# Freeze — leftover \(R\) is spectrally locked, not an FLRW scale factor

### Stationary geometric property of leftover \(S^3\) · \(a_0=c^2/R\) invariant in cosmic time as saturation clothes · \(\sqrt{6}\) still **[M]** · expanding \(R\) FIRE

**CPT Framework — AST / M12 / FT-A0-Z / FT-YIELD-GEOM / FT-NS3 / R-EFF**  
**Date:** 2026-09-22  
**Status:** **METHOD FREEZE** · `PASS_FT_R_LOCK_STATIONARY_NOT_FLRW`  
**Stimulus:** User — \(R\) cannot be an expanding FLRW scale factor; container radius is a spectrally locked, stationary geometric property of leftover \(S^3\), fixing \(a_0=c^2/R\) as an invariant threshold across cosmic time  
**Parents:** AST freeze (R-static / R-locked / R-epoch) · M12 · FT-A0-Z · FT-YIELD-GEOM · FT-NS3  
**Script:** `scripts/investigations/s3_geometry/ft_r_lock.py`  
**Data:** `scripts/investigations/s3_geometry/data/ft_r_lock.json`

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

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## 0. One-line

> **Agreed: leftover \(R\) does not Hubble-expand.** It is a stationary property of leftover \(S^3\) (spectrum \(\ell(\ell+2)/R^2\)). Then \(a_0=c^2/R\) is an **invariant** yield threshold, not \(a_0(z)\). That is the AST **R-locked / static** branch. High-\(z\) baryon-looking disks stay high-\(g\) compactness with **today’s** \(a_0\) (FT-A0-Z / Milgrom 2017). Clothes \(R=\sqrt{6}\,c/H_0\) give \(c^2/R\simeq 2.31\,a_0\) — rhyme, still **[M]**. The \(\ell=1\) gap is \(\omega=\sqrt{3}\,c/R\simeq 0.71\,H_0\) on those clothes (FT-NS3), **not** \(a_0\). Naive expanding \(R\) remains M12 **FIRE**. \(R\) unnamed, not FLRW. No \(H_0=69\) recycle. No CDM. No \(\mu(x)\).

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## 1. What is frozen

| | |
|--|--|
| Leftover \(R\) as FLRW \(a(t)\) | **FIRE** |
| \(a_0(z)\) from \(\dot R/R=H_0\) | **FIRE** (wrong sign + M12) |
| \(R\) as leftover \(S^3\) spectral radius | **LOCKED** (ontology) |
| \(a_0=c^2/R\) invariant saturation clothes | **KIND** (FT-YIELD-GEOM) |
| \(\sqrt{6}\) as a law for \(R\) | **still [M]** |
| Spectral gap \(=a_0\) | **NO** (gap \(\simeq 0.71\,H_0\), \(a_{\ell=1}\simeq 4\,a_0\)) |

**Finding L1 [E process].** This is the remaining AST fork: static / spectrally locked leftover \(R\), not present-epoch invert as cosmology, and not expanding. Hubble expansion, if real, is **not** \(\dot R/R\) of this container.

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## 2. Integrity

| Gate | Held |
|------|:----:|
| \(R\) not FLRW scale factor | **Yes** |
| \(a_0\) invariant *kind* | **Yes** |
| Derived \(a_0\) from spectrum | **No** |
| Expanding \(R\) | **FIRE** |
| Name \(R\) / chip / CDM / \(\mu(x)\) | **No** |

**Exit:** `PASS_FT_R_LOCK_STATIONARY_NOT_FLRW`

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## 3. One-line for agents

> **§FT-R-LOCK: leftover \(R\) is a stationary \(S^3\) spectral radius, not FLRW \(a(t)\); \(a_0=c^2/R\) invariant clothes; \(\sqrt{6}\) still [M]; gap \(\neq a_0\); expanding \(R\) FIRE.**