# Freeze — \(R_{\mathrm{eff}}\) Is an Unnamed Mach Length

### Ontology unresolved · \(\sqrt{6}\,c/H_0\) is M13 dual, not a derivation · stop premature naming · no promote

**CPT Framework — absolute-scale \(R\) honesty**  
**Date:** 2026-08-16  
**Status:** **THEORY FREEZE** · `PASS_R_EFF_HONESTY` · **supersedes** R-ONT-0 default *name* \(R_{\mathrm{spin2}}\) as “what \(R\) is”  
**Parents:** R-ONT-0…2 · AST / M13 three-gate · SC4 B-√6 · Mach M12–M14  
**Census:** `scripts/investigations/s3_geometry/r_eff_ontology_honesty_census.py` · `data/r_eff_ontology_honesty_census.json`

Tags: **[E honesty]** · **[Programme]** · **[M packaging]** · **[O residual]**

---

## 0. One-line

> **The framework does **not** presently know what physical object \(R\) is.** Mixed-mass Mach closure plus laboratory \(\{G,\alpha,m_e,m_p\}\) infers an effective length \(R_{\mathrm{eff}}\approx 3.28\times 10^{26}\,\mathrm{m}\) **under the published coefficient book** (\(k_1=3/8\) as \(K_{\mathrm{Green}}\) at \(\Omega x^2=1\)); writing \(R_{\mathrm{eff}}\approx\sqrt{6}\,c/H_0\) is the **same** M13/M14 relation (\(H_0\approx 68.97\)), not an independent derivation and **convention-conditional**; FLRW curvature, expanding \(S^3\), theorem-grade spin-2 coherence, and horizons are all **not** licensed names; \(\sqrt{6}\) stays a **brick**; stop naming \(R\) until a proposal explains **both** stationarity **and** the \(\sim\sqrt{6}\) Hubble match.

---

## 1. Integrity (held)

| Rule | Held |
|------|:----:|
| Ontology left unnamed | **Yes** |
| \(\sqrt{6}\,c/H_0\) not sold as derived \(R\) | **Yes** |
| R-ONT-1 FAIL_DERIVE not forgotten | **Yes** |
| \(\Omega_k=-1/6\) still rejected | **Yes** |
| M12 expanding-\(R\) still FIRE | **Yes** |
| No \(m_e\) mint | **Yes** |
| No promote | **Yes** |

This freeze **does not retire** \(R_{\mathrm{spin2}}\) as a *candidate*. It **denies** it as the present answer.

---

## 2. What is actually known

Before the \(\sqrt{6}\) Hubble anchor, mixed-mass closure with \(k_1=3/8\) is
\[
G
=
\frac{\alpha\,k_1\,\hbar^{2}}{R\,m_e^{2}m_p},
\qquad
k_1=\frac38.
\]
Laboratory constants therefore infer
\[
\boxed{
R_{\mathrm{eff}}
=
\frac{\alpha\,(3/8)\,\hbar^{2}}{G\,m_e^{2}m_p}
\approx 3.285\times 10^{26}\,\mathrm{m}
}
\]
(\(\approx 10.65\,\mathrm{Gpc}\approx 34.7\,\mathrm{Gly}\)).

Census: inverting \(R_{\mathrm{eff}}=\sqrt{6}\,c/H_0\) returns \(H_0\approx 68.97\,\mathrm{km\,s^{-1}\,Mpc^{-1}}\) — **exactly** the M13/M14 invert. So
\[
R_{\mathrm{eff}}\approx\sqrt{6}\,\frac{c}{H_0}
\]
is an **empirical dual** of that prediction, not a second derivation of \(R\). After R-XCHECK the dual is **convention-conditional**: the same algebra with the raw-book coefficient \(3\pi/4\) moves \(R_{\mathrm{eff}}\) by \(2\pi\) and \(H_0\) to \(\sim10.98\). The published number remains the length the **published** closure requires. `3/8` stays **[E]** in the named Green book; it is not a radius-independent constant.

**Honest object:** \(R_{\mathrm{eff}}\) is the effective global length **required by the published Mach closure**. Its physical ontology is **unresolved**.

---

## 3. What it cannot currently be called

| Name | Why not |
|------|---------|
| FLRW curvature radius | Implies \(\Omega_k=-1/6\) (R-ONT-0 **REJECT**) |
| Expanding \(S^3\) scale factor | M12 \(\dot G/G\sim H_0\) **FIRE** |
| Spin-2 coherence length **as a theorem** | Lock \(\omega_2=H_0\) **FAIL_DERIVE** (R-ONT-1) |
| Particle or event horizon | No identification banked |

The \(L=2\) eigenvalue \(\omega_2=\sqrt{6}\,c/R\) remains **[E]**. That does **not** identify the Mach \(R_{\mathrm{eff}}\) with a coherence length until the lock is a theorem.

---

## 4. Correct next posture

| Do | Do not |
|----|--------|
| Leave \(R=R_{\mathrm{eff}}\) as an independently inferred Mach length | Prematurely name it |
| Treat \(\sqrt{6}\) as a **brick** (R-ONT-2 held) | Call \(\sqrt{6}\,c/H_0\) the answer |
| Demand any new proposal explain **stationarity** *and* the \(\sim\sqrt{6}\) Hubble match | Reopen C3 as if it locked |
| Keep M13 three-gate observational (convention-conditional invert) | Recycle \(H_0=69\) as confirmation of a name, or retire G-H0 because the coefficient is spliced |

**R-ONT-0 default label \(R_{\mathrm{spin2}}\)** is hereby **demoted** from “what \(R\) is” to **one candidate among unnamed options**, consistent with R-ONT-2 (brick **[M]**, lock extra law).

---

## 5. Gates

| Gate | Status |
|------|--------|
| PASS_R_EFF_FROM_LAB | **PASS** |
| PASS_SQRT6_IS_M13_DUAL | **PASS** |
| PASS_ONTOLOGY_UNNAMED | **PASS** |
| PASS_SPIN2_NOT_THEOREM | **PASS** |
| PASS_FLRW_STILL_REJECTED | **PASS** |
| PASS_NO_PREMATURE_NAME | **PASS** |
| PASS_NO_ME_MINT | **PASS** |
| PASS_NO_PROMOTE | **PASS** |
| PASS_INTEGRITY | **PASS** |

**Status:** `PASS_R_EFF_HONESTY`

---

## 6. Kill board

| ID | Kill |
|----|------|
| **K-REFF-1** | “The framework knows \(R\) is the spin-2 coherence length” |
| **K-REFF-2** | \(\sqrt{6}\,c/H_0\) sold as independent of M13/M14 |
| **K-REFF-3** | \(R_{\mathrm{eff}}\) sold as FLRW \(R_k\) or horizon |
| **K-RONT-*** / **K-AST-*** | Held |

---

## 7. Next

| Option | Note |
|--------|------|
| **Default** | Live with unnamed \(R_{\mathrm{eff}}\) (published-book length); observe M13 gates |
| R-MCR / R-MCR-2 | **DONE** · no one-convention PRIMARY hit · remainder \(k\) **FAIL_DERIVE** · coefficient line closed |
| New proposal | Must do **stationarity + \(\sim\sqrt{6}\) to \(H_0\)** together, without shopping \(k\) |
| Not | Another name; another \(m_e\) mint |

---

## 8. One-line

> **§R-EFF: framework does not know what \(R\) is; \(R_{\mathrm{eff}}\) from published Mach+lab; \(\sqrt{6}\) brick = convention-conditional M13 dual; stop naming; no promote.**