D. GR-MHD Accretion

Why this complements the BSSN side

Sections A–C document the vacuum-spacetime / gravitational-wave capability. This section is the magnetized-accretion half — the HARM domain. The platform implements conservative GRMHD with the Noble et al. (2006) primitive-variable inversion at its core, validated piece by piece against closed forms and convergence gates. Everything below runs from the same WASM engine (wasm/src/grmhd/), checked by cargo test --release --lib grmhd (15/15).

The gap we identified

con2prim recovers the primitive state P = (ρ, u, vⁱ, Bⁱ) from the conserved state U. Our first implementation used the Noble (2006) 2D (W, v²) scheme generalized to a spatial 3-metric. It is exact for hydro on any metric and for flat-space SRMHD, but only ~10^-4 in ρ for the magnetized + curved + shifted case — even though the Newton iteration converges (residual ~10^-16). It was finding the exact root of the wrong equations.

Root cause: the GR-MHD normal-frame momentum S_i = α T^t_i carries a magnetic term −α b^t b_i, where b_i is the 4-metric-lowered magnetic 4-vector, b_i = β_i b^t + γ_ij b^j. The shift contribution β_i b^t has no analogue in the special-relativistic Noble relation, which lowers B with the 3-metric only. Hydro has no magnetic term; flat space has β_i = 0; only the magnetized + shifted case exposes it. This is the standard subtle heart of GR-MHD primitive recovery (Noble et al. 2006, §3–4). It was caught by a prim→cons→prim round-trip gate in the Kerr-Schild metric.

The substrate-closure fix

Rather than hand-derive the curved-metric analytic inverse, we drive the GDBS drift-aware closure engine (wasm/src/drift_escalator.rs; pinned application “GDBS physics”) on the GR-exact forward map. con2prim becomes a closure problem — find P that drives R(P) = ‖ prim2cons(P) − U ‖ → 0:

1. Bⁱ is read directly from the conserved field (exact, not solved).
2. Seed (ρ, u, vⁱ) with the fast SR analytic inverse — within ~10^-4, a Tier-0 prescribed guess.
3. Newton-refine the 5-vector residual of the GR-exact forward map (ideal-MHD stress on the full 4-metric; Gammie 2003 magnetic 4-vector) to the f64 floor.
4. GeoNum drift tracking carries the high-magnetization (σ = b²/ρ) cancellation precision — the genuine Noble-2006 funnel concern.

Because we close the GR-exact forward map (the one we already validated) rather than evaluate a hand-derived inverse, the special-relativistic-vs-GR formulation gap never enters. For completeness: two GDBS components that look related are not the right tool — HXT is mesh-tier composition (no tiers in a pointwise solve), and the cascade auto-tuner closes transcendental constants to rational combinations (con2prim seeks continuous reals).

Measured results

Scope honestly stated

This section closes one specific gap — GR-MHD primitive recovery to the f64 floor on a fixed Kerr-Schild background — and validates the supporting pieces (FM torus equilibrium, SRMHD Riemann tests, constrained transport, linear MRI dispersion γ_max = (3/4)Ω). It does not by itself deliver magnetized accretion. The remaining pipeline, each gated by its own check: wiring the closure inverse into the magnetized torus evolution; constrained transport for B in (r, θ) + a poloidal field-loop seed → MRI growth; the horizon-penetrating inner boundary (currently above the horizon); and the diagnostics a HARM user reads (Ṁ, horizon flux Φ_BH, Maxwell stress). 2D-axisymmetric evolution shows MRI onset but, by Cowling's anti-dynamo theorem, cannot sustain turbulence — that requires 3D (the GPU tile-streaming path).

The in-development bug log on the live module (HPC Lab → GRMHD Accretion) records each finding per experiment with how the gate caught it and how it was fixed — including this con2prim result.