# Results — leftover’s only scale eliminates other RAR organizers; \(a_{\mathrm{ext}}\) is not eliminated

### Isolated well: unique \(y=g_N/a_0\) · \(\Omega\), \(r/R\), \(\Phi/c^2\), \(M\), \(J\) fail the board · disk \(\Sigma\) is \(y\) in disk clothes · EFE same dimension · interpolator unpaid

**CPT Framework — FT-H0FREQ / FT-FAT-FORCE / FT-YIELD-GEOM / FT-COV-YIELD-EQ / FT-BULK-ROT / FT-S3-PHI / FT-CONST-PRIZE**  
**Date:** 2026-09-23  
**Status:** **BANKED** · `PASS_FT_RAR_ELIM_A_ONLY_KIND_NOT_EFE`  
**Stimulus:** User — opportunity for proof by elimination that acceleration is the only possible variable that organises the RAR  
**Parents:** `a_geom_ft_hubble_freq_results.md` · `a_geom_ft_fat_force_results.md` · `a_geom_ft_cov_yield_eq_results.md` · `a_geom_ft_bulk_rot_results.md` · `a_geom_ft_rar_curve_results.md`  
**Script:** `scripts/investigations/s3_geometry/ft_rar_elim.py`  
**Data:** `scripts/investigations/s3_geometry/data/ft_rar_elim.json`

Tags: **[E algebra]** · **[E honesty]** · **[Programme]** · **[O residual]**

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## 0. One-line

> **Yes as dimensional elimination for an isolated leftover well; not as a uniqueness theorem, and not against EFE.** Leftover supplies one scale \(a_0=c^2/R\) (clothes). A spherical Newton well is \((GM,r)\). The only local dimensionless argument is \(y=g_N/a_0=GM/(a_0 r^2)\). That is why a winning constitutive law must be a universal map \(y\mapsto g_{\mathrm{obs}}\) with no leftover \(M\) in the EL (FT-FAT-FORCE). Named competitors fail the same board: \(\Omega/H_0\sim 370\)–\(480\) in *both* MW and dSph (wrong clock, FT-H0FREQ); \(r/R\sim 10^{-6}\)–\(10^{-8}\) (compactness does not pay RAR, FT-S3-PHI); \(\Phi/c^2\sim 10^{-7}\)–\(10^{-9}\) (1PN, not the IR); \(M/M_{\mathrm{crit}}\sim 10^{-12}\); \(j/(cR)\) tinier. Ricci vanishes in the RAR outskirts (FT-COV-YIELD-EQ). Disk \(\Sigma G/a_0\) is \(g_N/a_0\) in thin-disk clothes (Gauss). \(a_{\mathrm{ext}}/a_0\) has the **same dimension** as \(g_N\) and is **not** eliminated — that is EFE, still CAT-EVID, Sciama parked. Interpolator not unique (FT-RAR-CURVE). Not a proof of all mathematics. Prize OPEN.

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## 1. What leftover is allowed to spend

Locked leftover \(R\), invariant \(c\). One acceleration \(a_0=c^2/R\). Isolated baryon well: \(G,M,r\). Buckingham:

\[
y=\frac{GM}{a_0 r^2}=\frac{g_N}{a_0}.
\]

Any other local argument needs another leftover scale, or fails to split Earth / MW / dSph the way the RAR does.

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## 2. Elimination board

| Place | \(y=g_N/a_0\) | \((a/c)/H_0\) | \(\Omega/H_0\) | \(r/R\) | \(\Phi/c^2\) |
|-------|----------------|---------------|----------------|---------|--------------|
| Earth | \(8.2\times 10^{10}\) | \(1.4\times 10^{10}\) | \(5\times 10^{14}\) | \(10^{-20}\) | \(7\times 10^{-10}\) |
| MW \(8.5\,\mathrm{kpc}\) | \(0.80\) | \(0.14\) | \(370\) | \(8\times 10^{-7}\) | \(2.8\times 10^{-7}\) |
| dSph \(10\,\mathrm{km\,s^{-1}}\), \(0.3\,\mathrm{kpc}\) | \(0.090\) | \(0.016\) | \(476\) | \(3\times 10^{-8}\) | \(1.6\times 10^{-9}\) |

**Finding E1 [E algebra].** Only the acceleration column (and \(a/cH_0\), which is \(y\) times clothes \(a_0/(cH_0)\)) runs Newton \(\to\) threshold \(\to\) deep IR. \(\Omega/H_0\) is *larger* in the dSph than in the MW.

**Finding E2 [E].** \(r/R\) and \(\Phi/c^2\) are tiny everywhere on the RAR. They cannot organise a factor-\(40\) boost at \(300\,\mathrm{kpc}\). \(M\) or \(J\) in the EL is the Ginzburg–Landau wreck (\(\times 500\), FT-FAT-FORCE).

| Competitor | Why it dies |
|------------|-------------|
| Orbital \(\Omega\) | Wrong clock (FT-H0FREQ, FT-BULK-ROT) |
| Radius \(r/R\) | Compactness mill; \(10^{-6}\) |
| Potential \(\Phi/c^2\) | 1PN; gravity-of-gravity not a halo |
| Mass \(M\), \(J\), mass type | Not universal in \(y\); wrong dSph end |
| Ricci / \(f(R)\) | Vacuum in the outskirts; analytic IR Newton |
| History \(\tau\), foam | Scatter, not present \(y\) |
| Disk \(\Sigma\) | \(g\simeq 2\pi G\Sigma\): same as \(y\) for a sheet |
| Fermion spin / Cosserat \(R\) | Compton-small or Gpc |

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## 3. What this is not

- Not uniqueness of McGaugh \(\nu\) versus Pythagorean \(g=\sqrt{g_N(g_N+a_0)}\) (FT-RAR-CURVE).
- Not elimination of **EFE** \(a_{\mathrm{ext}}/a_0\). Same dimension; vector sum \(L_a=c^2/\lvert\mathbf{a}_{\mathrm{int}}+\mathbf{a}_{\mathrm{ext}}\rvert\) remains licensed (FT-WBS-EFE-VEC). Sciama derivation stays parked.
- Not a SPARC residual census (literature “no second parameter” is CAT-EVID, not this mill).
- Not \(\sqrt{6}\) as SPARC \(a_0\).

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## 4. Integrity

| Gate | Verdict |
|------|---------|
| Isolated leftover \(\Rightarrow\) unique \(y=g_N/a_0\) | **YES** (dimensional) |
| \(\Omega\), \(r/R\), \(\Phi\), \(M\), \(J\), Ricci organise RAR | **NO** |
| Disk \(\Sigma\) is a second variable | **NO** (same as \(y\) for a sheet) |
| \(a_{\mathrm{ext}}\) eliminated | **NO** |
| Interpolator unique / prize closed | **NO** |
| Proof of all mathematics / promote | **NO** |

**Exit:** `PASS_FT_RAR_ELIM_A_ONLY_KIND_NOT_EFE`

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## 5. One-line for agents

> **§FT-RAR-ELIM: leftover’s only scale makes isolated RAR a map of \(y=g_N/a_0\); \(\Omega,r/R,\Phi,M,J,\)Ricci eliminated; disk \(\Sigma\) is \(y\); \(a_{\mathrm{ext}}\) not eliminated (EFE); interpolator unpaid; prize OPEN; no promote.**