Measure how the gravity correction changes as a spacetime mesh gets finer.
Recognition gravity replaces Regge's A·δ contribution per hinge
with A·sinh(δ). As the mesh gets finer, the predicted difference per unit volume
falls with the fourth power of the mesh scale. These scripts test that prediction on a simplicial mesh
of a curved spacetime window, computing the dihedral angles from the geometry and measuring how the correction changes.
A four-dimensional simplicial mesh
A four-dimensional window of Euclidean Schwarzschild (around a black hole) or static-patch de Sitter (a universe with a cosmological constant) is cut into 4-simplices by the Kuhn triangulation, up to 1.6 million of them. Edge lengths are proper lengths in the exact metric. Dihedral angles come from embedding each simplex isometrically in flat 4-space.
sinh(δ) instead of δ
Since sinh(δ) = δ + δ³/6 + … and deficits shrink like h², the correction density must shrink like h⁴. The script measures this power-counting prediction across successive refinements.
The null test runs first
The identical pipeline runs on flat space before any curved window. Every interior deficit must vanish to rounding (it does, at 10−15). If the mesh machinery itself manufactured curvature, the run aborts. In de Sitter, the plain Regge density is also checked against the known continuum value R/2κ = 1/12π ≈ 0.02653, with no adjustable parameter.
The measured convergence
Each point is one full triangulation at mesh scale h (log–log axes). The upper series is the plain Regge deficit content, following a slope-2 line. The lower series is the recognition correction |SRS − SRegge| per unit volume, following slope 4. A straight line on these axes represents a power law; its slope gives the exponent.
Run it yourself
One file, no dependencies beyond NumPy. The quick mode finishes in about a second; the full seven-level series takes ~20 seconds on a laptop.
$ python3 qg_regge_schwarzschild.py --quick FLAT-SPACE NULL TEST (identity metric, same pipeline): n = 6: interior hinges = 16064, max|delta| = 7.11e-15 [PASS] interior deficits vanish in flat space SCHWARZSCHILD SERIES (far window, r0 = 2.0 r_s): ... MEASURED exponents (least-squares over all levels): median|delta| ~ h^2.04 (power counting: 2) |S_RS - S_Regge|/Vol ~ h^4.27 (power counting: 4)
How the exponent changes with refinement
| Refinement | Schwarzschild r₀=2rₛ | Schwarzschild r₀=1.5rₛ | de Sitter outer | de Sitter inner |
|---|---|---|---|---|
| n = 4 → 8 | +4.271 | +4.267 | +4.292 | +4.251 |
| n = 6 → 12 | +4.142 | +4.136 | +4.156 | +4.128 |
| n = 8 → 16 | +4.094 | +4.089 | +4.105 | +4.083 |
Each entry is log₂ of how much the correction density drops when the mesh is halved. A clean h⁴ law reads exactly 4.000; the small excess above 4 shrinks toward it as the mesh refines. This is consistent with a leading fourth-power term and higher-order terms that diminish with refinement.
This measures the size of the recognition correction against Regge calculus on real triangulations: the measured convergence supports the h² and h⁴ laws, with a flat-space null test and the de Sitter continuum anchor as controls. It does not by itself prove Regge → Einstein-Hilbert convergence (that is the classical Cheeger–Müller–Schrader result, taken as external input), and it is a consistency demonstration for the strong-field theory, not a standalone derivation of it. Coefficients are mesh-dependent; the exponents are the invariant content.