# Results — leftover \(a_\star=c^2/R_{\sqrt{6}}\) loses to \(cH_0/6\) on SPARC

### 175 galaxies, 153 after RAR cuts · \(a_\star\simeq 2.78\times 10^{-10}\) is \(0.36\,\mathrm{dex}\) high · \(cH_0/6\simeq 1.13\times 10^{-10}\) matches SPARC \(a_0\) · not a \(\mu\) fit · \(\sqrt{6}\) still **[M]**

**CPT Framework — FT-SQRT6-PACK / FT-RAR-CURVE / FT-SS-JOIN / FT-AQUAL-LOCK / McGaugh, Lelli & Schombert 2016**  
**Date:** 2026-09-24  
**Status:** **BANKED** · `PASS_FT_SPARC_ASTAR_CH6_BEATS_SQRT6`  
**Stimulus:** User — how does honest \(a_\star=c^2/R_{\sqrt{6}}\simeq 2.31\,a_0\) perform on 175 SPARC galaxies vs \(a_0=cH_0/6\simeq 1.10\times 10^{-10}\,\mathrm{m\,s^{-2}}\)?  
**Parents:** `a_geom_ft_sqrt6_pack_results.md` · `a_geom_ft_rar_curve_results.md` · Lelli+2016 AJ 152 157 SPARC Table 1–2  
**Script:** `scripts/investigations/s3_geometry/ft_sparc_astar.py`  
**Data:** `scripts/investigations/s3_geometry/data/ft_sparc_astar.json` · SPARC `sparc_table1.dat` · `sparc_table2.dat`

Tags: **[E algebra]** · **[E honesty]** · **[CAT-EVID]** · **[M packaging]** · **[Programme]** · **[O residual]**

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## 0. One-line

> **\(cH_0/6\) sits on the SPARC RAR. Leftover \(\sqrt{6}\) clothes do not.** With lab \(H_0=70\,\mathrm{km\,s^{-1}\,Mpc^{-1}}\): \(a_\star=cH_0/\sqrt{6}=2.78\times 10^{-10}\,\mathrm{m\,s^{-2}}\) (\(2.31\times\) SPARC \(a_0\), \(+0.36\,\mathrm{dex}\)); \(cH_0/6=1.13\times 10^{-10}\) (\(0.94\times\) SPARC, \(-0.026\,\mathrm{dex}\)). On 3168 points from **153** of 175 SPARC galaxies (McGaugh 2016 cuts \(Q=1,2\), \(i\ge 30^\circ\), \(\Upsilon_d=0.5\), \(\Upsilon_b=0.7\)), McGaugh-shape rms is \(0.183\,\mathrm{dex}\) at SPARC \(a_0\), \(0.182\,\mathrm{dex}\) at \(cH_0/6\), and \(0.251\,\mathrm{dex}\) at \(a_\star\) (median residual \(-0.15\,\mathrm{dex}\): too much IR boost). Empirical IR \(a_0=g_{\mathrm{obs}}^2/g_N\) (1241 points with \(g_N<0.1\,a_{0,\mathrm{SPARC}}\)) has median \(1.27\times 10^{-10}\). McGaugh systematic \(\pm 0.24\times 10^{-10}\) covers \(cH_0/6\), not \(a_\star\). Not a \(\mu\) shop. Pythagorean join remains SS-FIRE (FT-SS-JOIN). \(\sqrt{6}\) still **[M]**. Prize OPEN.

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## 1. Scales (\(H_0=70\), lab clothes)

| Scale | Value (\(\mathrm{m\,s^{-2}}\)) | / SPARC \(1.20\times 10^{-10}\) | dex |
|-------|--------------------------------|----------------------------------|-----|
| SPARC \(a_0\) (McGaugh 2016) | \(1.20\times 10^{-10}\) | \(1\) | \(0\) |
| \(cH_0/6\) | \(1.13\times 10^{-10}\) | \(0.94\) | \(-0.026\) |
| leftover \(a_\star=c^2/R_{\sqrt{6}}\) | \(2.78\times 10^{-10}\) | \(2.31\) | \(+0.364\) |

User’s \(cH_0/6\simeq 1.10\times 10^{-10}\) is \(H_0\simeq 67\)–\(68\). The ranking does not change.

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## 2. SPARC performance (no interpolator fit)

Published McGaugh function \(g=g_N/(1-e^{-\sqrt{g_N/a}})\) and simple \(\nu\) (contrast), **only \(a\) changed**.

| Predictor | rms dex | median dex |
|-----------|---------|------------|
| Newton | \(0.521\) | \(+0.435\) |
| McGaugh shape × SPARC \(a_0\) | \(0.184\) | \(-0.013\) |
| McGaugh shape × \(cH_0/6\) | \(0.182\) | \(-0.005\) |
| McGaugh shape × leftover \(a_\star\) | \(0.251\) | \(-0.151\) |
| simple \(\nu\) × SPARC \(a_0\) | \(0.184\) | \(-0.020\) |
| simple \(\nu\) × \(cH_0/6\) | \(0.183\) | \(-0.012\) |
| simple \(\nu\) × leftover \(a_\star\) | \(0.253\) | \(-0.154\) |

McGaugh 2016 quoted \(0.13\,\mathrm{dex}\) about *their* fit on 2693 points; this mill uses all radial points after \(Q,i\) cuts (3168). Extra scatter is expected. The **scale** ranking is not sensitive to that.

IR estimator \(a_{0,\mathrm{emp}}=g_{\mathrm{obs}}^2/g_N\): median \(1.27\times 10^{-10}\) (1241 points, 121 galaxies). Near SPARC and \(cH_0/6\), not \(a_\star\).

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## 3. Honesty

This is CAT-EVID of the **scale**, not a leftover theorem. \(cH_0/6\) is the packing \(k=6\) in \(R=k\,c/H_0\), neighbour of the refused \(2\pi\) shop (FT-SQRT6-PACK). It **wins SPARC** and is **not** leftover Ricci-\(\sqrt{6}\). Leftover \(a_\star\) is the honest geometric clothes and **loses** SPARC by \(0.36\,\mathrm{dex}\) (already FT-RAR-CURVE). Do not retune \(R\) to SPARC \(a_0\). Do not adopt \(k=6\) as a derivation.

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## 4. Integrity

| Gate | Verdict |
|------|---------|
| \(a_\star\) closer to SPARC than \(cH_0/6\) | **NO** |
| \(cH_0/6\) inside McGaugh sys \(\pm 0.24\times 10^{-10}\) | **YES** |
| leftover \(a_\star\) inside that sys | **NO** |
| \(\mu\) fitted / SPARC \(I\) / \(R\) retuned | **NO** |
| Pythagorean reused | **NO** |
| Prize closed / promote | **NO** |

**Exit:** `PASS_FT_SPARC_ASTAR_CH6_BEATS_SQRT6`

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## 5. One-line for agents

> **§FT-SPARC-ASTAR: on 153/175 SPARC galaxies, \(cH_0/6\simeq 1.13\times 10^{-10}\) matches the RAR; leftover \(a_\star=c^2/R_{\sqrt{6}}\simeq 2.78\times 10^{-10}\) is \(0.36\,\mathrm{dex}\) high (rms \(0.25\) vs \(0.18\)); not a \(\mu\) fit; \(\sqrt{6}\) still [M]; prize OPEN; no promote.**