# Results — evolving leftover \(R\) makes \(a_0\) *larger* at high \(z\): wrong sign for baryon dominance

### Larger \(a_0\) \(\Rightarrow\) more leftover, not more Newton · Genzel disks already sit at \(g(R_{1/2})=(3\)–\(11)\,a_0\) *today* · \(a_0\sim 4a_0\) at \(z\sim 2\) excluded · compactness, not \(a_0(t)\) · M12 FIRE

**CPT Framework — FT-YIELD-GEOM / AST freeze M12 / FT-A0-KIND / R-EFF**  
**Date:** 2026-09-22  
**Status:** **BANKED** · `PASS_FT_A0_Z_WRONG_SIGN`  
**Stimulus:** User — if \(R\) evolves cosmologically, \(a_0\) should increase at high redshift, matching baryon-dominance of early galaxies  
**Parents:** FT-YIELD-GEOM · M12 · Milgrom 2017 (arXiv:1703.06110) · Genzel et al. 2017  
**Script:** `scripts/investigations/s3_geometry/ft_a0_z.py`  
**Data:** `scripts/investigations/s3_geometry/data/ft_a0_z.json`

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

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## 0. One-line

> **Wrong sign.** If leftover \(R\) was smaller in the past, \(a_0=c^2/R\) was **larger**. Larger \(a_0\) shrinks \(r_y=\sqrt{GM/a_0}\), raises \(V_{\mathrm{flat}}\propto a_0^{1/4}\), and drops \(g/a_0\) — **more** yield, more extra \(\Phi\), not baryon Newton. At \(z=2\): \(a_0\times 3\) if \(R\propto 1/(1+z)\), \(\times 5.2\) if \(a_0\propto H_{\mathrm{matter}}\). Milgrom 2017: Genzel et al. disks already have \(g(R_{1/2})=(3\)–\(11)\,a_0\) with **today’s** \(a_0\), so they look baryon-dominated because they are **high-\(g\) and compact**, like inner ETG rings; \(a_0\sim 4a_0\) at \(z\sim 2\) is excluded (\(a_0\propto(1+z)^{3/2}\) included). Expanding leftover \(R\) that runs \(a_0\) or \(G\) is M12 **FIRE**. \(R\) unnamed. No CDM. No \(\mu(x)\).

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## 1. Algebra of the slogan

| \(z\) | \(a_0/a_{00}\) if \(R\propto 1/(1+z)\) | \(r_y\) shrinks | \(V_{\mathrm{BTFR}}\) boost | if \(a_0\propto H\propto(1+z)^{3/2}\) |
|------|----------------------------------------|-----------------|------------------------------|----------------------------------------|
| 0 | 1 | 1 | 1 | 1 |
| 1 | 2 | 0.71 | 1.19 | 2.83 |
| 2 | **3** | 0.58 | 1.32 | **5.20** |
| 6 | 7 | 0.38 | 1.63 | 18.5 |
| 10 | 11 | 0.30 | 1.82 | 36.5 |

**Finding Z1 [E algebra].** Yield when \(g<a_0\). Raising \(a_0\) puts *more* of a given galaxy below threshold. That is the opposite of “appears Newtonian / baryon-dominated.”

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## 2. What Genzel actually is

Milgrom 2017 on Genzel et al. 2017 (\(z\sim 0.9\)–\(2.4\)):

- \(g(R_{1/2})=(3\)–\(11)\,a_0\) **with present \(a_0\)**
- leftover/phantom fraction at \(R_{1/2}\) only tens of percent — agrees without evolving \(a_0\)
- \(V_{\mathrm{flat}}\) from baryons is \(0.55\)–\(0.75\) of the inner peak \(\Rightarrow\) declining outer RC, as observed
- **excludes** \(a_0\sim 4a_0\) at \(z\sim 2\), hence \(a_0\propto(1+z)^{3/2}\)

**Finding Z2 [E honesty].** Early-disk “baryon dominance” is the **high-gradient** side of FT-A0-KIND (compact, high \(\Sigma\)), not a larger yield threshold. Same lesson as NGC 6505 / SLACS inner \(\theta_E\).

JWST “too big too early” is a **formation-time** issue (Sanders/McGaugh structure-growth), not \(a_0(z)\) in the RAR.

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## 3. Expanding \(R\)

If \(c\) is fixed and \(a_0\propto 1/R\), Hubble-growing leftover \(R\) runs \(a_0(t)\) and, in Mach clothes, \(\dot G/G\sim H_0\) (M12 **FIRE** vs LLR). AST freeze: \(R\) static, locked, or present-epoch only. Do not name leftover \(R\) as FLRW.

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## 4. Integrity

| Gate | Verdict |
|------|---------|
| \(R\) smaller in past \(\Rightarrow a_0\) larger | **YES** |
| That matches baryon-looking high-\(z\) disks | **NO** (wrong sign) |
| Compact high \(g\) with today’s \(a_0\) | **YES** (Milgrom 2017) |
| \(a_0\times 4\) at \(z\sim 2\) | **excluded** |
| Expanding \(R\) licensed | **NO** (M12) |
| CDM / \(\mu(x)\) / name \(R\) | **NO** |

**Exit:** `PASS_FT_A0_Z_WRONG_SIGN`

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## 5. One-line for agents

> **§FT-A0-Z: if leftover \(R\) grows, \(a_0\) was larger at high \(z\) — that *increases* leftover, not baryon Newton; Genzel is high \(g\) with today’s \(a_0\); \(a_0\propto H(z)\) excluded; M12 FIRE.**