Measure gravity's first nonlinear term. Watch two symmetries force zeros.
The Regge triangulation lab measured curvature convergence on Schwarzschild and de Sitter meshes. This lab probes the nonlinearity of discrete gravity: the third Taylor coefficient of the Regge action on a flat 3×3×3 periodic Freudenthal torus, along a fixed, published probe direction (seed 42, printed in full so your run is bitwise comparable). Around flat space the action starts at quadratic order; the cubic term is the first gravitational self-interaction. It comes out finite and reproducible. Two directions where symmetry forces the cubic to vanish are measured alongside as built-in nulls.
A 5-point stencil on 162 tetrahedra
Each unit cube splits into six tetrahedra (Freudenthal); vertex potentials ξ deform edge lengths conformally, Lij = L₀ij e(ξi+ξj)/2. Dihedral angles come from exact Cayley–Menger cofactors. The cubic S′′′(0;η) is estimated by central finite differences at four step sizes with Richardson extrapolation.
Symmetry says zero. The run says zero.
The J-cost action Σ L₀(cosh(ξu−ξv)−1) is even under ξ → −ξ, so every odd derivative vanishes identically; the symmetric stencil returns exactly 0. Along the constant direction the flat-background Regge action is identically zero on the whole ray. A nonzero reading in the stable window would mean the instrument is broken.
Why h = 10⁻⁵ explodes
The stencil divides by 2h³. Below h ≈ 10⁻³ floating-point cancellation grows as h⁻³ and swamps the signal: the raw table shows 13.86 at h = 10⁻⁵ on the Regge direction and −232 on the scale direction. That is roundoff, not physics, and it is the correct reason the table looks wild at small h. The measurement lives in the stable window h = 10⁻² to 10⁻³, where Richardson gives −0.2392893568.
The error V, from the run
Squares: deviation of the Regge cubic estimate from the Richardson value. Circles: raw magnitude along the scale direction, where the true answer is 0. Truncation error falls as h² (left slope), roundoff rises as h⁻³ (right slope). The J-cost series cannot be plotted on a log axis: it is exactly zero at every step size.
Raw table, both directions
| h | Regge cubic (seed-42 dir) | J-cost cubic (same dir) | Regge cubic (scale dir) |
|---|---|---|---|
| 10⁻² | −0.2392853326 | 0.0000000000 | −0.0000001873 |
| 10⁻³ | −0.2392893165 | 0.0000000000 | 0.0002941473 |
| 10⁻⁴ | −0.2131404429 | 0.0000000000 | 0.1643626063 |
| 10⁻⁵ | 13.8581331095 | 0.0000000000 | −232.4834327233 |
Richardson extrapolation over the stable pair (10⁻², 10⁻³): −0.2392893568. The two rows below the stable window are shown, not hidden: they are the roundoff wall every finite-difference computation hits, arriving exactly on the h⁻³ schedule.
Run it yourself
One file, numpy only. The probe direction is regenerated from the two printed lines
(np.random.seed(42); eta = np.random.randn(27) * 0.1), so every number should reproduce
bitwise on the same numpy generation.
$ python3 qg_regge_torus_cubic.py ==== Regge cubic along seed-42 direction ==== 1e-02 -0.2392853326 1e-03 -0.2392893165 Richardson h=1e-02 -> h=1e-03: -0.2392893568 ==== J-cost cubic along seed-42 direction (must be ~0) ==== 1e-02 0.0000000000 (and at every other h)
MEASURED, on a MODEL: the cubic value is a property of this specific mesh (3×3×3 periodic Freudenthal torus), the conformal deformation ansatz, and the published seed-42 direction. It is a reproducible benchmark of discrete-gravity nonlinearity, not a theorem about continuum general relativity, and no continuum limit is claimed here (the h in this lab is a probe amplitude, not a mesh size). The two zero checks are structural: the J-cost parity null and the conformal-scale null both land at machine floor in the stable window, and either one coming out nonzero there would falsify the symmetry claims this page rests on. The contrast the page draws, Regge cubic finite while the J-cost cubic vanishes by reciprocal symmetry, is exact in this instrument and is the point.