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]
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 \(\mu(x)\).
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.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
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.