and expected artifacts.

For a single pass/fail evidence bundle, run:

mhx validate all --outdir outputs/validation_suite

This command executes the active deterministic FAST validation gates and writes:

  • outputs/validation_suite/validation_suite.json

  • outputs/validation_suite/validation_suite.md

  • outputs/validation_suite/artifact_manifest.json

  • one subdirectory per validation case, each with its own validation.json and manifest.json.

FAST timing benchmark

MHX records performance as an artifact, not as a brittle absolute CI gate:

mhx benchmark timing --outdir outputs/benchmarks/timing --repeats 3 --warmups 1

This measures wall-clock time and Python allocation peaks for:

  • linear_tearing_fast: the active reduced-MHD smoke run.

  • resistive_decay_fast: the exact single-mode resistive validation case.

  • reconnection_scaling: analytic FKR/plasmoid/ideal-tearing scaling gates.

The command writes:

  • timing.json

  • timing.md

  • figures/timing_summary.png

  • manifest.json

The CI job uses --repeats 1 --warmups 0 to keep runtime low and uploads the result in outputs/ci/timing/. Timing comparisons should be made between artifacts from the same runner class; MHX does not currently fail CI on absolute speed.

Theory scaffolds

MHX includes analytic scaling estimates for benchmark planning and reports. They are not replacements for calibrated eigenvalue calculations.

For the constant-\(\psi\) Furth-Killeen-Rosenbluth tearing regime, MHX uses the Harris-sheet outer-region proxy

\[ \Delta' a = 2\left[(ka)^{-1} - ka\right], \]

and the order-unity-coefficient-free scaling

\[ \gamma \tau_a \sim S_a^{-3/5}(ka)^{2/5}(\Delta'a)^{4/5}. \]

For Sweet-Parker plasmoid estimates, MHX includes the Loureiro scaling

\[ \gamma_{\max}\tau_A \sim S^{1/4}, \qquad k_{\max}L \sim S^{3/8}. \]

For ideal tearing planning, MHX includes the Pucci-Velli aspect-ratio scaling

\[ a/L \sim S^{-1/3}. \]

The separate FKR-window gate checks the sampled constant-\(\psi\) regime by plotting \(\gamma\tau_a\) and \(\Delta'\delta\) versus \(ka\) at fixed \(S_a\). This prevents future numerical tearing comparisons from silently mixing FKR and large-\(\Delta'\) Coppi regimes.

The Harris Delta-prime gate numerically integrates the ideal outer tearing equation for a Harris sheet and recovers \(\Delta'a=2[(ka)^{-1}-ka]\). It validates the outer-region target but still does not solve the resistive FKR inner-layer eigenvalue problem.

The direct Harris-sheet tearing eigenvalue gate solves a finite-difference linear reduced-MHD eigenproblem for \(S=1000\), \(ka=0.5\), and \(d/a=10\). It checks the unstable tearing eigenvalue against a published growth rate \(\gamma\simeq0.0131\), verifies grid extrapolation in \(\Delta x^2\), checks the dense eigenpair residual, confirms tearing parity, and verifies a stable \(ka=1.2\) control has no positive-growth eigenvalue. This is a single reference eigenproblem, not yet a full FKR/Coppi dispersion scan.

The finite-domain tearing dispersion gate repeats the same eigenproblem over a small \(ka\) scan, verifies positive growth for sampled \(0<ka<1\), verifies stable controls above \(ka=1\), checks dense eigenpair residuals, and retains the \(ka=0.5\) literature anchor. This is the first dispersion-shaped tearing gate, but it is still FAST and finite-domain rather than a production asymptotic FKR/Coppi study.

The Harris eigenfunction-layer gate repeats the direct eigenproblem over a small \(S\) scan and measures half-maximum widths of the flux, flow, and current perturbation profiles. It gates monotonic flow-layer narrowing, outer-flux width stability, broad fitted-slope ranges, and eigen residuals. This is a shape/localization validation, not an asymptotic exponent claim.

The time-domain Harris eigenmode replay gate integrates the same discrete operator as \(\dot q=Lq\) from the selected tearing eigenvector. It fits \(\log\|q(t)\|_2\) and gates the recovered growth rate against the dense eigenvalue, while also checking RK4 amplitude error and final mode-shape alignment. This closes the diagnostic loop between eigenvalue selection and growth fitting, but remains a linear finite-domain validation rather than a nonlinear island-growth claim.

The linearized-RHS gate compares JAX’s matrix-free Jacobian-vector product against a centered finite difference of the reduced-MHD RHS. This is the operator-level scaffold for later calibrated tearing eigenmode benchmarks.

The reduced-MHD eigenmode gate applies the flattened JVP operator at the zero-state linear limit and checks the analytic resistive and viscous Fourier diffusion eigenvalues for the \(\psi\) and \(\omega\) blocks.

The cosine-equilibrium linearization gate moves the same JVP machinery to a nonzero current-sheet equilibrium \(\psi_0=A\cos y\). It checks two exact Poisson-bracket couplings: flow advection of equilibrium flux and magnetic tension from perturbed flux. This is a physics gate for the current-sheet terms that calibrated FKR eigenmode benchmarks will use next.

The periodic current-sheet eigenvalue gate then assembles the tiny dense spectrum of that same nonzero-equilibrium operator. It verifies mean/gauge modes, checks the selected dense eigenpair residual, and fails if the non-gauge spectrum has spurious positive growth. This remains a periodic operator-stability gate, not a calibrated FKR/Coppi growth-rate benchmark.

The periodic current-sheet time-domain replay gate selects a real decaying eigenmode of the same dense JVP, advances \(dq/dt=Lq\) with RK4, and compares the whole state vector to \(q(t)=\exp(\lambda t)q(0)\). This catches errors between operator assembly, time stepping, and decay/growth fitting before moving the workflow to nonlinear perturbation studies.

The nonlinear current-sheet bridge differentiates the complete RK4 saved trajectory map with JAX, then compares it to centered finite differences of full nonlinear trajectories. The expected \(O(\epsilon^2)\) convergence is an evidence gate for differentiable programming claims; it is the step between linear operator replay and future adjoint/inverse-design workflows.

The nonlinear energy-budget gate advances a multi-mode nonlinear reduced-MHD state and checks \(dE/dt=-\eta\langle j^2\rangle-\nu\langle\omega^2\rangle\) by comparing the measured total energy to integrated resistive/viscous dissipation. This is now the primary FAST nonlinear PDE consistency gate, but it is still not a Rutherford, Sweet–Parker, or plasmoid validation.

The diffusion-eigenvalue gate validates the Rayleigh-quotient/residual path on a known Fourier eigenpair before using the same matrix-free machinery for tearing operators.

The power-iteration gate validates a minimal dominant-eigenpair iteration on a known diagonal matrix-free operator. This keeps the eigensolver control path tested independently before later coupling to reduced-MHD tearing operators.

The Arnoldi gate validates a small Krylov Ritz-spectrum path on a known non-normal upper-triangular operator. It is the direct scaffold for future matrix-free tearing eigenmode calculations where the linearized reduced-MHD JVP is too large to assemble explicitly.

References used for the benchmark roadmap include Furth, Killeen, and Rosenbluth 1963, MacTaggart 2019, MacTaggart and Stewart 2017, Loureiro, Schekochihin, and Cowley 2007, and Pucci and Velli ideal tearing context.