# 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]**

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## 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.

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## 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 |

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## 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.

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## 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`

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## 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.**