A symbolic diagnostic framework for assessing whether prescribed perturbative spacetime geometries admit physically realizable matter sectors in general relativity.
This repository contains the reference implementation of the Matter Realizability Audit (MRA) described in the accompanying manuscript:
Salehizadeh, S., "Assessing Matter Realizability in Perturbative Cosmological Spacetimes," manuscript in preparation (2026).
The MRA is a hierarchical symbolic framework that separates geometric validation from matter-sector realizability through benchmark reconstruction, exact off-shell contracted second Bianchi identity verification, conservation diagnostics, closure analysis, residual-ledger construction, invariant decomposition, and realizability assessment.
All symbolic results reported in the accompanying manuscript are reproduced directly by the reference implementation contained in this repository.
Matter-Realizability-Audit/
├── README.md # This file
├── LICENSE # BSD 3-Clause License
├── CITATION.cff # Citation metadata (DOI-linked)
├── flrw_grand_audit_final.py # Reference implementation of the MRA
├── requirements.txt # Python package dependencies
└── run_audit.sh # Shell script to run the audit
Python 3.10 or later is required. The implementation was developed and tested under Python 3.12.3.
All dependencies are listed in requirements.txt. The only required
package is SymPy:
sympy>=1.12
The implementation was developed and tested with SymPy 1.14.0. Earlier versions of SymPy (1.12 or later) should be compatible but have not been independently verified.
No additional packages (NumPy, SciPy, etc.) are required. The audit is purely symbolic.
pip install -r requirements.txtpython3 flrw_grand_audit_final.pyOr using the provided shell script:
bash run_audit.shTo save output to a file:
bash run_audit.sh --saveThis writes all output to audit_output.txt while also printing to the
terminal.
Expected runtimes on a modern workstation (single core, Python 3.12, SymPy 1.14):
| Audit Stage | Approximate Time |
|---|---|
| Background geometry + Christoffel | 30–60 s |
| Benchmark validation (3 geometries) | 2–5 min |
| Off-shell Bianchi verification | 3–8 min |
| Linearized Einstein tensor δG | 5–15 min |
| Matter conservation diagnostics | 5–15 min |
| Closure + residual ledger | 5–10 min |
| Invariant decomposition (q², Π²) | 3–8 min |
| Total (full audit) | ~30–60 min |
Runtime depends on the host machine and SymPy version. The audit is single-threaded.
[PASS] Minkowski benchmark: G_munu = 0
[PASS] Schwarzschild benchmark: G_munu = 0
[PASS] FLRW benchmark: standard Friedmann Einstein tensor
[PASS] Off-shell Bianchi identity: nabla_mu G^mu_nu = 0 (all components)
Scalar closure:
drho_sol = 2*G*M*rho0 / (r*a(t))
dp_sol = (6*G*M*p0 + 2*G*M*rho0) / (3*r*a(t))
Closure Jacobian det: nonzero [non-degenerate]
Pi^2 = G**2*M**2*(32*H**2*omega**2*r**2*a(t)**4*sin(theta)**2 + 6)
/ (r**6*a(t)**6)
Symbolic independence verification:
dPi2/dv_r = 0 [PASS]
dPi2/dv_phi = 0 [PASS]
dPi2/ddrho = 0 [PASS]
dPi2/ddp = 0 [PASS]
Pi^2 > 0 [PASS] (strictly positive throughout perturbative domain)
GEOMETRIC SECTOR: Established (independent of matter realization)
MATTER DIAGNOSTICS: Established (consistent diagnostic sequence)
RESIDUAL STRUCTURE: Empirical (non-vanishing post-closure residual)
INVARIANT q^2: Empirical (realization-dependent)
INVARIANT Pi^2: Empirical (nonzero, velocity-independent,
realization-class robust)
GENERAL REALIZABILITY: Open Question
The audit is applied to an axisymmetric Kerr-like frame-dragging perturbation of a spatially flat FLRW background:
g_munu = g^(0)_munu + epsilon * h_munu
with nonzero perturbative components:
h_tt = 2*G*M / (a(t)*r)
h_tphi = h_phit = 2*G*M*a(t)*omega*sin^2(theta)
All other components vanish. Here M is the characteristic mass scale, omega parametrises the frame-dragging strength, and a(t) is the cosmological scale factor.
All symbolic results in the manuscript — including the explicit form of
Π², the scalar closure solutions, and the independence verification
checks — are reproduced exactly by running the audit. To verify a
specific result, search the output for the corresponding diagnostic label
(e.g. Pi2, drho_sol, C_G_full).
If you use this software in your research, please cite both the accompanying manuscript and the software archive:
Manuscript:
Salehizadeh, S., "Assessing Matter Realizability in Perturbative
Cosmological Spacetimes," manuscript in preparation (2026).
Software archive:
Salehizadeh, S., Matter Realizability Audit (MRA) v1.0.0,
Zenodo (2026), DOI: 10.5281/zenodo.20934770
Citation metadata is also provided in CITATION.cff.
This project is licensed under the BSD 3-Clause License.
See the LICENSE file for details.