Results — Einstein curvature cannot pay the aquadratic density
Vacuum Ricci is 0 where the RAR lives · leftover \(^3R=6/R^2\) is a wall · EH is analytic and second-order · \(\lvert R\rvert^{3/2}\) still vanishes in Ricci-flat outskirts · prize OPEN
CPT Framework — FT-WK-32 / FT-WE / FT-COV-YIELD-EQ / FT-GEJ-A0-COMPUTE / FT-S3-PHI / FT-AQUAL-LOCK Date: 2026-09-24 Status: BANKED ·PASS_FT_GR_CURV_VACUUM_RICCI_NOT_AQUAL
Stimulus: User — why pure GR curvature cannot pay the density
Parents: a_geom_ft_wk_32_results.md · a_geom_ft_w_of_e_results.md · a_geom_ft_cov_yield_eq_results.md · a_geom_ft_gej_a0_compute_results.md · a_geom_ft_s3_phi_results.md
Script: scripts/investigations/s3_geometry/ft_gr_curv.py
Data: scripts/investigations/s3_geometry/data/ft_gr_curv.json
Tags: [E algebra] · [E honesty] · [Programme] · [O residual]
0. One-line
Because the RAR outskirts are Ricci-flat, and Einstein–Hilbert is analytic Laplace there. Outside the baryons, \(G_{\mu\nu}=0\) \(\Rightarrow\) \(\nabla^2\Phi=0\) \(\Rightarrow\) Coulomb, not \(\lvert\nabla\Phi\rvert^3/a_0\). Leftover background \(^3R=6/R^2\) is a homogeneous wall (GE-J home, \(O(a_0)\) along the leftover normal), not a spherical \(\nu(y)\). Curvature is second derivatives of the metric; yield is first (\(\lvert\nabla\Phi\rvert\sim c^2/R\)). Analytic \(f(R)\) is FT-WE again. \(\lvert R\rvert^{3/2}\) evaluated on a Ricci-flat well is still zero. Weyl/tidal bends light at 1PN and does not source extra \(\Phi\) in vacuum. That is why the prize is constitutive in acceleration, not a curvature density. Do not write \(f(R)\). Prize OPEN.
1. Three independent kills
Vacuum. The RAR is extra \(\Phi\) at \(50\)–\(300\,\mathrm{kpc}\), where \(\rho=0\). Einstein’s equation there is Laplace. Needed BTFR boost at \(300\,\mathrm{kpc}\) is \(\sim 40\). Ricci of the well cannot supply it because it is identically zero. Wrong derivative. \(G_{\mu\nu}\) is linear in \(\partial^2 g\). AQUAL IR is cubic in \(\partial\Phi\). No local curvature scalar is \(\lvert\nabla\Phi\rvert^3\). Comparing leftover \(1/R\) to \(g/c^2\) is a first-derivative yield, not an EH term. Wrong home for leftover curvature. Umbilic \(^3R=6/R^2\) is leftover-wide. Israel readout is a wall along the leftover normal, \(0.18\)–\(2.3\,a_0\) by convention, not a disk RAR. In-slice de Sitter \(g_\Lambda=\Omega_\Lambda H_0^2 r\) is \(8\times 10^{-6}\,a_0\) at \(8.5\,\mathrm{kpc}\) (FT-S3-PHI).| \(r\) (kpc) | needed \(\nu\) | well Ricci (vacuum) | leftover \(^3R\) | |-------------|----------------|---------------------|------------------| | 8.5 | \(1.12\) | \(0\) | \(6/R^2\) (wall) | | 50 | \(6.6\) | \(0\) | wall | | 300 | \(39\) | \(0\) | wall |
2. Why \(\lvert R\rvert^{3/2}\) does not sneak AQUAL in
If one puts a fractional power on curvature to mimic \(X^{3/2}\), the well’s Ricci is still \(0\) in the outskirts, so that density is still \(0\). If the leftover background \(^3R=6/R^2\) is the argument, the power is a constant, i.e. a wall / \(\Lambda\)-like term, not \(\nu(y)\). Analytic \(f(R)\) at leftover umbilic is Newton for small wells (FT-PRESTRAIN). Metric \(f(R)\) also risks \(c_{\mathrm{gw}}\) extra modes.
Weyl \(\sim GM/(c^2 r^3)\) is free in vacuum and already in GR light deflection. It does not rewrite Poisson.
3. Integrity
| Gate | Verdict | |------|---------| | EH / well Ricci pays AQUAL IR | NO | | Leftover \(^3R\) is a \(y\)-map | NO (wall) | | Curvature is the yield argument | NO (wrong order) | | \(\lvert R\rvert^{3/2}\) pays vacuum RAR | NO | | \(f(R)\) shop / prize closed | NO |
Exit:PASS_FT_GR_CURV_VACUUM_RICCI_NOT_AQUAL
4. One-line for agents
§FT-GR-CURV: Einstein curvature cannot pay the aquadratic density because RAR outskirts are Ricci-flat (Laplace), leftover \(^3R\) is a wall, and yield is \(\lvert\nabla\Phi\rvert\) not \(R_{\mu\nu}\); \(\lvert R\rvert^{3/2}\) still vanishes there; prize OPEN; no promote.