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]
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.
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.
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 \(a_\star\).
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.
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
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.