Section A. Measured BSSN Benchmarks

A.1 Gauge-wave convergence

Apples-with-Apples gauge wave test (Alcubierre et al. 2004), 1D-equivalent amplitude 0.01, wavelength 1.0, evolved for 60 RK4 steps with CFL=0.25 and Kreiss-Oliger 6th-order dissipation strength ε_KO=0.10. Formulation: pure BSSN. Slicing: 1+log; shift: gamma-driver with η=2.0. Three resolutions sampled (browser-CPU WASM; the GPU-resident path supports 256³, the tiled path 512³-1024³).

Convergence order

For a scheme of order p evolving a smooth solution, the constraint norm scales as ‖C‖_h ∼ h^p, so the ratio of norms at adjacent refinements gives p directly.

Observed orders 4.05 and 3.88 are consistent with the 4th-order centered finite-difference spatial scheme combined with RK4 time integration. The internal test suite at wasm/tests/bssn_convergence.rs typically reports 3.0-3.8 on alpha-field slices; constraint-norm ratios sample the scheme accuracy more directly and recover the design order.

bssn_gauge_wave_resolution_scan.json bssn_convergence_order.json

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A.2 Minkowski fixed-point

Flat spacetime initial data with no perturbation. The exact solution is preserved by the BSSN equations; constraints must remain at floating-point round-off. The internal test suite at wasm/tests/bssn_validation.rs:42-43 gates this at Ham L2 < 10^-6.

Configuration: 16³ grid, 50 RK4 steps, 1+log slicing, ε_KO=0.10, Hamiltonian damping disabled to expose the floor.

The constraint floor sits at machine epsilon. Algebraic constraints (det γ-tilde = 1, Tr A-tilde = 0) are enforced post-RK4 by the dispatch_algebraic pass at wasm/src/gpu/bssn/engine.rs:69; the residual is bounded by f64 representation error in the projection operator.

bssn_minkowski_constraint_floor.json

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A.3 Robust-noise long-time stability

Random Gaussian perturbation at amplitude 10^-10 on a Minkowski background, evolved for 100 RK4 steps. Per Alcubierre et al. (2004) the test passes if constraints remain bounded; growth, even exponential within a finite window, is acceptable. Failure modes the test gates against: unbounded blow-up and premature gauge collapse.

The sub-grid regulator (wasm/src/gpu/bssn/regulator.rs:1-50) did not trigger on clean noise data, which is the designed behavior: it watches the maximum gradient across the puncture sphere and only activates when the field becomes locally under-resolved. The 67.9× growth in Ham L2 reflects nonlinear coupling at the noise amplitude and is well within the bounded regime for BSSN with 1+log slicing.

bssn_robust_noise_stability.json

A.4 Substrate drift on constraint norms

Each measured Hamiltonian constraint norm is routed through Rust GeoNum (v2.2) via geonum_evaluate. The returned trust band is the substrate's native precision claim on that value.

The PRECISE band is defined as drift < 1 shade, where one shade is the substrate equivalent of one ULP. All three resolutions return sub-shade drift; the measured constraint norms are substrate-trusted to better than one mantissa unit.

Standard NR codes (HARM, HARM3D, Cactus, Einstein Toolkit, BAM) do not propagate this per-value precision tag. Constraint violation is the canonical accuracy proxy; the substrate-validated trust band is additive to that diagnostic.

geonum_drift_on_constraints.json

A.5 Analytic gravitational waveform pipeline

End-to-end LIGO data analysis pipeline using the IMRPhenomD analytic waveform model (Khan, Husa et al., Phys. Rev. D 93, 044007, 2016) as the template. Fiducial event configured as a GW150914 analog (m₁ = 36 M_⊙, m₂ = 29 M_⊙, aligned spins χ ∼ 0, luminosity distance 410 Mpc).

The analytic waveform path is production-grade for matched-filter detection and parameter estimation against PhenomD / TaylorF2 references. The numerical-relativity path - BSSN evolution to ψ₄ to strain extraction to cross-validation against an SXS catalog reference - is the Phase I deliverable described in Section B.

analytic_waveform_phenomd.json

A.6 Timing summary

Browser-CPU WASM timing on a commodity laptop without GPU dispatch:

Scaling is approximately 6× wall increase per refinement (16 to 24 to 32), consistent with O(N⁴) cost: N³ spatial work multiplied by a time step that scales with N under CFL=0.25.

The GPU resident path (run_bssn_gpu, wasm/src/gpu/bssn/engine.rs) supports 256³ resident and 512³-1024³ via the GMDBS toroid streaming substrate (wasm/src/gpu/bssn/tiled.rs). This is the path relevant for production HPC comparisons. The WASM-CPU timing above is the browser-reproducible baseline.

summary.json

A.7 Out of scope (Phase I deliverables)

The following are not measured in this section and are the technical content of Section B: