Chop spacetime into triangles. Measure what the correction does.
Recognition gravity says the action on a triangulated spacetime is not quite Regge's
A·δ per hinge; it is A·sinh(δ). The two agree exactly as the mesh
refines, and the difference must die like the fourth power of the mesh scale. That is a number you can measure.
These scripts build a genuine simplicial mesh of a curved spacetime window, compute every dihedral angle
geometrically, and measure the law. No step is asserted by construction.
A real mesh, not a cartoon
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, the honest geometric way.
sinh(δ) instead of δ
Since sinh(δ) = δ + δ³/6 + … and deficits shrink like h², the correction density must shrink like h⁴. Power counting predicts it; the script measures it.
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. And in de Sitter, the plain Regge density is watched converging to the known continuum value R/2κ = 1/12π ≈ 0.02653, a sanity anchor with nothing adjustable.
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, sliding down a slope-2 line. The lower series is the recognition correction |SRS − SRegge| per unit volume, sliding down slope 4. Straight lines on these axes are power laws; the slopes are the whole claim.
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)
What the numbers say
| 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, the signature of a genuine leading-order power law with higher-order contamination dying off.
This measures the size of the recognition correction against Regge calculus on real triangulations: the h² and h⁴ laws are MEASURED here, 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.