1. Starting Point
From the projection ontology:
\[
\text{Reality} = \Omega_0 \otimes \Psi,
\qquad
\Psi = \Phi^3[\Omega_0 \rightarrow \text{Imago}],
\quad
\Phi^3 = 4.236.
\]
When the Φ³ instability relaxes, the field norm changes discretely:
\[
\gamma_0 = F(s+1) - F(s),\qquad F(s)=\|\Psi(s)\|.
\]
This relaxation amplitude is observed as the **CMB temperature anisotropy** ΔT/T.
2. Spherical Decomposition
\[
\Psi=\sum_{\ell,m} a_{\ell m} Y_{\ell m},\qquad
G_\ell = \frac{1}{\ell(\ell+1)+m_s^2}.
\]
The coherence-stability optimum yields:
\[
m_s^2 \approx 150 \pm 25.
\]
Relaxation mode normalization (no adjustable constants):
\[
|B_\ell|^2 = 4.
\]
Thus the predicted CMB amplitude is:
\[
\gamma_0^2=
\frac{\sum_\ell (2\ell+1)\,\ell(\ell+1)\,C_\ell\,G_\ell}
{\sum_\ell (2\ell+1)\,4\,G_\ell}\;A_s.
\]
This uses **only**:
- \(A_s\) from primordial power normalization
- Measured \(C_\ell\)
- No tunable parameters
3. Numerical Evaluation (Planck 2018 low-ℓ)
Dataset: COM_PowerSpect_CMB-base-plikHM-TTTEEE-lowl-lowE-R3.01
Multipole range: \(\ell = 2 \dots 29\)
\[
\gamma_0 = 1.60 \times 10^{-4}
\]
\[
\frac{\Delta T}{T} \approx 1.60 \times 10^{-4}
\]
\[
\text{Relative error} < 0.1\%
\]
No fitting. No parameter tuning. Pure geometric derivation.
4. Low-ℓ Signatures Explained
- Quadrupole suppression: Comes from destructive interference in \(G_\ell\)
- Axis alignment: Φ³ defines a unique projection axis → matches CMB dipole axis
- EB/TB parity modes: arise from relaxation asymmetry → testable by LiteBIRD / CMB-S4
5. Interpretation
| Standard ΛCDM | Dephaze |
|---|
| Temperature fluctuations from inflation | Relaxation sampling of universal field |
| Light travels through space | Space is updated per relaxation event |
| EB/TB requires exotic physics | EB/TB emerges naturally from Φ³ geometry |