# Results — RAR transition: geometric saturation as a plottable curve, not a unique \(W\)

### Distinguished zero-parameter candidate \(g=\sqrt{g_N(g_N+a_0)}\) · hard floor FIRE · leftover \(\sqrt{6}\) scale \(0.36\,\mathrm{dex}\) high · SPARC mean does not pick the interpolator · KiDS sits on the same IR · constitutive prize OPEN

**CPT Framework — FT-YIELD-GEOM / FT-CONST-PRIZE / FT-GAP-B-OUTER / FT-A0-KIND / Book G**  
**Date:** 2026-09-22  
**Status:** **BANKED** · `PASS_FT_RAR_CURVE_CANDIDATE_NOT_UNIQUE`  
**Stimulus:** User — turn geometric saturation from a conceptual threshold into an exact, falsifiable curve plotted against SPARC and KiDS  
**Parents:** FT-YIELD-GEOM · FT-CONST-PRIZE · McGaugh, Lelli & Schombert 2016 · Mistele et al. 2024 ApJL · FT-GAP-B-OUTER · FT-A0-EXP  
**Script:** `scripts/investigations/s3_geometry/ft_rar_curve.py`  
**Data:** `scripts/investigations/s3_geometry/data/ft_rar_curve.json` · `ft_rar_curve.csv` · `ft_rar_curve.png`

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

---

## 0. One-line

> **Yes, plot it; no, saturation does not mint a unique law.** \(r_{\mathrm{loc}}=c^2/g\) hitting leftover \(R\) fixes a *threshold*. A hard floor \(g_{\mathrm{obs}}\ge a_0\) is **FIRE** (dSph \(g\sim 0.027\,a_0\); KiDS even deeper). The extra identification that a saturated well has radius \(\sqrt{r_N R}\) is deep MOND, \(g_{\mathrm{obs}}^2=g_N a_0\). The unique two-term inverse-square interpolator of those ends is
> \[
> \frac{1}{r_{\mathrm{obs}}^2}=\frac{1}{r_N^2}+\frac{1}{r_N R}
> \quad\Leftrightarrow\quad
> g_{\mathrm{obs}}=\sqrt{g_N(g_N+a_0)}.
> \]
> That is a **distinguished candidate**, not a derived \(W\). Plotted against the published SPARC mean (\(\pm 0.13\,\mathrm{dex}\) point scatter) and four KiDS \(300\,\mathrm{kpc}\) bins: with Milgrom \(a_0\) clothes the shape sits \(0.034\,\mathrm{dex}\) rms from McGaugh 2016 (inside scatter; simple \(\nu\) is \(0.011\,\mathrm{dex}\) — SPARC does **not** pick the interpolator). Leftover clothes \(a_{\mathrm{geom}}=c^2/R_{\sqrt{6}}\simeq 2.31\,a_0\) (\(0.36\,\mathrm{dex}\)) is outside SPARC \(a_0\). KiDS tracks the same IR as stars (\(\langle|\mathrm{dex}|\rangle\simeq 0.08\) vs Newton \(1.65\)) — photons see this \(\Phi\) curve. Hard step at \(a_0\) is \(0.20\,\mathrm{dex}\) low at the join. Not a \(\mu(x)\) shop. Prize still OPEN.

---

## 1. What geometry actually fixes

| Object | Status |
|--------|--------|
| UV: \(r_{\mathrm{loc}}\ll R\Rightarrow g_{\mathrm{obs}}=g_N\) | **fixed** (Book G) |
| Threshold: \(g\sim c^2/R\) | **fixed as kind** (\(\sqrt{6}\) still **[M]**) |
| Hard floor \(g_{\mathrm{obs}}\ge a_0\) | **FIRE** (\(\times 37\) vs a \(10\,\mathrm{km\,s^{-1}}\), \(1\,\mathrm{kpc}\) dSph) |
| IR: \(r_{\mathrm{obs}}=\sqrt{r_N R}\) | **extra identification** (geometric mean), not forced by \(r_{\mathrm{loc}}=R\) alone |
| Interpolator \(f(g_N/a_0)\) | **not unique** (constitutive prize) |

**Finding C1 [E algebra].** Saturation without the geometric-mean IR is a floor, and the sky kills it. Saturation *with* that IR still leaves a one-variable family of transitions. Plotting a distinguished member is licensed. Calling it *the* leftover law is a \(\mu\) shop.

The two-term inverse-square sum is distinguished because \(g=c^2/r_{\mathrm{loc}}\) so \(1/r^2\) is the Gauss/flux variable, and those two terms are exactly the UV and IR inverse-square well radii. It is not uniqueness.

---

## 2. Shape test (Milgrom \(a_0\) clothes) and scale test (leftover \(\sqrt{6}\))

Join \(g_N=a_0\), \(g_{\mathrm{obs}}/a_0\):

| Curve | \(g_{\mathrm{obs}}/a_0\) | dex vs McGaugh |
|-------|-------------------------|----------------|
| McGaugh 2016 mean | \(1.582\) | \(0\) |
| simple \(\nu\) (contrast) | \(1.618\) | \(+0.010\) |
| geo Pythagorean | \(1.414\) | \(-0.049\) |
| hard step / floor | \(1.000\) | \(-0.199\) |

SPARC-window rms vs McGaugh (\(\log_{10}(g_N/a_0)\in[-2,2]\)): Pythagorean \(0.034\,\mathrm{dex}\), simple \(0.011\), step \(0.085\), leftover-\(\sqrt{6}\) Pythagorean \(0.090\), Newton \(0.45\). Published point scatter is \(0.13\,\mathrm{dex}\); that is **not** the error on the mean. The step’s \(0.20\,\mathrm{dex}\) join offset is a shape **tension**. Transition width (\(g_{\mathrm{obs}}/g_N\) from \(1.3\) to \(3\)): McGaugh \(1.12\) decades, Pythagorean \(1.06\), step \(0.73\).

Leftover scale: \(a_{\mathrm{geom}}/a_0=2.31\) (\(0.36\,\mathrm{dex}\)). McGaugh 2016 systematic on \(a_0\) is \(\sim 20\%\) (\(\sim 0.08\,\mathrm{dex}\)). \(\sqrt{6}\) clothes do **not** pay SPARC \(a_0\) (FT-A0-EXP rhyme stands; not a \(2\pi\) shop).

**Finding C2 [E honesty].** With the *sky’s* \(a_0\), the distinguished curve is compatible with the published SPARC mean at the level of point scatter. So is simple \(\nu\). SPARC does not award uniqueness. With leftover \(c^2/R_{\sqrt{6}}\), the scale fails.

---

## 3. KiDS (photons)

Mistele 2024 ApJL, isolated stacks at \(300\,\mathrm{kpc}\), \(g_N/a_0\sim (1.6\)–\(25)\times 10^{-4}\) — two decades below SPARC, deep IR.

| Curve | mean \(\lvert\mathrm{dex}\rvert\) vs four bins |
|-------|-----------------------------------------------|
| Newton | \(1.65\) |
| hard floor | \(1.48\) (wrong sign: too high) |
| geo Pythagorean at \(a_0\) | \(0.085\) |
| simple \(\nu\) / McGaugh | \(0.078\) |
| leftover \(\sqrt{6}\) Pythagorean | \(0.097\) |

**Finding C3 [E].** KiDS tests the IR amplitude and that photons see the same curve as stars. It does **not** test the transition width. That is why it already killed lone \(m(a)\) (FT-GAP-B-OUTER) without minting \(\mu(x)\). Gray band at \(g_N\lesssim 10^{-12}\,\mathrm{m\,s^{-2}}\) is the McGaugh *function* extrapolated, not SPARC points.

Figure: `scripts/investigations/s3_geometry/data/ft_rar_curve.png`.

---

## 4. Integrity

| Gate | Verdict |
|------|---------|
| Saturation \(\Rightarrow\) unique \(g_{\mathrm{obs}}(g_N)\) | **NO** |
| Distinguished candidate plottable vs SPARC/KiDS | **YES** |
| Hard floor | **FIRE** |
| Hard step as SPARC mean | **tension** (\(0.20\,\mathrm{dex}\) at join) |
| Pythagorean shape at Milgrom \(a_0\) vs SPARC scatter | **compatible**, not unique |
| Leftover \(\sqrt{6}\) as SPARC \(a_0\) | **NO** (\(0.36\,\mathrm{dex}\)) |
| KiDS = SPARC IR (\(\Phi\)) | **YES** |
| \(\mu(x)\) shop / SPARC \(I\) / CDM | **NO** |
| Constitutive prize closed | **NO** |

**Exit:** `PASS_FT_RAR_CURVE_CANDIDATE_NOT_UNIQUE`

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## 5. One-line for agents

> **§FT-RAR-CURVE: saturation is a threshold; distinguished \(g=\sqrt{g_N(g_N+a_0)}\) is plottable vs SPARC/KiDS, not a unique \(W\); hard floor FIRE; \(\sqrt{6}\) scale \(0.36\,\mathrm{dex}\) high; SPARC does not pick the interpolator; KiDS same IR; prize OPEN.**